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The lateral-torsional buckling of doubly symmetric wide flange sections De Vall, Ronald H. 1968

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THE LATERAL-TORSIONAL BUCKLING OF DOUBLY SYMMETRIC WIDE FLANGE SECTIONS  by  RONALD H. DE VALL  B.A.Sc.  (Civil  Eng.)  The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1966  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n t h e Department of C I V I L ENGINEERING  We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d standard  THE UNIVERSITY  OF BRITISH COLUMBIA  M a r c h , 1968  I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of t h e r e q u i r e m e n t s 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 C o l u m b i a , I agree thav 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 s t u d y .  I  further  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 s c h o l a r l y poses may be g r a n t e d by the Head o f ' m y Department or by h i s It  the  pur-  representatives.  i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n  s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n .  R. H. De V a i l  Department of C i v i l  Engineering  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , B . C . March 1968  i  ABSTRACT  In t h i s t h e s i s , a s t i f f n e s s m a t r i x which i n c l u d e s the  non-linear  e f f e c t s of p r i n c i p a l p l a n e s h e a r s , moments and a x i a l l o a d s on l a t e r a l and t o r s i o n a l d e f l e c t i o n s i s developed f o r a d o u b l y symmetric wide f l a n g e s e c t i o n . Initially,  an e x a c t e i g h t by e i g h t l i n e a r m a t r i x i s developed f o r an  element of c o n s t a n t s e c t i o n p r o p e r t i e s .  The e i g h t a l l o w a b l e d e f l e c t i o n s  the independent r e p r e s e n t a t i o n of t h e d e f l e c t i o n s of e i t h e r f l a n g e a t end.  allows  either  The n o n - l i n e a r e f f e c t s a r e i n c l u d e d i n the d i f f e r e n t i a l e q u a t i o n s by  c o n s i d e r i n g t h e e f f e c t of the p r i m a r y s t r e s s e s on t h e e q u i l i b r i u m of a d i s placed element. Two a p p r o x i m a t i o n s a r e t h e n i n t r o d u c e d .  The f i r s t  n u m e r i c a l t e c h n i q u e f o r s o l v i n g the d i f f e r e n t i a l e q u a t i o n s .  c o n s i s t s of a The second c o n -  s i s t s of a s i m p l i f i c a t i o n of the boundary c o n d i t i o n s i n s o l v i n g t h e d i f f e r e n t i a l equations.  U s i n g t h e s e two a p p r o x i m a t i o n s , the n o n - l i n e a r p o r t i o n of  the  m a t r i x i s then b u i l t . Several s t r u c t u r e s are then analyzed. s e v e r a l elements.  Each s t r u c t u r e i s d i v i d e d  T h i s a l l o w s beams of n o n - c o n s t a n t  section properties  into  to be  a n a l y z e d , and i n c r e a s e s t h e a c c u r a c y o f the r e s u l t s of t h e a p p r o x i m a t e m a t r i c e s . The r e s u l t s o f t h e s e a n a l y s e s a r e t h e n compared t o r e s u l t s and t a b u l a t e d . cases t e s t e d .  It  theoretical  i s seen t h a t t h e m a t r i x g i v e s good agreement f o r  all  ii  TABLE OF CONTENTS Page ABSTRACT  i  TABLE OF CONTENTS  ii  LIST OF FIGURES  iii  DEFINITION OF SYMBOLS  v  ACKNOWLEDGEMENTS  vii  CHAPTER I  INTRODUCTION  1  CHAPTER I I  DEVELOPMENT OF LINEAR MATRIX  3  CHAPTER I I I  DEVELOPMENT OF NON-LINEAR DIFFERENTIAL EQUATIONS  13  CHAPTER IV  METHODS OF APPROXIMATIONS  20  CHAPTER V  ILLUSTRATION  22  CHAPTER VI  DEVELOPMENT OF LATERAL STABILITY MATRIX  31  i  A p p l i c a t i o n of A p p r o x i m a t i o n s  31  ii  C a l c u l a t i o n of N o n - l i n e a r S t i f f n e s s M a t r i x f o r Type One Loads  33  C a l c u l a t i o n of N o n - l i n e a r M a t r i x f o r Type Two Loads ,  40  N u m e r i c a l Examples  45  iii  iv  CHAPTER V I I  CONCLUSIONS  LIST OF REFERENCES  OF METHODS OF APPROXIMATIONS  53  iii  LIST OF FIGURES Page Fig.  1  Sign Conventions  3  Fig.  2  Beam Segment  4  Fig.  3  P r i m a r y Modes f o r R i g h t Hand End D e f l e c t i o n s  6  Fig.  4  S u p e r p o s i t i o n of P r i m a r y Modes  8  Fig.  5  Linear Stiffness Matrix K  9  Fig.  6  Test R e s u l t s f o r K  Fig.  7  Fig.  8  b  Fig.  9  o  11  o E f f e c t of F l a n g e Warping  12  S i g n C o n v e n t i o n f o r P r i n c i p a l S h e a r , Moment and A x i a l Load  13  E l e m e n t a l Beam S e c t i o n s i n D i s p l a c e d P o s i t i o n Under the A c t i o n of P r i m a r y S t r e s s e s  14  Fig.  10  R e p l a c i n g D i s t r i b u t e d Loads by F i x e d End R e a c t i o n s  20  Fig.  11  O r d i n a r y Beam S t i f f n e s s D e f l e c t i o n s , F o r c e s and C u r v a t u r e s  22  Fig.  12  N o n - l i n e a r Beam Column M a t r i x f o r Type 1 L o a d s , F i x e d End C o n d i t i o n s  25  Fig.  13  N o n - l i n e a r Beam Column Terms f o r Type 2 Loads  25  Fig.  14  N o n - l i n e a r Beam Column M a t r i x f o r Type 2 Loads  26  Fig.  15  Fig.  16  Complete N o n - l i n e a r Beam Column M a t r i x , F i x e d End Conditions N o n - l i n e a r Beam Column Terms f o r Type 1 L o a d s , P i n n e d End C o n d i t i o n s  26 27  Fig.  17  N o n - l i n e a r Beam Column M a t r i x f o r Type 1 L o a d s , P i n n e d End C o n d i t i o n s  28  Complete N o n - l i n e a r Beam Column M a t r i x , End C o n d i t i o n s  28  Fig.  Fig.  18  19  Pinned  P l o t of % E r r o r v s . Number of Elements f o r Beam-Column M a t r i c e s f o r 3 Column Types  30  Fig.  20  Type 1 Terms f o r N o n - l i n e a r M a t r i x f o r  67 = 1  35  Fig.  21  Type 1 Terms f o r N o n - l i n e a r M a t r i x f o r  65 = 1  38  Fig.  22  N o n - l i n e a r M a t r i x f o r Type 1 Loads  Fig.  23  Component D e f l e c t i o n s f o r 6 7 = 1  ,  39 40  iv  LIST OF FIGURES ( C o n t d . ) Page Fig.  24  F o r c e Components Due t o End D e f l e c t i o n s f o r 6  Fig.  25  Type 2 Terms f o r N o n - l i n e a r M a t r i x f o r 67 = 1  42  Fig.  26  Component D e f l e c t i o n s f o r 6g = 1  41  Fig.  27  F o r c e Components Due t o End D e f l e c t i o n s f o r 65 = 1  43  Fig.  28  V e r t i c a l F l a n g e D e f l e c t i o n s as F u n c t i o n s of <> f  44  Fig.  29  Type 2 Terms i n N o n - l i n e a r M a t r i x f o r 6g = 1  46  Fig.  30  N o n - l i n e a r M a t r i x f o r Type 2 Loads  47  Fig.  31  The Complete N o n - l i n e a r M a t r i x f o r Loads of Type 1 and 2  48  Fig.  32  T a b l e of R e s u l t s f o r T e s t  50  Fig.  33  P l o t of A c c u r a c y v s . Number o f Elements Used.  Structures  7  = 1  40  52  DEFINITION OF SYMBOLS  moment of i n e r t i a of f l a n g e about s t r o n g  axis  moment of i n e r t i a of s e c t i o n abou. : z a x i s 1  moment of i n e r t i a of s e c t i o n about y a x i s p o l a r moment of a r e a of  inertia  section  a r e a of web torsional  constant  Youngs modulus s h e a r modulus JG depth of 2  section  2C/EIh l a t e r a l d e f l e c t i o n (along y a x i s ) v e r t i c a l d e f l e c t i o n (along z a x i s ) torsional  deflection  f l a n g e shear torque distributed  load  distributed  torque  s t i f f n e s s matrix deflection i n n direction s t i f f n e s s matrix force i n n d i r e c t i o n l e n g t h of  element  p r i n c i p a l a x i a l l o a d i n element p r i n c i p a l moment a t cj: of  element  p r i n c i p a l shear i n element M - VL/2 + Vx = moment i n element @ p o i n t o normal s t r e s s i n element  x  DEFINITION Oi SYMBOLS ( C o n t d . ) ?  M^  =  f l a n g e moment  0  =  P/A  T  =  shear s t r e s s i n element  p  =  p r e s s u r e due t o n s t r e s s e s  X  =  M^/2EI = parameter i n s o l u t i o n  v  =  n  <j>  =  n*"^ term i n s e r i e s s o l u t i o n of <)j  n  =  dummy i n t e g r a l parameter  p  =  dummy i n t e g r a l  £;  =  dummy i n t e g r a l parameter  '  =  d i f f e r e n t i a t i o n with respect to x  o  n n  expansion  term m s e r i e s s o l u t i o n of y 1  R.H.S  R i g h t hand s i d e  L.H.S  L e f t hand s i d e  parameter  vii  ACKNOWLEDGEMENTS  The a u t h o r w i s h e s t o thank h i s s u p e r v i s o r , D r . R. F. H o o l e y ,  for his  i n v a l u a b l e a s s i s t a n c e and encouragement d u r i n g t h e development of t h i s work. 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 N a t i o n a l R e s e a r c h C o u n c i l of Canada f o r f i n a n c i a l support, facilities.  March 1968 Vancouver,  B.C.  and t o the U . B . C . Computing C e n t e r f o r the use of  its  THE LATERAL-TORSIONAL BUCKLING OF DOUBLY SYMMETRIC WIDE FLANGE SECTION  CHAPTER  I  INTRODUCTION 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 of beams has been a t o p i c of and r e s e a r c h f o r y e a r s .  