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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 . S c . ( C i v i l Eng.) The U n i v e r s i t y of B r i t i s h Co lumbia , 1966 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 accept t h i s t h e s i s as conforming to the r e q u i r e d s tandard 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 the requi rements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Co lumbia , I agree thav 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 s t u d y . I f u r t h e r 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 copy ing of t h i s t h e s i s f o r s c h o l a r l y p u r -poses may be granted by the Head of 'my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copy ing 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 lowed w i thout 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 E n g i n e e r i n g 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 n o n - l i n e a r e f f e c t s of p r i n c i p a l p lane s h e a r s , moments and a x i a l loads 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 doubly symmetric wide f l a n g e s e c t i o n . I n i t i a l l y , an exact 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 constant 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 a l l o w s the independent r e p r e s e n t a t i o n of the d e f l e c t i o n s of e i t h e r f l a n g e at e i t h e r end. The n o n - l i n e a r e f f e c t s are i n c l u d e d i n the d i f f e r e n t i a l equat ions by c o n s i d e r i n g the e f f e c t of the pr imary s t r e s s e s on the e q u i l i b r i u m of a d i s -p l a c e d e lement . Two approx imat ions are then i n t r o d u c e d . The f i r s t c o n s i s t s of a n u m e r i c a l techn ique 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 . 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 the d i f f e r e n -t i a l e q u a t i o n s . Us ing these 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 . S e v e r a l s t r u c t u r e s a re then a n a l y z e d . Each s t r u c t u r e i s d i v i d e d i n t o s e v e r a l e lements . T h i s a l l o w s beams of n o n - 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 to be a n a l y z e d , and i n c r e a s e s the accuracy of the r e s u l t s of the approximate m a t r i c e s . The r e s u l t s of these ana l yses a re then compared to t h e o r e t i c a l r e s u l t s and t a b u l a t e d . I t i s seen tha t the m a t r i x g i v e s good agreement f o r a l l cases t e s t e d . i i TABLE OF CONTENTS Page ABSTRACT TABLE OF CONTENTS LIST OF FIGURES DEFINITION OF SYMBOLS ACKNOWLEDGEMENTS 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 OF METHODS OF APPROXIMATIONS 22 CHAPTER VI DEVELOPMENT OF LATERAL STABILITY MATRIX 31 i A p p l i c a t i o n of Approx imat ions 31 i i 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 i i i 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 i v Numer ica l Examples 45 CHAPTER VI I CONCLUSIONS 53 LIST OF REFERENCES i i i i i i v v i i i i i LIST OF FIGURES Page F i g . 1 S i g n Convent ions 3 F i g . 2 Beam Segment 4 F i g . 3 P r imary Modes f o r R i g h t Hand End D e f l e c t i o n s 6 F i g . 4 S u p e r p o s i t i o n of Pr imary Modes 8 F i g . 5 L i n e a r S t i f f n e s s M a t r i x K 9 b o F i g . 6 Test R e s u l t s f o r K 11 o F i g . 7 E f f e c t of F lange Warping 12 F i g . 8 S i g n Convent ion f o r P r i n c i p a l Shear , Moment and A x i a l Load 13 F i g . 9 E lementa 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 Pr imary S t r e s s e s 14 F i g . 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 F i g . 11 Ord inary Beam S t i f f n e s s D e f l e c t i o n s , Forces and Curvatures 22 F i g . 12 N o n - l i n e a r Beam Column M a t r i x f o r Type 1 Loads , F i x e d End C o n d i t i o n s 25 F i g . 13 N o n - l i n e a r Beam Column Terms f o r Type 2 Loads 25 F i g . 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 F i g . 15 Complete N o n - l i n e a r Beam Column M a t r i x , F i x e d End C o n d i t i o n s 26 F i g . 16 N o n - l i n e a r Beam Column Terms f o r Type 1 Loads , P inned End C o n d i t i o n s 27 F i g . 17 N o n - l i n e a r Beam Column M a t r i x f o r Type 1 Loads , P inned End C o n d i t i o n s 28 F i g . 18 Complete N o n - l i n e a r Beam Column M a t r i x , P inned End C o n d i t i o n s 28 F i g . 19 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 F i g . 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 F i g . 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 F i g . 22 N o n - l i n e a r M a t r i x f o r Type 1 Loads , 39 F i g . 