The f o u n d a t i o n s of t h e t h e o r y of l a t e r a l  interest  torsional  b u c k l i n g of t h i n r e c t a n g u l a r s e c t i o n s were l a i d by P r a n d t l and M i c h e l l e [ 5 , i n 1899 i n the s t u d y of the l a t e r a l b u c k l i n g of t h i n r e c t a n g u l a r H. R e i s s n e r [6]  sections.  l a t e r s t r e s s e d the e f f e c t o f d e f l e c t i o n s i n the major  principal  a x i s i n the P r a n d t l - M i c h e l l e t h e o r y and i n t r o d u c e d m o d i f i c a t i o n s t o account them.  I n 1910, S . Timoshenko  6]  [1] developed t h e d i f f e r e n t i a l e q u a t i o n t h a t  for in-  c l u d e d the w a r p i n g e f f e c t of the f l a n g e s of I s e c t i o n s deformed i n t o r s i o n . 1929, Wagner  [6] d e t e r m i n e d t h a t t h i n open s e c t i o n s may b u c k l e i n a pure  s i o n a l mode under an a p p l i e d a x i a l  tor-  load.  S i n c e t h e n many o t h e r r e s e a r c h e r s have c o n t r i b u t e d of l a t e r a l b u c k l i n g .  In  to the knowledge  The u s u a l form t h i s work has t a k e n i s t h e d i r e c t  solution  of t h e d i f f e r e n t i a l e q u a t i o n f o r s i m p l e cases and t h e use of n u m e r i c a l methods i n t h e more c o m p l i c a t e d c a s e s . difficulty  The drawback i n the above approaches i s  i n a p p l y i n g them t o the g e n e r a l c a s e .  the  As the c o m p l e x i t y of t h e l o a d  and s u p p o r t c o n d i t i o n s i n c r e a s e s , the problem becomes i n t r a c t i b l e . The purpose of t h i s t h e s i s w i l l be t h e p r e s e n t a t i o n of a method  for  t h e e l a s t i c a n a l y s e s of l a t e r a l t o r s i o n a l b u c k l i n g of doubly symmetric wide f l a n g e s e c t i o n s under p r i n c i p a l moment, shear and a x i a l l o a d s .  The method of  s o l u t i o n u t i l i z e s a s t i f f n e s s m a t r i x c o n s i s t i n g of two p a r t s - an exact l i n e a r p o r t i o n and an a p p r o x i m a t e n o n - l i n e a r p o r t i o n . a Numbers i n s q u a r e b r a c k e t s r e f e r to the R e f e r e n c e s .  The m a t r i x i s  elastic  developed  2.  u s i n g the assumptions t h a t d e f l e c t i o n s i n the p r i n c i p a l p l a n e remain s m a l l ,  the  s t r e s s e s remain e l a s t i c , t h e r e i s no d i s t o r t i o n of the c r o s s - s e c t i o n , and the l o a d s m a i n t a i n t h e i r o r i g i n a l d i r e c t i o n of a p p l i c a t i o n .  These m a t r i c e s , once  o b t a i n e d , can be used t o b u i l d any case of v a r y i n g s u p p o r t , c o n d i t i o n s of beams w i t h v a r y i n g s e c t i o n  properties.  f i x i t y and l o a d  3.  CHAPTER  II  DEVELOPMENT OF LINEAR MATRIX  I n order to develop a s t i f f n e s s m a t r i x ,  the g o v e r n i n g  differential  e q u a t i o n s of the s e c t i o n must f i r s t be o b t a i n e d . "For the development of the l i n e a r c a f e d i f f e r e n t i a l e q u a t i o n s , a r i g h t - h a n d e d c o - o r d i n a t e system was used as i n F i g . 1 ( a ) , w i t h the u s u a l bending moment-shear s i g n c o n v e n t i o n used f o r bending of the top f l a n g e , as i n F i g .  1(b).  R o t a t i o n p r o d u c i n g p o s i t i v e y d e f l e c t i o n of t h e upper f l a n g e was chosen as p o s i t i v e as shown i n F i g . 1 ( c ) .  A p o s i t i v e end t o r q u e T i n d u c e s p o s i t i v e  t o r s i o n i n the s e c t i o n and a n e g a t i v e shear i n the top f l a n g e .  Ely  Z  w  (a) (b)  (c) (d)  Fig.  1  SIGN CONVENTIONS  =-w  pure  To o b t a i n the d i f f e r e n t i a l e q u a t i o n f o r t o r s i o n , i t i s o n l y  necessary  t o c o n s i d e r t h e a p p l i e d t o r q u e a t any p o i n t a l o n g t h e s e c t i o n . The t o r q u e due i ti iti 2 t o shear i n t h e f l a n g e i s = - Vh = - (h<j> /2) (EIh)= -<j> E l h / 2 . The t o r q u e i due t o p u r e t o r s i o n i s C<> j . T h e r e f o r e t h e e q u a t i o n f o r t o r q u e T a c t i n g a t any section i s •  C«>  III  2  - Elh 2  <> j -  0)  as f i r s t d e v e l o p e d by Timoshenko [ 1 ] . E q u a t i o n (1) reduces t o III  2  2  2  - a $ = - a T  <(>  where a  = _2C  C  Elh One i n t e g r a t i o n II  7  gives  2  2  - a <> { = - a_T [x + A]  tj)  (2)  To o b t a i n t h e e q u a t i o n of t h e s e c t i o n under t h e a c t i o n o f a d i s t r i b u t e d t o r q u e , e q u a t i o n (1) i s d i f f e r e n t i a t e d once t o g i v e i i 2 m i i Ctj>  -  Elh  cj>  =  T  =  -  q  (3)  2 The e q u a t i o n g o v e r n i n g p u r e l a t e r a l d e f l e c t i o n i s t h e w e l l known mi  2EIY _ = - u>  (4)  R a t h e r t h a n s o l v e e q u a t i o n s (2) and (4) f o r any c o n d i t i o n o f l o a d or v a r i a t i o n of s e c t i o n p r o p e r t i e s ,  i t i s better  t o s p l i t t h e beam i n t o a number  of segments; each segment h a v i n g c o n s t a n t s e c t i o n p r o p e r t i e s , l a t e r a l shears. be r e l a t i v e l y built.  t o r q u e s and  The s o l u t i o n of e q u a t i o n s (2) and (4) f o r such a segment w i l l  s i m p l e , and a s t i f f n e s s m a t r i x f o r the segment can e a s i l y be  By u t i l i z i n g s e v e r a l elements t o r e p r e s e n t a s t r u c t u r e under  distributed  l o a d o r v a r y i n g s e c t i o n p r o p e r t i e s , v e r y l i t t l e a c c u r a c y w i l l be l o s t . The segment used i s shown i n F i g . 2 .  5.  *  2|  ^1  f/  F  Fig.  In t h i s f i g u r e , indirectly  2  e  BEAM SEGMENT  t o r s i o n a l d e f l e c t i o n s and f o r c e s a r e a c c o u n t e d f o r  t h r o u g h t h e d i f f e r e n c e s i n l a t e r a l s h e a r d e f l e c t i o n and f o r c e s .  The r e l a t i o n s h i p between the e i g h t independent d e f l e c t i o n s and t h e i r  corres-  ponding f o r c e s a r e g i v e n by kS = f where 6 i s an 8x1 m a t r i x r e p r e s e n t i n g deflections 6  ...  l s  6 , f i s an 8x1 m a t r i x r e p r e s e n t i n g the f o r c e s f j , 8  and k i s an 8*8 m a t r i x j o i n i n g the two. and u s i n g F i g .  1(c),  the  ...  f , 8  By e x a m i n i n g the segment i n F i g . 2,  the f o l l o w i n g r e l a t i o n s a r e o b t a i n e d . tj> h/2  = y  -  = y  + <|> h/2  -  EI(y  - *  h/2)  -  EI(y  +*  h/2)  i  6  2  x = 0  f  2  x = 0  fi  3  = y  + <> j h/2  f3 = +  Ei(y  fi  4  = y  -  fit  = +  El(y  f  5  = +  EI(y  - <r  h/2)  f  6  = +  El(y  + $  h/2)  •  »  fi  6  = y  <> f h/2  - $h/2 + cb'h/2  'h/2) i  C<f> /h  'h/2) + C4> /h (5)  x = L  x •=> L  h  = y  + * h/2  f ?  fi  = y  -  fft  8  f h/2  «= -  Ei(y EI(y  + <r 'h/2) + ctf'/h -  <> f  h/2) -  C<j> /h  I  f  k  8 8 x  e q u a l to z e r o , torsion.  It  w  e  r  e  o b t a i n e d by a l l o w i n g 6  n  = 1, with a l l other  6's  t h e c a l c u l a t i o n s would e n t a i l w o r k i n g w i t h combined b e n d i n g and  i s therefore  proposed to s o i v e f o u r b a s i c pure b e n d i n g cases and  f o u r b a s i c pure t o r s i o n a l cases which can oe superposed to g i v e any d e f l e c t e d shape.  desired  The f o u r of t h e s e modes a s s o c i a t e d w i t h the r i g h t hand end  d e f l e c t i o n s are given i n F i g . 3.  > S =S FS3=S =S =S = o |  2  (a)  4  6  5  shear  V V  8=8 =8 =8 =8 =8 =o  1  12  3 4 7 8 (b) end rotation  AY ->  W W W (c)  0  V-Vi  W W W  torsion  Fig.  (d)  3  into i n further  3(a),  (b)  warping  are solved using e q . ( 4 ) .  the w e l l known beam s t i f f n e s s detail.  1  PRIMARY MODES FOR RIGHT HAND END DEFLECTIONS  The shapes i n F i g . s o l u t i o n gives  W  0  equations,  The shapes p r e s e n t e d i n F i g .  shapes and can be s o l v e d u s i n g The shape i n F i g . other allowable d e f l e c t i o n s  3(c)  Since  their  they w i l l not be gone  3(c),  (d)  are  torsional  eq.(2). represents  fixed.  a unit torsional rotation with  The end c o n d i t i o n s  f o r t h i s case a r e :  all  7.  x = 0  x = L  <> j = 0  <>j = 1  i  4-  •  = 0  The s o l u t i o n of e q . ( 2 ) <> f = B  B  where  o  T °  o  s i n h ax + D  = - T  o  f o r t h i s case i s  cosh ax + T _o C  r  , ., [x + AJ  D = T ( c o s h aL - 1) o o Ca s i n h aL  o aC  = 0  A  o  = -  ( c o s h aL -  1)  a s i n h aL  = aC s i n h aL [ 2 - 2 cosh aL + aL s i n h aL]  The shape i n F i g . 3(b)  i s o b t a i n e d by a p p l y i n g e q u a l and o p p o s i t e  moments to t h e upper and l o w e r f l a n g e s of one e n d , and r e s t r a i n i n g a l l o t h e r allowable deflections. • .  x = 0  The end c o n d i t i o n s f o r t h i s case a r e :  <|> = 0  '  x = L  t  i  4> «= 0  •j» h = - 1 2  '  The s o l u t i o n of e q . ( 2 )  f o r t h i s case i s :  d> = Bi s i n h ax + Di cosh ax + Ti _ C 1  1  where Ca Ti= -  2C h  <> j = 0  Ti ~  , '. , [x + A i J  r  [ s i n h aL - aL ] A i = s i n h aL - aL [ a ( l - cosh aL)] a ( l - cosh aL)  (7)  cosh aL [ ] [ 2 - 2 cosh aL + aL s i n h aL]  U s i n g t h e above s o l u t i o n s ,  the i n d i v i d u a l columns of the s t i f f n e s s  m a t r i x may be o b t a i n e d by s u p e r p o s i t i o n .  