23 Component D e f l e c t i o n s f o r 6 7 = 1 40 i v LIST OF FIGURES (Contd. ) Page F i g . 24 Force Components Due to End D e f l e c t i o n s f o r 6 7 = 1 40 F i g . 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 F i g . 26 Component D e f l e c t i o n s f o r 6g = 1 41 F i g . 27 Force Components Due to End D e f l e c t i o n s f o r 65 = 1 43 F i g . 28 V e r t i c a l F lange D e f l e c t i o n s as Func t ions of <f> 44 F i g . 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 F i g . 30 N o n - l i n e a r M a t r i x f o r Type 2 Loads 47 F i g . 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 F i g . 32 Table of R e s u l t s f o r Test S t r u c t u r e s 50 F i g . 33 P l o t of Accuracy v s . Number of Elements Used. 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 a x i s moment of i n e r t i a of s e c t i o n abou.1: z a x i s 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 i n e r t i a a rea of s e c t i o n area of web t o r s i o n a l constant Youngs modulus shear modulus JG depth of s e c t i o n 2 2C/EIh l a t e r a l d e f l e c t i o n (a long y a x i s ) v e r t i c a l d e f l e c t i o n (a long z a x i s ) t o r s i o n a l d e f l e c t i o n f l a n g e shear torque d i s t r i b u t e d l o a d d i s t r i b u t e d torque s t i f f n e s s m a t r i x d e f l e c t i o n i n n d i r e c t i o n s t i f f n e s s m a t r i x f o r c e 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 oad i n element p r i n c i p a l moment at 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 x o normal s t r e s s i n element DEFINITION Oi? SYMBOLS (Contd. ) M^ = f l a n g e moment 0 = P/A o T = shear s t r e s s i n element p = p r e s s u r e due to n s t r e s s e s X = M^/2EI = parameter i n s o l u t i o n expansion v = n term m s e r i e s s o l u t i o n of y n <j>n = n*"^ 1 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 parameter £; = 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 w i t h r e s p e c t to x R.H.S R igh t hand s i d e L .H .S L e f t hand s i d e v i i ACKNOWLEDGEMENTS The author wishes to thank h i s s u p e r v i s o r , D r . R. F. Hoo ley , f o r h i s 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 the development of t h i s work. G r a t i t u d e i s a l s o expressed to the N a t i o n a l Research C o u n c i l of Canada f o r f i n a n c i a l s u p p o r t , and to the U .B .C . Computing Center f o r the use of i t s f a c i l i t i e s . March 1968 Vancouver , B .C . 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 i n t e r e s t and r e s e a r c h f o r y e a r s . The foundat ions of the theory 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 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 , 6] i n 1899 i n the study 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 s e c t i o n s . H. R e i s s n e r [6] l a t e r s t r e s s e d the e f f e c t of d e f l e c t i o n s i n the major p r i n c i p a l a x i s i n the P r a n d t l - M i c h e l l e theory and i n t r o d u c e d m o d i f i c a t i o n s to account f o r them. In 1910, S . Timoshenko [1] developed the d i f f e r e n t i a l equat ion that i n -c luded the warp ing 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 . In 1929, Wagner [6] determined that t h i n open s e c t i o n s may b u c k l e i n a pure t o r -s i o n a l mode under an a p p l i e d a x i a l l o a d . S i n c e then many other r e s e a r c h e r s have c o n t r i b u t e d to the knowledge of l a t e r a l b u c k l i n g . The u s u a l form t h i s work has taken i s the d i r e c t 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 f o r s imp le cases and the use of n u m e r i c a l methods i n the more c o m p l i c a t e d c a s e s . The drawback i n the above approaches i s the d i f f i c u l t y i n a p p l y i n g them to the g e n e r a l c a s e . As the complex i t y of the load and support 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 the p r e s e n t a t i o n of a method f o r the e l a s t i c ana lyses 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 e l a s t i c l i n e a r p o r t i o n and an approximate n o n - l i n e a r p o r t i o n . The m a t r i x i s developed a Numbers i n square 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 . 2. u s i n g the assumptions tha 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 lane remain s m a l l , the s t r e s s e s remain e l a s t i c , the re 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 loads 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 to b u i l d any case of v a r y i n g s u p p o r t , f i x i t y and load 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 p r o p e r t i e s . 