As an example, columns 6 and 7 can  be o b t a i n e d , by u s i n g t h e shapes i n F i g . 3 as i n d i c a t e d i n F i g .  4.  8.  Fig.  4  SUPERPOSITION OF PRIMARY MODES  S i m i l a r o p e r a t i o n s y i e l d t h e o t h e r columns of t h e s t i f f n e s s  matrix.  P r e s e n t a t i o n of the m a t r i x i s s i m p l i f i e d by i n t r o d u c i n g the  following  functions. S '  :  = (aL)  -  3  s i n h aL/12<f>  2 5  2  = (aL)  (cosh aL — -1) /6d>  5 3 = aL (aL cosh aL - s i n h aL)/4c(> S^ = aL ( s i n h aL - aL)/2<J> 4> = 2 - 2 cosh aL + aL s i n h aL a  2  •  = 2C/EIh  2  where S j , S 2 , S 3 , Sn and <> J a r e the same as the s t a b i l i t y  functions given  Gere and W e a v e r . [ 2 ]  the f o r c e s g i v e s  Use of t h e s e f u n c t i o n s  complete l i n e a r m a t r i x K , shown i n F i g . 5 .  to represent  in the  2  1  2L [1 + S ]  2  2L [1 - S ]  2L [1 + S ]  3  3L[1 - S ]  3L[1 + S ]  6[1 + S ]  4  3L[1 + S ]  3L[1 - S ]  6[1 - S ]  6[1 + S ]  5  L [l + S ]  L [l  - S J  3L[1 - S ]  3L[1 + S ]  2L [1 + S ]  6  L [l  L  [1 + S j  3L[1 + S ]  3L[1 - S ]  2L  7  3L[-1 + S ]  3L[-1 - S ]  6[-l -  +6[-l +  8  3L[-1 - S ]  3L[-1 +'S ]  6[-l +  1  2  3  2  2 3  2  2  3  2  SYMMETRIC .  x  2  x  x  2 2  4  2  2  2  3  2  2 2  - S J  2  2  2  2  2  ^ 1  3  Fig. 5  2  S] x  6[-l - S ] x  2  [1-S ]  2L [1 + S ]  3  3  3L[-1 + S ]  3L[-1 - S ]  6[1 + S ]  3L[-1 - S ]  +3L[-1 + S ]  6[1 - S ]  5  6  7  2  2  4  LINEAR STIFFNESS MATRIX K  Q  2  2  x  x  8  •10.  T h i s m a t r i x r e p r e s e n t s t h e e x a c t l i n e a r case w i t h two l i m i t a t i o n s : t h e l o a d s must be a p p l i e d a t the node p o i n t s of the s t r u c t u r e and the s e c t i o n p r o p e r t i e s between nodes must remain c o n s t a n t . A s t r u c t u r e m a t r i x was g e n e r a t e d by standard, methods and the  results  f o r v a r i o u s l o a d c a s e s were compared t o e x i s t i n g t h e o r e t i c a l s o l u t i o n s . s t r u c t u r e s were a n a l y z e d , a c a n t i l e v e r and a r e s t r a i n e d beam.  The  Two  cantilever  had a l l degrees of freedom f i x e d a t one e n d , and a l l f r e e a t the o t h e r .  The  r e s t r a i n e d beam had a l l degrees of freedom f i x e d a t one e n d , but o n l y t h e f l a n g r o t a t i o n s were f i x e d at t h e o t h e r e n d . t h e r e s t r a i n e d beam.  T h i s a l l o w e d p l a c i n g an end t o r q u e on  The r e s u l t s a r e g i v e n i n F i g . 6.  From F i g . 6 i t can be seen t h a t the m a t r i x g i v e s t h e same r e s u l t s as t h e s t r e n g t h of m a t e r i a l s s o l u t i o n .  T h i s i s t o be e x p e c t e d s i n c e no a p p r o x i -  m a t i o n to t h e s t r e n g t h of m a t e r i a l s o l u t i o n was used i n the d e r i v a t i o n . I n some beams, most of t h e t o r q u e can be c a r r i e d i n pure t o r s i o n . If  t h e beam i s r e p r e s e n t e d w i t h many s h o r t elements w h i c h tend t o c a r r y most of  t h e t o r s i o n i n f l a n g e b e n d i n g , the q u e s t i o n a r i s e s as t o whether the m a t r i c e s c o n t a i n s u f f i c i e n t a c c u r a c y t o c o n v e r t the weak pure t o r s i o n r e s i s t a n c e of  the  element t o t h e predominant pure t o r s i o n r e s i s t a n c e of t h e main s t r u c t u r e .  In  other words,  if  there i s i n s u f f i c i e n t accuracy i n the computation procedure,  t h e f l a n g e s h e a r may overshadow t h e pure t o r s i o n terms i n s h o r t elements and produce erroneous r e s u l t s when summed i n t o a l a r g e  structure.  I n o r d e r t o i n v e s t i g a t e t h i s p r o b l e m s e v e r a l s t r u c t u r e s of l e n g t h were a n a l y z e d .  varying  Each s t r u c t u r e was f u l l y r e s t r a i n e d at one e n d , and had  the f l a n g e r o t a t i o n s r e s t r a i n e d a t t h e o t h e r e n d .  For each of t h e s e  structures  a p l o t of t o r q u e c a r r i e d by shear over t o t a l t o r q u e (Vh/T) a g a i n s t x was made, where the r e s u l t s came from s t r e n g t h of m a t e r i a l s c a l c u l a t i o n s . The r e s u l t s a r e g i v e n i n F i g .  7.  T Cantilever >  I =41.6 in  4  J = 1.25 in  E= 3 0 0 0 0 k/in " 1  G = 1 0 0 0 0 k/in^  h = 10  L = 240" @ 3 Segments  X  S t r e n g t h of Material F l a n g e Moment (Kip inches)  M a t r i x Value F l a n g e Moment (Kip inches)  S t r e n g t h of Material Flange D e f l e c t i o n s (inches)  M a t r i x Value Flange D e f l e c t i o n s (inches)  70.5 22.32 6.55 0.  70.496 22.501 6.57 0.  0. 0.1270 0.3850 0.678  0. 0.128 0.3843 0.6780  0 80" 160" 240"  L = 240" @ 1 Segment 0 240"  70.5 0.  70.496 0.  0. 0.6780  0. .678  L = 360  Restrained  Properties as above 15 Segments @ 24"  10 Segments @ 3 6 "  X  S t r e n g t h of Material F l a n g e Moment (Kip inches)  0 36" 72" 108" 144" 180"  69.8 41.5 24.15 13.26 5.86 0.  M a t r i x Value F l a n g e Moment (Kip inches) 69.795 41.473 24.155 13.244 5.846 0. Fig.  6  S t r e n g t h of Material F l a n g e Moment (Kip inches)  M a t r i x Value F l a n g e Moment (Kip inches)  69.8  69.807  24.15  24.161  5.36  5.87  -  -  TEST RESULTS FOR K  o  12. I =41.6in E = 3 0 0 0 0 k / i n 4  "»T  J =1.25 i n  4  G =1000  2  k/in  2  h= 10  > X  1.0  L = 3'  .9  L = 6'  .8 .7 Vh T  .6 .5 .4 .3 .2 .1  L = 35  L = I40'  ->x  0 .IL  Fig.  It  .3 L  .2L  7  .4L  .5L  EFFECT OF FLANGE WARPING  c a n . b e seen from F i g . 7 t h a t the e f f e c t s of the f l a n g e s i n  i n g t o r s i o n f o r members of t h i s t y p e i s c o n s i d e r a b l e and i n t h e c a s e of members, t h e f l a n g e s c a r r y v i r t u a l l y  the e n t i r e t o r q u e .  carryshort  T h i s would i n d i c a t e  t h a t c a u t i o n s h o u l d be e x e r c i s e d i n r e p r e s e n t i n g s t r u c t u r e s w i t h a l a r g e number of e l e m e n t s . Fig.  However,  a thirty  f o o t beam of t h e same type as r e p r e s e n t e d  in  7 was a n a l y z e d a c c u r a t e l y u s i n g two f o o t elements (see F i g . 6) so the p r o -  blem i s not o v e r l y  serious.  The l i n e a r m a t r i x d e v e l o p e d i n t h i s s e c t i o n , or v a r i a t i o n on i t , s h o u l d be used i n t h e a n a l y s i s of g r i d frameworks composed of wide f l a n g e s e c t i o n s , as i t  c o n s i d e r s the e f f e c t of f l a n g e w a r p i n g .  T h i s i s i m p o r t a n t , as  f l a n g e w a r p i n g may account f o r a l a r g e p a r t of t h e t o r s i o n a l s t r e n g t h of a wide flange section.  13.  CHAPTER  III  DEVELOPMENT OF NON-LINEAR DIFFERENTIAL EQUATIONS  The team element may be s u b j e c t e d t o moments, s h e a r s and a x i a l l o a d s i n t h e major p r i n c i p a l axes as shown i n F i g . 8 .  L  >  V  M= M-~  Fig.  8  +  v  <-  x  SIGN CONVENTION FOR PRINCIPAL SHEAR, MOMENT AND AXIAL LOAD  When t h i s c o n d i t i o n e x i s t s ,  the element b e h a v i o u r  i s no l o n g e r  and a s t r u c t u r e composed of t h e s e elements may r e a c h a c o n d i t i o n of  linear,  instability.  To i n v e s t i g a t e t h i s c o n d i t i o n , elements of t h e web and f l a n g e under the a c t i o n of P, M, and V were examined i n a d i s p l a c e d p o s i t i o n , as shown i n F i g . 9 . From symmetry,  the s h e a r c e n t e r of t h e s e c t i o n c o i n c i d e s w i t h  c e n t r o i d , and i t s l a t e r a l d e f l e c t i o n i s measured by y ,  as shown i n F i g .  the 9(a).  L a t e r a l d e f l e c t i o n s of p o i n t s o t h e r t h a n the c e n t r o i d a r e found from the t i o n y i = y + <J>n.  Due t o t h e p r e s e n c e of P, M and V,  the d i f f e r e n t i a l elements  a r e under the a c t i o n of s t r e s s e s a and x as shown i n F i g s . where ' o = (P _ MQ) _ (A I ) y  „ ~ ~ I M  o  and  T = V  A w  rela-  9(b),  (c),  (d),  (e)  14.  (a)  (b) web  (c) flange  (e)  Fig.  9  curature = z' =4>' p flange  ELEMENTAL BEAM SECTIONS IN DISPLACED POSITION UNDER THE ACTION OF PRIMARY STRESSES The shear s t r e s s T i s assumed c o n s t a n t over the web and t h e b e n d i n g  moment M i s g i v e n by M = M - VL o — + Vx The s t r e s s e s a and T may be c o n s i d e r e d as g e n e r a t i n g l a t e r a l p r e s s u r e s i n the y d i r e c t i o n of p  and p , as shown i n F i g s .  9(b),  (c),  (d),  which  a c t on t h e element where i  (a - Mn )• ' ( ° I ) l • 1  y  p  = TtcJ. T  +  fc  (a " (  0  2xt £ n  ' —  2T(J)  t  -  MQ. )  I  (y  i  +  ? i $>  n )  t  (8)  ) (9)  15. where  t = b i n the web = w i n the f l a n g e The s t r e s s e s a may a l s o be thought of as g e n e r a t i n g v e r t i c a l p r e s s u r e s  a c t i n g on elements i n the top f l a n g e of v a l u e .p  , as shown i n F i g .  9(e),  where p °  = (a - Mh ) ( ° 21)  ." *  . ^ ^ n  p  e  n  By i n t e g r a t i n g t h e s e p r e s s u r e s over t h e i r r e s p e c t i v e a r e a s and d i v i d i n g by d x , t h e f o r c e s and t o r q u e s per u n i t l e n g t h can be o b t a i n e d . They a r e g i v e n by  L a t e r a l force/Length  =  + h/2 . (p dx . -n <= - h/2 J  + h/2 Torque/Length = 1 _ j (p dx n = - h/2  + p ) dn dx  0  (11)  T  + w/2 + p^)  &  ndn dx + 1_ dx  pdp dx p = - w/2  •+ w/2 V e r t i c a l f o r c e / L e n g t h = JL_ ^ . p^ dx p = - w/2  dp  (13)  Now, t h e l a t e r a l f o r c e / u n i t l e n g t h becomes the R . H . S . of e q . ( 4 ) give:  (12)  to  +h/2 ^Ely""  -  ' |n = - h/2 1  ( o  o - ^  V  c  (  "  y  +  L  +  (14)  d r i  J  The t o r q u e / u n i t l e n g t h becomes the R . H . S . of eq.(3) t o 2 '"' E l h <ji 2  + h/2  11  CtJ) ' =  ,,  t-(a - MnHy ( ° I.) = -h/2 y '  + w/2  , n) n t + 2 cp t n ] dn X  + w/2 (°  p = - w/2  ,, + tj>  give:  2 i ) y  <  f ' P  e  d  P  +  f p = - w/2  ( °  21) $ y  P  e d  P  ( ) 1 5  16.  The v e r t i c a l f o r c e / u n i t  l e n g t h a f f e c t s the z d e f l e c t i o n of the c e n -  t r o i d of the s e c t i o n i n the f o l l o w i n g manner: + w/2 EI z y  r- / 2 w  /  (a - Mh ) , . } o -zj : <J> pedp +  P = - w/2  By m u l t i p l y i n g o u t , of t h e s e c t i o n , e q s .  (14),  •  2  M M  E l h <> f 2 EI z y where  I  P  I  I  + M<> j  I  ^V  reduce t o t h e g o v e r n i n g  properties  differential  •  f  + 2cj>V  t I  - Cd>  I I  = - a I $ °  (17) I I  +My  (18)  P  =0 -  (19)  = p o l a r moment of i n e r t i a about  E q u a t i o n (19) the s e c t i o n .  (16)  (16)  follows:  I  = - Py  1  , pedp  i n t e g r a t i n g , and u s i n g the symmetry  (15),  e q u a t i o n of t h e s e c t i o n as 2EIy  (a + Mh ) } o — ' $  P = - w/2  y  }  1  M M  I  It  centroid.  i s the e q u a t i o n g o v e r n i n g  the v e r t i c a l d e f l e c t i o n s  of  s t a t e s t h a t the p r i n c i p a l axes f o r c e s M, P and V have no  e f f e c t on t h e v e r t i c a l d e f l e c t i o n s when the element undergoes a l a t e r a l or t o r s i o n a l displacement. e f f e c t on the y ,  It  s h o u l d be n o t e d though t h a t t h e r e w i l l be some  z and <> f d e f l e c t i o n s due t o v e r t i c a l d e f l e c t i o n , but i n  d e r i v a t i o n t h e v e r t i c a l d e f l e c t i o n s a r e assumed to be s m a l l and t h e i r  this  effect  i s t a k e n as z e r o . The e x a c t s o l u t i o n f o r t h e d i f f e r e n t i a l e q u a t i o n s f o r t h e v a r i o u s end c o n d i t i o n s r e q u i r e d by the s t i f f n e s s m a t r i x would be d i f f i c u l t I n s t e a d an i t e r a t i v e If  to o b t a i n .  t e c h n i q u e w i l l be d e v e l o p e d .  the beam i s r e p r e s e n t e d by s e v e r a l e l e m e n t s , t h e s e w i l l be much  s h o r t e r than the s t r u c t u r e .  T h i s means the d e f l e c t i o n s of the element  to i t s l o c a l c o - o r d i n a t e s w i l l be much s m a l l e r t h a n the s t r u c t u r e and c o n s e q u e n t l y the element w i l l be much s t i f f e r  relative  deflections  than t h e s t r u c t u r e .  Because  of t h i s , the c r i t i c a l P , M and V f o r the s t r u c t u r e w i l l be much lower t h a n the  17. c r i t i c a l P , M and V f o r t h e e l e m e n t .  Thus t h e P , M and V i n each element w i l l  have o n l y a s m a l l e f f e c t i n m o d i f y i n g ?:he d e f l e c t i o n s ; shape, p r e v i o u s l y  consequently  o b t a i n e d , w i l l be q u i t ^ c l o s e t o t h e f i n a l d e f l e c t e d shape.  By p l a c i n g t h e l i n e a r d e f l e c t i o n s , w h i c h were p r e v i o u s l y i n t o the R.H.S.  the l i n e a r  of e q s . ( 1 7 )  obtained,  and (18) we o b t a i n new l i n e a r e q u a t i o n s , i n w h i c h  the e f f e c t of M, P and V w i l l be a p p r o x i m a t e l y accounted f o r ; s o l v i n g  these  new e q u a t i o n s f o r homogeneous boundary c o n d i t i o n s y i e l d s i n c r e m e n t s i n y and <j>. T h i s p r o c e s s can be r e p e a t e d u s i n g t h e newly o b t a i n e d y and <> } t o get a f u r t h e r r e f i n e m e n t on t h e l i n e a r y and cf>. T h i s may be s i m p l y w r i t t e n as  2 E I  y +i = n  2  " "  p  y  n  "  + M+n'  2 V  +  ' '  E l h ch — —  V  ' '  - C| , , = - o H n+1 o p n  T  <> 20  +My  "  (21)  n  n+i where n = 0 , 1 , 2 . . . and y^ and <j> r e p r e s e n t Q  the l i n e a r  deflections.  S i n c e t h e boundary c o n d i t i o n s a r e s a t i s f i e d by t h e l i n e a r y  o  deflections  and <> j , t h e sequence of new s o l u t i o n s y and <h , n = l , 2 . . . a s remarked o n n Y  Y  above must s a t i s f y homogeneous boundary  conditions.  Upon t e r m i n a t i o n o f t h e i t e r a t i o n p r o c e d u r e , be o b t a i n e d by summing t h e y^ and <)j y = y  + y i + y2 • • •  D  • = +  0  + *1 + *  2  •••  y +  By u s i n g t h i s t e c h n i q u e ,  t h e f i n a l r e s u l t s may  f u n c t i o n s o b t a i n e d , as'shown i n e q s . (22)  n  ( ?  n  '  2 )  t h e f i n a l y and <> f obtained s a t i s f y the  r e q u i r e d end c o n d i t i o n s , and t h e terms i n t h e s t i f f n e s s m a t r i x can be found by s u i t a b l e d i f f e r e n t i a t i o n of y and cj>. As has been p r e v i o u s l y  i n d i c a t e d , t h e use of s e v e r a l elements t o  r e p r e s e n t a s t r u c t u r e reduces t h e e f f e c t of M, P and V on the element d e f l e c tions.  Indeed,  t h i s e f f e c t c a n be made as s m a l l as we p l e a s e Dy t a k i n g  18.  s u f f i c i e n t elements; i n these c i r c u m s t a n c e s , then, i t one i t e r a t i o n of e q s . results.  (20)  and (21) w i l l g i v e s u f f i c i e n t a c c u r a c y i n the  S i n c e the l i n e a r f o r c e s have a l r e a d y been found from y and S , •'o o  o n l y remains to f i n d the f o r c e s due to y j n o n - l i n e a r terms of i n t e r e s t , Kj.  and qb^.  These f o r c e s w i l l be  it  the  and the m a t r i x o b t a i n e d from them w i l l be c a l l e d  T h i s m a t r i x may be thought of a r i s i n g from a known d i s t r i b u t e d l o a d ,  to a p r e v i o u s l y tial  can be m a i n t a i n e d t h a t  o b t a i n e d s e t of y and <j>, b e i n g a p p l i e d to t h e l i n e a r  due  differen-  equations. .  tii  It  s h o u l d be noted t h a t the use of y  II  and y  t o f i n d s h e a r s and  moments i m p l i e s t h a t t h e c o - o r d i n a t e system i n w h i c h t h e f o r c e s on t h e beam a r e r e p r e s e n t e d t r a n s l a t e s and r o t a t e s w i t h the member. a r e tangent and p e r p e n d i c u l a r  In other words,  to t h e f i n a l d e f l e c t e d beam shape.  the  T h i s means  t h a t t h e f o r c e s on t h e beam end must be t r a n s f e r r e d i n t o the s t r u c t u r e ordinate system.  forces  co-  S i n c e t h e f o r c e s found from the d i f f e r e n t i a l e q u a t i o n need i  o n l y be m o d i f i e d by the c o s i n e of a n g l e s <> j or y , they remain b a s i c a l l y unchanged f o r s m a l l d e f l e c t i o n t h e o r y .  However, s i n c e t h e p r i n c i p a l f o r c e s M,  P and V a r e a l s o r e p r e s e n t e d i n t h e s e a x e s , they must a l s o be t r a n s f o r m e d  into  i  s t r u c t u r e c o - o r d i n a t e s by the use of t h e s i n e of cf> or y . d e f l e c t i o n theory s i n e 0 = 8 ,  Since for  the components w i l l be t h e f o r c e s of  m u l t i p l i e d by the d e f l e c t i o n of i n t e r e s t .  small interest  The j o i n t f o r c e s must be s u i t a b l y  a d j u s t e d t o account f o r t h e p r e s e n c e of t h e s e components.  These component  f o r c e s may be thought of as p o i n t l o a d s , and t h e m a t r i x due t o t h e i r w i l l be c a l l e d K . 2  effects  S i n c e t h e s e f o r c e s a r e due o n l y to t h e l i n e a r end d e f l e c t i o n s  of the e l e m e n t , they a r e u n a f f e c t e d by element l e n g t h or assumed end c o n d i t i o n s f o r the s o l u t i o n of t h e n o n - l i n e a r d i f f e r e n t i a l For c o n v e n i e n c e of r e f e r e n c e , be known as l o a d s of the f i r s t of t h e second t y p e .  equations.  the e f f e c t i v e d i s t r i b u t e d loads  will  t y p e and the p o i n t l o a d s w i l l be known as l o a d s  The complete n o n - l i n e a r p o r t i o n of the m a t r i x i s  then  19. K, + K „ . t o w h i c h must be added the l i n e a r m a t r i x K . lo  .  