3. CHAPTER I I 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 govern ing d i f f e r e n t i a l equat ions 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 convent ion 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 roduc ing p o s i t i v e y d e f l e c t i o n of the 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 torque T induces p o s i t i v e pure 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 . Z E l y =-w w (a) (b) (c) (d) F i g . 1 SIGN CONVENTIONS 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 on ly necessary to c o n s i d e r the a p p l i e d torque at any p o i n t a l o n g the s e c t i o n . The torque due i t i i t i 2 to shear i n the f l a n g e i s = - Vh = - (h<j> /2) (EIh)= -<j> E l h / 2 . The torque i due to pure t o r s i o n i s C<j> . Therefore the e q u a t i o n f o r torque T a c t i n g at any s e c t i o n i s • 2 I I I C«> - E l h <j> -2 as f i r s t developed by Timoshenko [ 1 ] . E q u a t i o n (1) reduces to I I I 2 2 2 <(> - a $ = - a T where a = _2C C 0) 7 E l h One i n t e g r a t i o n g i v e s I I 2 2 tj) - a <{> = - a_T [x + A] (2) To o b t a i n the e q u a t i o n of the s e c t i o n under the a c t i o n of 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 to g i v e i i 2 m i i Ctj> - E l h cj> = T = - q (3) 2 The e q u a t i o n govern ing pure l a t e r a l d e f l e c t i o n i s the w e l l known mi 2EIY _ = - u> (4) Rather than s o l v e equat ions (2) and (4) f o r any c o n d i t i o n of 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 b e t t e r to s p l i t the beam i n t o a number of segments; each segment h a v i n g constant s e c t i o n p r o p e r t i e s , torques and l a t e r a l s h e a r s . The s o l u t i o n of equat ions (2) and (4) f o r such a segment w i l l be r e l a t i v e l y 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 b u i l t . By u t i l i z i n g s e v e r a l elements to represent a s t r u c t u r e under d i s t r i b u t e d load or v a r y i n g s e c t i o n p r o p e r t i e s , very l i t t l e accuracy w i l l be l o s t . The segment used i s shown i n F i g . 2. 5. 2| * 1^ Ff/e F i g . 2 BEAM SEGMENT In t h i s f i g u r e , 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 are accounted f o r i n d i r e c t l y through the d i f f e r e n c e s i n l a t e r a l shear 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 c o r r e s -ponding f o r c e s a re g i ven 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 the d e f l e c t i o n s 6 l s . . . 6 8 , 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 , . . . f 8 , and k i s an 8*8 m a t r i x j o i n i n g the two. By examining the segment i n F i g . 2, and u s i n g F i g . 1 ( c ) , the f o l l o w i n g r e l a t i o n s are o b t a i n e d . = y - tj> h/2 6 2 = y i + <|> h/2 fi3 = y + <j> h/2 fi4 = y - <f> h/2 • - $h/2 fi6 » = y + cb'h/2 h = y + * h/2 fi8 = y - f h/2 x = 0 f 2 x = L - E I ( y - E I (y f3 = + E i ( y f i t = + E l ( y f 5 = + E I (y f 6 = + E l ( y - E i ( y f ? f f t - * + * i - <r + $ + <r h/2) h/2) 'h/2) - C<f> /h 'h/2) + C4> /h h/2) h/2) 'h/2) + ctf'/h «= - E I (y - <f> h/2) - C<j> /h x = 0 (5) x •=> L I f k 8 x 8 w e r e ob ta ined by a l l o w i n g 6 n = 1 , w i t h a l l o ther 6 ' s equa l to z e r o , the c a l c u l a t i o n s would e n t a i l work ing w i t h combined bending and t o r s i o n . I t i s t h e r e f o r e proposed to s o i v e four b a s i c pure bending cases and four b a s i c pure t o r s i o n a l cases which can oe superposed to g i ve any d e s i r e d d e f l e c t e d shape. The four of these 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 a re g i v e n i n F i g . 3 . > S|=S2FS3=S4=S5=S6= o (a) shear V V 1 8=8 =8 =8 =8 =8 =o 1 2 3 4 7 8 (b) end rotation -> W W W 0 V - V i (c) torsion AY W W W 0 W 1 (d) warping F i g . 3 PRIMARY MODES FOR RIGHT HAND END DEFLECTIONS The shapes i n F i g . 3 ( a ) , (b) a re s o l v e d u s i n g e q . ( 4 ) . S i n c e t h e i r s o l u t i o n g i v e s the w e l l known beam s t i f f n e s s e q u a t i o n s , they w i l l not be gone i n t o i n f u r t h e r d e t a i l . The shapes presented i n F i g . 3 ( c ) , (d) a re t o r s i o n a l shapes and can be s o l v e d u s i n g e q . ( 2 ) . The shape i n F i g . 