1  A l t h o u g h t h e n u m e r i c a l t e c h n i q u e as d e s c r i b e d s i m p l i f i e s the of t h e d i f f e r e n t i a l e q u a t i o n s , i t  s t i l l e n t a i l s the s o l u t i o n of a second  d i f f e r e n t i a l e q u a t i o n as w e l l as s e v e r a l i n t e g r a t i o n s . posed t o overcome t h i s work w i t h a f u r t h e r be d e s c r i b e d i n the next  chapter.  solution  It  i s therefore  order pro-  a p p r o x i m a t i o n or s i m p l i f i c a t i o n to  20.  CHAPTER IV  METHODS OF APPROXIMATIONS.  B e f o r e p r e s e n t i n g the n e x t a p p r o x i m a t i o n used i n the s o l u t i o n of  the  d i f f e r e n t i a l e q u a t i o n , i t may p r o v e v a l u a b l e to i n v e s t i g a t e t h i s same a p p r o x i m a t i o n a p p l i e d t o a s i m p l e r and more f a m i l i a r p r o b l e m . I n t h e a n a l y s i s of beams under the a c t i o n of d i s t r i b u t e d l o a d s , one method of t r e a t m e n t e n t a i l s d i v i d i n g t h e beam i n t o s e v e r a l segments by d u c i n g new j o i n t s  ±  *  a l o n g the member as i n F i g .  w v v  10(a).  w v  v  v  v  m '  >'  „  -  v  i/ „ </ - I - ^  „  UJ  L C  3  wL 2  w  intro-  '—A  wL 2  D-  wL< 12  12  wL 12  -><  „ L  Or  Fig.  10  j  wL  wL  I  <-  wL  wL  I 1 5^  wL  -5<f-  REPLACING DISTRIBUTED LOADS BY FIXED ENp REACTIONS  wL  21. The d i s t r i b u t e d l o a d on each segment i s then r e p l a c e d by f i x e d end r e a c t i o n s a c t i n g at the j o i n t s ,  as i n F i g . 1 0 ( b ) .  When t h e s e a r e p l a c e d on the  beam as j o i n t l o a d s , t h e moments c a n c e l at t h e i n t e r i o r j o i n t s o n l y a t t h e end j o i n t s ,  as i n F i g .  10(c).  and a r e  present  These a r e u s u a l l y i g n o r e d as they  have n e g l i g i b l e e f f e c t on t h e a n a l y s i s r e s u l t s i f s u f f i c i e n t elements a r e used. For beams w i t h an a r b i t r a r y  load d i s t r i b u t i o n ,  not i n g e n e r a l c a n c e l at each i n t e r i o r j o i n t .  t h e end moments w i l l  However, the end s h e a r s  are  p r o p o r t i o n a l t o t h e l e n g t h of the e l e m e n t , whereas the end moments a r e p r o p o r t i o n a l to the l e n g t h squared.  T h i s means t h a t as the l e n g t h L goes to  t h e end moments d e c r e a s e f a s t e r than t h e end s h e a r s .  Therefore,  by d e c r e a s i n g  L, w h i c h means i n c r e a s i n g the number of e l e m e n t s , the moments approach  zero.  S i n c e an i n c r e a s e i n the number of elements a l s o causes the d i s t r i b u t e d on a d j a c e n t members to approach each o t h e r i n v a l u e , approach z e r o ,  zero,  loads  the end moments not  they approach each o t h e r but w i t h a s i g n d i f f e r e n c e .  only  A good  a p p r o x i m a t i o n i s t h e n o b t a i n e d by u s i n g s e v e r a l e l e m e n t s , i g n o r i n g the s m a l l moment r e s u l t a n t s , and u s i n g o n l y the end s h e a r s .  T h i s amounts to r e p l a c i n g  the f i x e d end r e a c t i o n s of each element by p i n - e n d  reactions.  This leads to the next approximation i n  the s o l u t i o n of the  differ-  e n t i a l equation for l a t e r a l t o r s i o n a l b u c k l i n g : A p p l y the t y p e one l o a d s of t h e R . H . S . s i m p l y s u p p o r t e d element r a t h e r t h a n a f i x e d  of e q s . ( 2 0 )  and (21)  element.  T h i s w i l l e l i m i n a t e end moments a c t i n g a t the j o i n t s t y p e one.  to a  The t y p e two l o a d s e x i s t unchanged by t h i s  due t o l o a d s of  approximation.  22.  CHAPTER V  ILLUSTRATION OF METHODS OF APPROXIMATIONS  As an i l l u s t r a t i o n of t h e above method and a p p r o x i m a t i o n , the b u c k l i n g m a t r i x of a beam column w i l l be developed w i t h the type one l o a d s a c t i n g on a f i x e d element and t h e n on a s i m p l y s u p p o r t e d e l e m e n t . m a t r i x w i l l be developed i n two p a r t s : a f i r s t a second p a r t due to type two l o a d s . Weaver  Each n o n - l i n e a r  p a r t due to type one l o a d s and  The e x a c t m a t r i x i s g i v e n by Gere and  [2]. For a beam element under c o n s t a n t a x i a l l o a d , the d i f f e r e n t i a l equa-  tion  is Ely  = - P.y  f o r t h e element of F i g . 1 1 ( a ) . ended element f i r s t ,  ' -  (23)  E q u a t i o n (23) w i l l be a p p l i e d to a f i x e d  u t i l i z i n g the s i g n c o n v e n t i o n of F i g .  1(b)  for  Properties -  E,I  shear,  moment and l o a d .  A  y"=M/EI  46 L.2 x  -6 y  • i _ - 6 ^ I2x ~2 ~~3 +  M/EI  ^ r r n -4 L Fig.11  y =  — L  + 2_ L  6x  ORDINARY BEAM STIFFNESS DEFLECTIONS, FORCES AND CURVATURES  .  x  23.  The e q u a t i o n of  y o  1 1  y  = - 6 -7 L  f o r 61 e q u a l s u n i t y  is  , 12x +•-—3 L  S u b s t i t u t i o n of e q . ( 2 4 )  (24)  i n t o the R . H . S .  of e q . (23) g i v e s  E I y i " = " P [ - 6 _ , 12x] L Integration  L  gives  Ely'i"  Ely/  = " P[-6x r 2 L  » y i  6x^ +  L  = - P[- 3 x ! [  El  L  = - P [ - x _  y  i  4L  A  +  ]  ]  (26)  J  2x1  2L.  +  = - p[- x!_ 1  +  3  +  A  x  +  J  B  L  L Ei  .  (  (  7  )  ]  3 +  A x _  +  B  x  +  l  (  2L  +  2  2  g  )  J  * ! _ + A|!  +  B^!  +  C  X  +  D  ]  10L  (  2  9  )  J  I t i s t o be remembered t h a t t h e y^ i n t h e above e q u a t i o n i s t h e y^ due t o l o a d s o f t y p e o n e . For a f i x e d e l e m e n t , t h e end c o n d i t i o n s a r e : @x = 0  yj = 0 y[  = 0  @x = L  .y  1  = 0  y[  = 0  U s i n g t h e s e end c o n d i t i o n s and s o l v i n g f o r t h e c o n s t a n t s A = 6/5L B = - 1/10 C = 0 D = 0  gives  24.  Eqs.(26)  and (27) t h e n g i v e end s h e a r s and moments as  @x = 0  M = Ely!  ii  II  = P/10  V = ElyJ "  @x = L  M = EIy  1  = - 6P/5L  = - P/10  V = EIy[ "  = -  6P/5L  11  For  6 = 1, y 2  d  =. - 4  +  S u b s t i t u t i o n of e q s . ( 3 0 )  Integration  = - P (( - 4 L  i n t o eqs.(23)  3  0  )  (  3  1  )  gives  6x ) 2 )  +  (  L  L  Elyi  6x  L  gives I I I  Elyj  = - P ( - 4x ( L  ^ i '  =  "  P  ["|- - + X  i  3  2  3x_  )  2  4+A x 1  4  (  + B  1  )  2  1  1  The end c o n d i t i o n s f o r a f i x e d element a r e  y[  x = L  = 0  S o l v i n g f o r the constants y i e l d s A  = 11/10  :  B  :  = -2L/15  Ci = 0 D  l  = 0  2  (33)  )  •"'••'i'^^T ^* "^) @x = 0 y ! = 0  3  1 )  y  x  = 0  yi = 0  (35)  )  Substitution shears  of A j , B j , C' and Dj i n t o eqs. (32) and (33) y i e l d s the 1  and moments o f i n t e r e s t . t t  it  M = Ely  (?;•; = 0  = 2PL/15  @x = L  V = E l y / ' = - 11 P/10 The calculations  ead f o r c e s f o r ,$ and SL, e q u a l t o one can be found 3  t o t h e ones above.  6P " 5L  IIP 10  P 10  2PL 15  P ~ 10  Fig.  12  The m a t r i x  + JL 10  +  using  When t h e r e s u l t s a r e p l a c e d i n m a t r i x  on a f i x e d ended element.  5L  = PL/30  V = E l y / ' = .- P/10  they g i v e the p o r t i o n o f the n o n - l i n e a r m a t r i x acting  M = Ely  similar form,  due t o l o a d s of the f i r s t  type  i s g i v e n i n F i g . 12.  P 10 + 6P 5L  +  ^ 30  ^ 30 1  1  P  10 2PL 15  NON-LINEAR BEAM COLUMN MATRIX FOR TYPE 1 LOADS,' FIXED END CONDITIONS  For t h e p o r t i o n o f t h e n o n - l i n e a r m a t r i x due t o l o a d s of the second type, the v a l u e o f the l i n e a r end d i s p l a c e m e n t  Fig.  13  need be c o n s i d e r e d .  NON-LINEAR BEAM COLUMN TERMS FOR TYPE 2 LOADS  See F i g . 1 3 .  26.  For 6^ = 1 , as i n F i g . 1 3 ( a ) , t h e r e a r e no components of P a c t i n g i n t h e shear d i r e c t i o n , so t h e r e a r e no c o n t r i b u t i o n s i n t h e second p o r t i o n of  the  m a t r i x f o r t h i s d e f l e c t e d shape. For 6  2  =  1J  a  s  i  F i g . 13(b),  n  the l o a d P w h i c h the j o i n t must  provide  has a component i n t h e shear d i r e c t i o n of v a l u e P x s i n e (0) = P0 = P a t the L . H . S of t h e s t r u c t u r e . of the m a t r i x f o r 6  2  T h i s v a l u e must be e n t e r e d i n t h e shear f o r c e  = 1.  position  The R.H.S has no component.  By t r e a t i n g t h e o t h e r d e f l e c t i o n s the same way, t h e second p o r t i o n of the n o n - l i n e a r m a t r i x i s b u i l t up as shown i n F i g . 1 4 .  Fig.  14  0  +p  0  0  0  0  0.  0  0  0  0  -p  0  0  0  0  NON-LINEAR BEAM COLUMN MATRIX FOR TYPE 2 LOADS  When t h e m a t r i c e s p r e s e n t e d i n F i g s .  12 and 14 a r e added, the  result  i s the complete second o r d e r m a t r i x as shown i n F i g . 1 5 .  6P 5L P 10  + & 5L P 10 Fig.  15  . P 10  + f  2PL 15  P 10  + •£ 6P 5L  2PL 15  COMPLETE NON-LINEAR BEAM COLUMN MATRIX, FIXED END CONDITIONS  27.  In summary, t h i s m a t r i x wat- found by using shape to g e n e r a t e a l o a d t o use i n eq.'23).  the l i n e a r  deflected  T h i s l o a d was a p p l i e d t o an  element f i x e d a t t h e ends. It exact  i s of i n t e r e s t t o note a t t h i s p o i n t  matrix f o r buckling,  imate m a t r i x i n F i g . 16.  containing  the r e l a t i o n between t h e  s i n e and c o s i n e  f u n c t i o n s , and the approx-  I f t h e s e r i e s expansions f o r the s i n e and c o s i n e a r e  s u b s t i t u t e d i n t o the exact m a t r i x , the second term of t h i s expansion g i v e s the approximate m a t r i x d e r i v e d  above.  E q u a t i o n (23) w i l l now be a p p l i e d t o a p i n ended element. Fig. loads.  12 s t i l l  represents  the l i n e a r d e f l e c t e d shapes and t h e type one  To s o l v e f o r the f o r c e s , i t i s o n l y n e c e s s a r y t o i n t e g r a t e eq.(23) t i  t w i c e s i n c e t h e known end c o n d i t i o n s constants  of y  =0  a t the ends w i l l s o l v e the two  o f i n t e g r a t i o n , and then one d i f f e r e n t i a t i o n w i l l  However, because o f the simple type one loads used t o determine the end s h e a r s ,  Fig.  16  g i v e the end s h e a r s .  f o r these c a s e s ,  s t a t i c s can be  as i n F i g . 16.  NON-LINEAR BEAM COLUMN TERMS FOR TYPE 1 LOADS, PINNED END CONDITIONS  28.  S i m i l a r l y , t h e end f o r c e s f o r 63 and 64 equal t o one can be o b t a i n e d . Combining t h e end f o r c e s i n t o m a t r i x form g i v e s the p o r t i o n o f the n o n - l i n e a r m a t r i x due to l o a d s o f the f i r s t  p L  0  0 F i g . 17  type a c t i n g on a pinned element.  -p  ., I L  0  0  0  P L  0  0  0 0  + P  0  NON-LINEAR BEAM COLUMN MATRIX FOR TYPE 1 LOADS, PINNED END CONDITIONS  The p o r t i o n o f the m a t r i x due t o l o a d s o f the second changed.  See F i g . 1 7 .  type remains un-  T h e r e f o r e , t h e complete m a t r i x i s found by adding the m a t r i x o f  F i g . 1 7 and F i g . 1 4 .  T h i s y i e l d s a r e l a t i v e l y s i m p l e m a t r i x , as shown i n  Fig. 18.  p  0  + L-  P  0  0  0  0  0  + T  P  0  P  0  0  0  0  L  L  Fig.  18  strut  0  COMPLETE NON-LINEAR BEAM COLUMN MATRIX, PINNED END CONDITIONS.  I t i s of i n t e r e s t pin-ended  L  t o note t h a t t h i s i s the n o n - l i n e a r m a t r i x f o r a  under a x i a l l o a d .  29.  The e f f e c t of t h e a p p r o x i m a t i o n s w i l l now be c o n s i d e r e d .  Three s t r u c -  t u r e s , a p i n n e d c o l u m n , a f i x e d column w i t h a p i n a t t h e c e n t e r and c. f i x e d column were a n a l y z e d u s i n g t h e m a t r i c e s p r e s e n t e d i n F i g . column was a n a l y z e d u s i n g a v a r y i n g number of e l e m e n t s .  18 and F i g . . 5 . The per cent  Each  er.rors  i n t h e l e s u l t s of each a n a l y s i s were p l o t t e d a g a i n s t t h e number of elements used f o r each column, as shown i n F i g . 1 9 . It  s h o u l d be noted t h a t the f o r c e s due t o l o a d s of t h e second t y p e  i n the m a t r i x a r e of no i m p o r t a n c e i n members w i t h c o n t i n u o u s  deflections.  T h i s a r i s e s from t h e f a c t t h a t the a d j o i n i n g ends of elements i n a  continuous  s t r u c t u r e have t h e same d e f l e c t i o n s but the end f o r c e s a r e of o p p o s i t e s i g n . The c o n t r i b u t i o n s each o t h e r o u t .  of end f o r c e s to each j o i n t from each member t h e n c a n c e l However, i f  a p i n e x i s t s i n the s t r u c t u r e ,  continuity  of  s l o p e no l o n g e r e x i s t s and the end f o r c e components of a d j a c e n t elements may n o t be s e l f c a n c e l l i n g .  T h i s i m p l i e s t h a t the type one l o a d s a r e a l l t h a t  necessary to analyze continuous s t r u c t u r e s , structures with i n t e r i o r  pins.  but t h a t they w i l l f a i l to  The r e s u l t s of a n a l y s i s of v a r i o u s  p i n n e d column u s i n g m a t r i c e s w i t h and w i t h o u t second t y p e b e a r t h i s  1  1  analyze  interior  the e f f e c t s of l o a d s of  the  out.  In conclusion then, i t y =y  is  i s proposed t o . c a l c u l a t e y  1  on t h e b a s i s of  = 0 a t each end and j u s t use t h e c a l c u l a t e d end s h e a r s i n t h e m a t r i x i  r a t h e r than using  =  Yj = 0 as the end c o n d i t i o n s and u s i n g the a s s o c i a t e d  c a l c u l a t e d s h e a r s and moments i n the m a t r i x .  The type two t e r m s , w h i c h a r e  added to t h e above r e s u l t s , a r e the same f o r e i t h e r s e t of assumed end c o n ditions.  Accuracy  Fig. 19  Plots  of  fix-fix p i n - pin  Matrices  PLOT OF % ERROR VS. NO. OF ELEMENTS MATRICES FOR 3 COLUMN T Y P E S  FOR  B E A M — COLUMN  31.  CHAPTER VI  •  i  DEVELOPMENT OF LATERAL STABILITY MATRIX  A p p l i c a t i o n of Only one  Approximations  i t e r a t i o n w i l l be used to develop the m a t r i x .  p o r t i o n of the m a t r i x , c o r r e s p o n d i n g t o the d e f l e c t i o n s y ^ and been o b t a i n e d .  The n o n - l i n e a r terms of the m a t r i x due  to y  found by s o l v i n g the d i f f e r e n t i a l e q u a t i o n f o r y^ and tions o f y  1  = y  =<f' =<f>  1  from y^ and <f> .  1  1  = 0 a t each end.  ^  Shears and  Once a g a i n , the type two  torques a r e then found  = <J>j  =0  a t each end as boundary  l o a d terms a r e independent  of whichever  However, the use of s t a t i c s t o determine  pinned element was  d i s c a r d e d as the type one  a s s o c i a t e d w i t h the beam column.  and then s o l v e d f o r the end  E l h 4>, —2— 1  2 i i  •Elh A — - — l y  con-  the end shears on a  were i n t e g r a t e d  twice  conditions.  Q  II  - C6, 1  II  + M<f> + 2V$ o  = - o I <j> o p o and  II 1  boundary  I n s t e a d , the g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s ,  it = - Py  I n t e g r a t i n g t w i c e , eqs.(36) 2EIy  conditions.  (21) become f o r n = 0,  ti i t  2 ""  neces-  l o a d s were more complex than those  a f t e r s u b s t i t u t i o n o f the l i n e a r d e f l e c t i o n s i n the R.H.S  (20) and  be  u s i n g boundary c o n d i -  d i t i o n s are used.  1  <J> w i l l now 1  i  s a r y to use y^ = y^ = 0 and  2EIy  and  o  I t i s deduced from the p r e v i o u s s e c t i o n t h a t i t i s not  i  Equations  linear  <j>, has a l r e a d y  l  J  The  (37)  i  (36)  Q  i  i  + My  t  (37)  o  J  become  II  = -'Py -  + ffW  Q  (38)  dxdx + 2V/<j> dx + Ax + B o  = - o I (j) + //My 1 o p o o J  II  dxdx + Dx + E  -  (39)  32. i  Integrating  t  t  i  dxdx by p a r t s , and remembering M  fM§  = V and V  =0,  gives  i t  JM<$> dxdx = JMd) dx o o Integration  o f /Mcfi^dx by p a r t s  ft, Vdx o  (40)  gives  T  /Mcf> dx  = Mtt  o  -  o  V/(J>  S u b s t i t u t i o n of (41) i n t o  o  dx  v  (40)  (41) J  gives  ii  J7:McJ>  dxdx = Mcf> - 2VJ"<j> dx o o  0  A similar integration  (42)  gives  it  //My  dxdx = M y - 2V/y dx  Q  Q  (43)  Q  Now, s u b s t i t u t i o n of e q s . ( 4 3 )  and (42) i n t o e q s . ( 3 8 )  1i  2EI  Q  + McJ> + Ax + B  1  -  Ccf), = 1  (44)  o  i i  2  E l h tt, — —  a I <> J + My o p o o  - 2V/y dx + Ex + D o  II  The end c o n d i t i o n s of <f>j = cj)^ equations  gives  i  = - Py  y i  and (39)  (44) and ( 4 5 ) .  (45)  II  y i  =  =  y i  =  u  w  e  r  e  then a p p l i e d to  These e q u a t i o n s l e n d themselves t o t h i s a p p r o a c h , as  the L . H . S of b o t h e q u a t i o n s become z e r o a t x = 0 , L , and t h e R . H . S o f t h e e q u a t i o n s c o n t a i n s o n l y t h e v a l u e s of t h e l i n e a r d e f l e c t i o n s a t x = 0 , L, t h e i n t e g r a t i o n of y ^ , and f o u r unknown c o n s t a n t s .  These c o n s t a n t s A , B , C , D a r e  e a s i l y found and t h e n o n - l i n e a r m a t r i x f o r c e s due t o l o a d s of t y p e one a r e then o b t a i n e d by a p p l i c a t i o n of t h e d i f f e r e n t i a l e q u a t i o n s ( 4 6 ) . i  III  Shear = E I y  = - Py  x  Q  i  + M<J> + Vcj>  + A (46)  i  2  l  T  III  Torque = C A , - E l h cb, T  i  i  t  = + a I <> f -My o p o 'o T  +Vy o 1  - E  33.  ii  C a l c u l a t i o n of N o n - l i n e a r S t i f f n e s s M a t r i x f o r Type One  The  Loads  terms of the n o n - l i n e a r m a t r i x w i l l be c a l c u l a t e d f i r s t  for  67  ii f o r which y^ i s g i v e n by one h a l f y0  = 3  of F i g . 3(a)  or,  6x  ~z - —  (47)  . L - L  I n t e g r a t i n g eq.(47) t h r e e times  fy  3 dx = x_ 2L  Z  - x 2  gives  k  4L  (48)  3 D  A c o n s t a n t s h o u l d be added t o e q . ( 4 8 ) , but b i n e d w i t h D i n eq.(45).  From F i g . 4,  the ends of the element f o r 6  x = 0  The  yi  end  y  7  the v a l u e s of the l i n e a r d e f l e c t i o n s a t  = 1 are:  x = L  = 0  1/2  = 0  1/h  = 0  0  = 0  0  c o n d i t i o n s f o r s o l u t i o n of eqs.  11 11 = y i =+!=<!>! = 0  Using eqs.(50),  i t i s taken t o be com-  @x =  (49)  (44)  and  (50)  (49) w i t h the d i f f e r e n t i a l  equation  2EI[0] = - P[0] + M[0]  x = L  2EI[0] = - P [ l / 2 ] + M [ l / h ] + A[L] + B  (51) and  (52)  are  O,L  x = 0  S o l v i n g eqs.  (45)  + A[0] + B  f o r A and  B gives  (44)  gives  (51)  (52)  34.  and (50)  U s i n g eqs.(49)  x = 0  with the d i f f e r e n t i a l equation  (45)  gives  E l h [ 0 ] - C[0] = - a I [0] + M[0] - 2 V [ 0 ] + E [ 0 ] + D o p  (53)  —z—  x = L  E l h [ 0 ] - C [ 0 ] = - a I [1/h] + M [ l / 2 ] - 2V[L/4] + Ei L + D o p  (54)  From e q s . ( 5 3 ) and ( 5 4 ) , VL E = c l (M + ^ ~ ) . „ o p o 2 + V hL 2L 2  D = 0  Substituting i n (49)  tions  SHEAR  t h e v a l u e s of A, B, D, E back i n t o eqs.(46) gives @x = 0  Q = - P [ 0 ] + M[0] + V [ 0 ] + A  = I (P - (M + ^ ) L(2 -fi-j- 2  @x = L  h @x = 0  1(2  VL + -f^ ))  — T  matrix  2L  T = o I [ 0 ] - M[0] + V [ l / 2 ] - E o p  hL By u s i n g s u i t a b l e  (55)  )  T = a I [ 0 ] - M[0] + V [ 0 [ - E o p  hL  (?x = L  ) )  Q = - P [ 0 ] + M[0] + V [ l / h ] + A V + 1 (P - (M  TORQUE  and u s i n g t h e r e l a -  2L  s i g n s , and t h e r e l a t i o n Q = T/h, t h e column o f t h e s t i f f n e s s  f o r 67 = 1 was b u i l t  from r e l a t i o n s  ( 5 5 ) and i s shown i n F i g .  20.  35.  jjr . 1 [P 2L [2  o  M  _1 [P. 2L [2  o  VL 2~] + 1 h ] h [  [V^ -  1 t l + I QL 2 [h  L (2  JL [V + 1 (P 2 [h  F i g . 20  L (2  M  o  _L  VL  + —, ,  r  VHjD ] + ) ]  T  +  + V ]  2L M  o  2 ]  . VL 2 2L  ,a I  2 ) ]-].[+ h ) ]- h [  h  o  hL  J L _ 2 ] - _1 [ o p h ] h [ hL  vr  VL  M  o p hL  + y_ J 2 ]  M  M  , VL  o  + —  2 2L  ,  ] ]  + ^  2 1 f+ o p o h [ hL 2L g  I  TYPE 1 TERMS FOR NON-LINEAR MATRIX FOR 6  ] ]  7  = 1  36.  For §6 ' 1 )  11  y  =  y  i  g i v e n by one h a l f  s  = - 1 + L  L  3 .  = - x  or  3x 2  I n t e g r a t i n g t h r e e times  fy  of F i g . 3(b)  gives h  + x  ,  From F i g . 4, the v a l u e s of the l i n e a r  d e f l e c t i o n a t the ends of the element f o r  6g = 1 a r e :  y = 0  x=0  x = L  o t  y  •'o  .  i  y = 0 ^o  J  4> = 0  -  O  end  still  y  *  = +  o  = 0  D  equation  (44) and  (45)  are  II  (58)  U s i n g eqs.(57) and  (58) w i t h the d i f f e r e n t i a l  equation  x = 0  2EI[0] = - P[0] + M[0]  + A[0] + Bj  x = L  2EI[0] = - PfO] + M[0]  + A L  From eqs.(59) and  (60), Bj and Aj  Aj = 0  :  + B  (44)  gives (59) (60)  l  are  Bj = 0  U s i n g eqs.(57) and x = 0  (57)  o  c o n d i t i o n s f o r s o l u t i o n of the d i f f e r e n t i a l  11 yi = yi  1/2  4> = + 1/h  <j> = 0 o The  =o  (58) w i t h the d i f f e r e n t i a l  EIh [0] i 2 2  equation  - C[0] = - a I [0] + M[0] on °  (45) g i v e s  - 2V[0]  P  + E,[0] + D,  i  *  (61)  2  [L ] - 2V [24] ~ + E, L + D, 124 J l l 2  E l h [0] - C[0] = - a I [0] + M[0] ° P  x = L  .*"2—  From eqs.(61) and T>  1  = 0  (62) E  = -  VL/12  (62)  37,  S u b s t i t u t i o n o f the v a l u e s o f A ^ . B j , C j ^ D j b a c k relations  i n t o eqs. (46) and u s i n g  (57) g i v e s  SHEAR  x = 0  Q = - P [ 0 ] + M[0] + V[0]  =0 x = L  Q = - P [ l / 2 ] + (M + ~ ) o z  - - £ TORQUE  x = 0  >  <  M  o  +  f  (1/h) + V[0]  >  (63)  T = + a I [ 0 ] - M[0] + V[0] - E Q  1  p  = + VL 12  x = L  T = +  a  l [1/h] + (M + ~ op o 2  = + o I - (M + ? o P o 2 h  ) [ 1 / 2 ] + V[0] - E, 1  ) + VL — x  z  By u s i n g the r e l a t i o n s i n (63) w i t h s u i t a b l e s i g n s , and t h e r e l a t i o n Q = T/h, the column of t h e m a t r i x due t o type one l o a d s f o r 65 = 1 was b u i l t ,  as shown  i n F i g . 21. Similarly,  the o t h e r s t i f f n e s s  d e f l e c t i o n s were t r e a t e d .  p l e t e n o n - l i n e a r p o r t i o n o f the m a t r i x due t o l o a d s o f type one i s be w r i t t e n as K, = Pk. + Mk, + Vk. , as i n F i g . 22. lp lm lv 1  The comand can  The p o r t i o n o f the ^  m a t r i x due t o l o a d s of type two must s t i l l be c a l c u l a t e d .  6  1  0  2  0  3  0 + 1 [- VL] h [ 12]  4  0 - 1 I- VL] h [ 12]  5  0  6  0  7  8  - 1 t- P + 2 [ 2  - 1 2  F i g . 21  [- P + [ 2  M  M  o  +  VL 2 ]  h  ]  VL r o I .M + 1 [- o p + o 2 - VL] h [ 2 12] h  VL VL a I M + + 1 L- o p + o 2 - VL] 2 ] o h 2 12] h ] h [ +  r  TYPE 1 TERMS FOR NON-LINEAR MATRIX FOR 6  6  = 1  Pk . = P  Pi  Mk  = M(f )  1  0  0  0  0  0  0  0  0  2  0  0  0  0  0  0  0  0  3  - a + b  - a - b  -(a+b)/L  (-a+b)/L  0  0  (a+b)/L  (a-b)/L  4  - a - b  - a - b  (-a+b)/L  -(a+b)/L  0  0  (a-b)/L  (a+b)/L  5  0  0  0  0  0  0  0  0  6  0  0  0  0  0  0  0  0  7 .  0  0 .  (a+b)/L  (a-b)/L  a - b  a + b  -(a+b)/L  (-a+b)/L  8  0  0  (a-b)/L  (a+b)/L  a + b  a - b  (-a+b)/L  (-a-b)/L  1  0  0  0  0  0  0  0  0  2  0  0  0  0  0  0  0  0  3  0  +L  +1  0  0  0  -1  0  4  -L  0  0  -1  0  0  0  +1.  5  0  0  0  0  0  0  0  0  6  0  0  0  0  0  0  0  0  7  0  0  -1  0  0  -L  +1  0  8  0  0  0  +1  +L  0  0  -1  X  0  0  0  0  0  0  0  0  2  0  0  0  0  0  0  0  0  3  +1  -5  ri  -1  -1  0  4  +5  -1  0  +1  +1  5  0  0  0  0  0  0  0  0  6  0  0  0  0  0  0  0  0  7  -1  -1  0  +1  -5  0  8  +1  +1  +5  -1  a = 1/4  b = I /Ah P  -6/L  +6/L  -6/1 0  +6/L Fig.  22  -6/L  -6/L  NON-LINEAR MATRIX FOR TYPE 1 LOADS  +6/L 0  +6/L 0  40.  iii  C a l c u l a t i o n o f N o n - l i n e a r M a t r i x f o r Type Two Loads  The  e f f e c t o f t h e type two l o a d s can b e s t be c a l c u l a t e d by s p l i t t i n g  each s t i f f n e s s d e f l e c t i o n i n t o i t s l a t e r a l and t o r s i o n a l component. For  6  7  = 1, t h e component l i n e a r d e f l e c t i o n s a r e g i v e n i n F i g . 23.  Ay  (a)  Fig.  By  applying  23  COMPONENT DEFLECTIONS FOR 6  M, P and V and t a k i n g  ?  = 1  t h e i r components about t h e R.H.S,  of t h e s e c t i o n as shown i n F i g . 24, g i v e s  Fig.  24  FORCE COMPONENTS DUE TO END DEFLECTIONS FOR  6 = 1  41.  In F i g . 24(a), joint deflections.  t h e r e a r e i;o components a c t i n g i n any of t h e a l l o w a b l e  Therefore: Shear  =0  Moment = 0 Torque For F i g . 2 4 ( b ) ,  =0  t h e components a c t i n g on the c r o s s s e c t i o n which must  be s u p p l i e d by t h e j o i n t a r e Shear  = + V/h  Moment = - (M + o 2  )/h  Torque = 0 The L.H.S. o f t h e d e f l e c t e d shapre f o r 6 all  the d e f l e c t i o n s are zero.  7  = 1 has no components, as  The column i n t h e m a t r i x  f o r l o a d s o f the second  type f o r <J = 1 i s g i v e n i n F i g . 25. 7  ° F o r <$6 = 1, t h e component  l i n e a r d e f l e c t i o n s a r e g i v e n i n F i g . 26.  (a) Fig.  26  (b)  COMPONENT DEFLECTIONS FOR 6  6  = 1  A p p l y i n g M, P and V, and t a k i n g t h e i r components a l o n g deflections  gives:  (i)  F o r t h e L.H.S. - no components, no f o r c e s .  (ii)  F o r t h e R.H.S. - see F i g . 27.  the allowed  42.  5" 2h  2  O  2h  + £_ 2h  2h  Fig.  25  TYPE 2 TERMS FOR NON-LINEAR MATRIX FOR S  7  = 1  43.  Fig.  27  FORCE COMPONENTS DUE TO END DEFLECTIONS FOR 6  6  = 1  For F i g . 27(a) components a c t i n g a l o n g t h e a l l o w e d d e f l e c t i o n s a r e M/2, P/2.  In s t i f f n e s s matrix s i g n Shear  convention, =  - P/2  Moment Torque  = =  0 (M  + VL/2)/2  For F i g . 27(b) t h e s l o p e changes c o n t i n u o u s l y from top t o bottom. T h e r e f o r e i n t e g r a t i o n a l o n g t h e c r o s s s e c t i o n must be used. e f f e c t o f a a t z on an element o f a r e a t d z , where o = + P - M z = a -Mz T o r A I I z z r  A force  =aztdz  s l o p e @ z i n x d i r e c t i o n i s z/h T h e r e f o r e , t h e component o f f o r c e a t z i s  oztdz h  Consider the  44.  Integrating with respect  t o z t o g e t the shear and torque  + h/2 . (a z t  Shear  z = - h/2  2  M z -t) . r v cz I h { z )  (  + h/2  M z t) , r ( dz  Ih"  1  z  ( z = - h/2  a I o z  )  s u f f e r the same a n g u l a r d i s p l a c e m e n t s as the  s e c t i o n as a whole, the end s l o p e s See  , M M + VL/2 + r = o  2  Torque  Because t h e f l a n g e s  gives  of the f l a n g e s must be taken i n t o account.  F i g . 28.  1  h  Fig.  The  28  =e  VERTICAL. FLANGE DEFLECTIONS AS FUNCTIONS OF <j>  s l o p e o f t h e z d e f l e c t i o n i n the x d i r e c t i o n i n the f l a n g e i s  t  £<j>  = £/h a t p o i n t  £.  The d i f f e r e n t i a l f o r c e a t p o i n t  £is  A f o r c e = oedE, The  v e r t i c a l shear component i s  + w/2 (a e£  M h£e) r  w/2  (  21  ") d? = 0 z )  45.  The  torque i s  + w/2 2 . (a e£  ;^r^ - - w/2  3  2  - M h£ e)  - - ~  d ?  21  + w/2  M £ e  ^i~  +  >  2  (  o I  Z  - w/2  However, t h e bottom f l a n g e has the same c o n f i g u r a t i o n w i t h a moment s t r e s s on i t o f o p p o s i t e s i g n .  T h e r e f o r e t h e n e t torque c o n t r i b u t i o n o f the  f l a n g e s under moment s t r e s s i s z e r o .  However, t h e a x i a l , l o a d  contribution i s  the same f o r t h e bottom f l a n g e as f o r the t o p . T h e r e f o r e , t h e t o t a l c o n t r i b u t i o n of t h e shape i n F i g . 27(b) i s g i v e n by Shear  = (M + VL/2)/h o  Moment = 0 Torque = - 2 a I - a I = - o I o o z o p h h h A f t e r combining  these f o r c e s , the column i n the m a t r i x f o r 65 = 1 due  to type two l o a d s i s g i v e n i n F i g . 29. from s i m i l a r c a l c u l a t i o n s .  K  9  ^  The r e m a i n i n g s i x columns can be found  T h i s m a t r i x , c a l l e d K , can be w r i t t e n as 2  = Pk  pz  + Mk  m  + Vk 2  v  2  See F i g . 30. The  complete  non-linear matrix f o r l a t e r a l t o r s i o n a l s t a b i l i t y i s  g i v e n i n F i g . 31 and i s o b t a i n e d by adding  iv  Numerical  the m a t r i c e s  and K . 2  Examples  The m a t r i x i n F i g . 31 was used  t o c a l c u l a t e the c r i t i c a l  loads of  s e v e r a l s t r u c t u r e s , and t h e r e s u l t s were compared t o the t h e o r e t i c a l A determinant  p l o t method o f s o l u t i o n was used.  solutions.  That i s , the d e t e r -  minant o f the s t r u c t u r e m a t r i x f o r i n c r e a s i n g v a l u e s of M, P and V were  46.  P  J? -  4  p  i_  4  F i g . 29  ,  +  +  ,  0 1  o p +  M  o  2  a i M o P + o 2  + —• M . + — 2 + o 2 2h 2h  +  f2 2h  -  M  o  +  — 2  2h  TYPE 2 TERMS IN NON-LINEAR MATRIX FOR 6  6  = 1  '  Pk  P2  Mk  V k  ni2  v  = P  •« M C{7 ) o hL T  " <T2h> V  2  1  2  3  4  5  6  7  8  1  0  0  0  0  0  0  0  0  2  0  0  0  0  0  0  0  0  3  a - b  a + b  0  0  0  0  0  0  4 5 6 7 8  a + b  a - b  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  - a  - a - b  0  0  0  0  0  0  - a - b  - a  0  0  1  0  0  + L/2'  - L/2  0  0  0  0  2  0  0  + L/2  - L/2  0  0  0  0  3  0  - L  0  0  0  0  0  0  + L  0  0  0  0  '. 0  0  0  0  0  0  0  0  0  - L/2  + L/2  0  0  0  0  0  0  - L/2  + L/2  0  0  0  0  0  + L  0  0  0  0  0  0  - L  0  0  0  1  0  0  - 3  + 3  0  0  0  0  2  0  0  - 3  + 3  0  0  0  0  3  0  + 6  0  0  0  0  4 5 6 7 8  -6  0  0  0  0  0  0  0  0  0  0  0  - 3  + 3  0  0  0  0  0  0  - 3  + 3  0  0  0  0  0  + 6  0  0  0  0  -6  0  4 5 6 7 8  a  =.1/4  b = I /Ah P  -  6/L 6/L ..  + 6/L + 6/L  F i g . 30  + b  + b  + 6/L + 6/L  -  6/L 6/L  NON-LINEAR MATRIX FOR TYPE 2 LOADS  1  2  3  4  .  5  6  7  8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  o"  0  -(a+b)/L  (-a+b)/L  0  0  (a+b)/L  (a-b)/L  0  0  (-a+b)/L  -(a+b)/L  0  0  (a-b)/L  (a+b)/L  0  0,  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  (a+b)/L  (a-b)/L  0  0  -(a+b)/L  (-a+b)/L  0  0  (a-b)/L  (a+b)/L  0  0  (~a+b)/L  -(a+b)/L  0  0  + L/2  - L/2  0  0  0  0  0  0  + L/2  - L/2  0  0  0  0  0  0  + 1  0 •  o  0  1  0  0  0  0  1  0  0  0  + 1  0  0  0  0  0  0  - L/2  • + L/2  0  0  0  0  0  0  - L/2  + L/2  0  0  1  0  0  0  + 1  0  0  0  + 1  0  0  0  0  0  - 3  + 3  0  0  0  0  0  0  - 3  + 3  0  0  0  0  + 1  + 1  1  0  + 6/L  -  -  - 6/L  0  -  -  -  - 6/L  0  1  0  + 6/L  0  0  0  0  0  1  1  1  + 1  -  + 1  -  1  -  -  b = I. /Ah P  -  1  + 1  + 1  0  0  0  - 3  + 3  0  0  0  0  - 3  + 3  0  - 6/L  + 1  + 1  + 6/L  0  + 6/L  o  -  - 1  0  - 6/L  1  2  a = 1/4  0  F i g . 31  THE COMPLETE NON-LINEAR MATRIX FOR LOADS OF TYPE 1 AND 2  49.  c a l c u l a t e d and p l o t t e d . e q u a l l e d zero was  The v a l u e of M,  taken as the c r i t i c a l  P and V a t which the load.  determinant  T h i s t e c h n i q u e was  an e i g e n v a l u e approach s i n c e i t a v o i d s the problem of c o n v e r g i n g e i g e n v a l u e which a r i s e s when an e i g e n v a l u e t e c h n i q u e  p r e f e r r e d to to a r e a l  i s a p p l i e d . t o systems  c o n t a i n i n g unsymmetric m a t r i c e s . A t a b l e of the r e s u l t s i s g i v e n i n F i g s . 3 2 and  32(a).  A plot  of  the number of elements used a g a i n s t the a c c u r a c y o b t a i n e d f o r f o u r cases i s g i v e n i n F i g . 33. The  e f f e c t of a p p l y i n g v e r t i c a l l o a d s a t p o i n t s o t h e r than the  " t r o i d can be accounted  f o r by m o d i f y i n g  the s t i f f n e s s m a t r i x of the  cen-  structure.  50.  10 E L E M E N T S  USED IN EACH  STRUCTURE  -° pinned  fixed  PURE EULER BUCKLING  I=9.7in L=20' E = 30,000k/in 4  STRUCTURE  2  ERROR  ->  1.2%  P * P  fixed  -o pinned PURE TORSIONAL BUCKLING STRUCTURE  ERROR -o <  < Ip^OOOin  4  l=500ir  4  J=5in  4  E = 3 0 , 0 0 0 k/in G = 11,500 k/in A = 10 i n h = 9.5 in 2  Fig. 32  TABLE OF RESULTS FOR TEST STRUCTURES  2  L = 20*  2%  3.5%  51.  10 E L E M E N T S  USED IN EACH  <;  pinned LATERAL  STRUCTURE  1  )  £ fixed  TORSIONAL BUCKLING STRUCTURE  ERROR  M  |M . 5 %  i 1%  2.5%  i  i  i  i  i  i  i  i  r~r  i  i  i  w  i  i  i  i  i  i  w  r  w  4%  1=21 i n  4  I = l5lin p  4  A = 7.l25in  2  J = .l48in  4  E = 30,OOOk/in G= II , 5 0 0 k / i n 2  Fig. 3 2 a  TABLE  OF R E S U L T S  FOR T E S T  STRUCTURES  2  52.  Fig. 3 3  PLOT OF ACCURACY  VS. N U M B E R OF E L E M E N T S  USED  CHAPTER V I I  CONCLUSIONS  An 8 x 8 m a t r i x f o r the exact l i n e a r treatment o f doubly  symmetric  •wide f l a n g e beams under t o r s i o n and l a t e r a l d i s p l a c e m e n t was developed. m a t r i x a l l o w s each f l a n g e independent t o t h e o t h e r  This  a t e i t h e r end t o assume t r a n s l a t i o n s and r o t a t i o n s flange.  An approximate m a t r i x a c c o u n t i n g f o r the e f f e c t o f p r i n c i p a l p l a n e forces  on t h e l a t e r a l d e f l e c t i o n s was developed.  m a t r i x , i t makes p o s s i b l e l o a d s o f wide f l a n g e d The and  When added t o t h e l i n e a r  the d e t e r m i n a t i o n o f t h e l a t e r a l - t o r s i o n a l b u c k l i n g  beams.  non-linear  m a t r i x was based on s m a l l  no d i s t o r t i o n o f t h e c r o s s - s e c t i o n .  Applied  p r i n c i p a l plane  external  deflections  l o a d s must m a i n t a i n  t h e i r d i r e c t i o n of a p p l i c a t i o n . To o b t a i n  the n o n - l i n e a r  were developed by c o n s i d e r i n g p r i n c i p a l forces.  s t i f f n e s s matrix, d i f f e r e n t i a l  a displaced  These were then s o l v e d  equations  element under t h e a c t i o n of the t o f i n d t h e end f o r c e s which were  entered i n the matrix. To developed.  ease t h e s o l u t i o n o f t h e e q u a t i o n , a n u m e r i c a l t e c h n i q u e was This  e n t a i l e d s u b s t i t u t i o n of the l i n e a r d e f l e c t e d  R.H.S. o f the d i f f e r e n t i a l e q u a t i o n t o produce a known l o a d , the L.H.S. f o r t h e new y^ then p l a c e d  and <J> ^.  initial  and then s o l v i n g  The new d e f l e c t i o n s were y^ and  i n t o t h e R.H.S. and t h e p r o c e s s r e p e a t e d .  were t o be those o f a f i x e d element. the  shape i n t o the  ^ were  The end c o n d i t i o n s  E f f e c t i v e end p o i n t  used  l o a d s a c t i n g due t o  l i n e a r end d e f l e c t i o n s gave a second l o a d s e t .  Two a p p r o x i m a t i o n s were then u t i l i z e d t i o n of the equations.  The f i r s t  e n t a i l e d using  to f u r t h e r s i m p l i f y the s o l u only  one c y c l e o f the i t e r a -  54.  t i o n scheme.  The  second  was  to a p p l y the e f f e c t i v e l a t e r a l l o a d s to an  element w i t h no end moment r e s t r a i n t . T e s t s of the element a g a i n s t known s o l u t i o n s i n d i c a t e t h a t good a c c u r a c i e s can be o b t a i n e d , but depend on the number of elements used analysis.  A c c e p t a b l e a c c u r a c i e s were o b t a i n e d u s i n g ten elements,  e r r o r encountered The  now  end  the  largest  5%.  advantages of t h i s m a t r i x are s e v e r a l .  l o a d , support and may  being  the  ii-  Cases i n v o l v i n g  general  c o n d i t i o n s of s e c t i o n s w i t h v a r y i n g s e c t i o n p r o p e r t i e s  be s o l v e d by s i m p l y b r e a k i n g the s t r u c t u r e up i n t o s e v e r a l elements  and a p p l y i n g the p r e s e n t e d  matrix.  LIST OF  REFERENCES  Timoshenko, S.P., " S t r e n g t h of M a t e r i a l s , P a r t I I " D. Van Nostrand Company, I n c . , New York, N.Y. pp. 255-265, 1956  Gere, J.M., and Weaver, W., " A n a l y s i s of Framed S t r u c t u r e s " D. Van Nostrand Company, I n c . , New York, N.Y. pp. 430, 1965  Timoshenko, S.P., and Gere, J.M. , "Theory of E l a s t i c M c G r a w - H i l l , I n c . , New York, N.Y. pp. 251-270, 1961  Stability"  B l e i c h , F., " B u c k l i n g S t r e n g t h of M e t a l S t r u c t u r e s " M c G r a w - H i l l , I n c . , New York, N.Y. pp. 149-160, 1952  Timoshenko, S.P., " H i s t o r y of S t r e n g t h of M a t e r i a l " M c G r a w - H i l l , I n c . , New York, N.Y. pp. 393, 1953  G o o d i e r , J.N., "Some O b s e r v a t i o n s on E l a s t i c S t a b i l i t y " P r o c e e d i n g s o f the F i r s t N a t i o n a l Congress of A p p l i e d Mechanics  

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