3(c) r e p r e s e n t s a u n i t t o r s i o n a l r o t a t i o n w i t h a l l o ther a l l o w a b l e d e f l e c t i o n s f i x e d . The end c o n d i t i o n s f o r t h i s case a r e : 7. where x = 0 <j> = 0 i 4- = 0 x = L <j> = 1 • = 0 The s o l u t i o n of eq . (2 ) f o r t h i s case i s <f> = B s i n h ax + D cosh ax + T r , . , o o _o [x + AJ C B = - T D = T (cosh aL - 1) A = - (cosh aL - 1) o o o o o aC T = aC s i n h aL Ca s i n h aL a 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 ob ta ined by a p p l y i n g equal and o p p o s i t e moments to the upper and lower f l a n g e s of one end, and r e s t r a i n i n g a l l o ther a l l o w a b l e d e f l e c t i o n s . The end c o n d i t i o n s f o r t h i s case a r e : • . x = 0 <|> = 0 ' x = L <j> = 0 t i 4> «= 0 ' where •j» h = - 1 2 The s o l u t i o n of eq . (2 ) f o r t h i s case i s : d> = Bi s i n h ax + Di cosh ax + Ti r , ' . , 1 1 _ [x + A i J C Ca T i [ s i n h aL - aL ] A i = s i n h aL - aL ~ [ a ( l - cosh aL) ] a ( l - cosh aL) cosh aL Ti= - 2C [ ] h [2 -2 cosh aL + aL s i n h aL] (7) Us ing the 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 obtained, by u s i n g the shapes i n F i g . 3 as i n d i c a t e d i n F i g . 4. 8. F i g . 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 the o ther columns of the s t i f f n e s s m a t r i x . 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 f o l l o w i n g f u n c t i o n s . 3 S : = (aL) 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 2 • 2 a = 2C/EIh where S j , S 2 , S 3 , Sn and <J> are the same as the s t a b i l i t y f u n c t i o n s g i ven i n Gere and Weaver . [2] Use of these f u n c t i o n s to rep resent the f o r c e s g i ves the complete l i n e a r m a t r i x K , shown i n F i g . 5 . 1 2 2L [1 + S 3] 2 2 2L [1 - S 3] 2 2L [1 + S 3] 3 3L[1 - S 2] 3L[1 + S 2] 6[1 + S x] SYMMETRIC . 4 3L[1 + S 2] 3L[1 - S 2] 6[1 - S x] 6[1 + S x] 5 L 2 [ l + S 4] L 2 [ l - S J 3L[1 - S 2] 3L[1 + S 2] 2 2L [1 + S 3] 6 L 2 [ l - S J 2 L [1 + S j 3L[1 + S 2] 3L[1 - S 2] 2 2L [1-S 3] 2 2L [1 + S 3] 7 3L[-1 + S 2] 3L[-1 - S 2] 6 [ - l - +6[-l + Sx] 3L[-1 + S 2] 3L[-1 - S 2] 6[1 + S x] 8 3L[-1 - S 2] 3L[-1 +'S2] 6 [ - l + ^ 1 6 [ - l - S x] 3L[-1 - S 2] +3L[-1 + S 2] 6[1 - S x] 1 2 3 4 5 6 7 8 F i g . 5 LINEAR STIFFNESS MATRIX K Q •10. Th is m a t r i x rep resents the exact l i n e a r case w i t h two l i m i t a t i o n s : the loads must be a p p l i e d at 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 generated by standard, methods and the r e s u l t s f o r v a r i o u s load cases were compared to 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 . Two 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 c a n t i l e v e r had a l l degrees of freedom f i x e d at one end, 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 at one end, but on ly the f l a n g r o t a t i o n s were f i x e d at the other end. Th is a l lowed p l a c i n g an end torque on the r e s t r a i n e d beam. The r e s u l t s are g i v e n i n F i g . 6. From F i g . 6 i t can be seen tha t the m a t r i x g i v e s the same r e s u l t s as the 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 . Th is i s to be expected s i n c e no a p p r o x i -mat ion to the 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 . In some beams, most of the torque can be c a r r i e d i n pure t o r s i o n . I f the beam i s represented w i t h many s h o r t elements which tend to c a r r y most of the 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 to 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 accuracy to conver t the weak pure t o r s i o n r e s i s t a n c e of the element to the predominant pure t o r s i o n r e s i s t a n c e of the main s t r u c t u r e . In o ther words , i f the re i s i n s u f f i c i e n t accuracy i n the computat ion p rocedure , the f l a n g e shear may overshadow the 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 s t r u c t u r e . In order to i n v e s t i g a t e t h i s problem s e v e r a l s t r u c t u r e s of v a r y i n g l e n g t h were a n a l y z e d . 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 end, 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 at the o ther end. For each of these s t r u c t u r e s a p l o t of torque c a r r i e d by shear over t o t a l torque (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 re g i v e n i n F i g . 7. I =41.6 in 4 E= 3 0 0 0 0 k/in 1 " T J = 1.25 in G = 1 0 0 0 0 k/in^ h = 10 L = 240" @ 3 Segments S t r e n g t h of S t r e n g t h of M a t e r i a l M a t r i x Va lue M a t e r i a l M a t r i x Va lue F lange Moment F lange Moment F lange D e f l e c t i o n s F lange D e f l e c t i o n s X (K ip inches) (K ip inches) ( inches) ( inches) 0 70 .5 70.496 0 . 0 . 80" 22.32 22.501 0 .1270 0 .128 160" 6 .55 6.57 0.3850 0 .3843 240" 0 . 0 . 0 .678 0.6780 L = 240" @ 1 Segment 0 7 0 . 5 70.496 0 . 0 . 240" 0 . 0 . .678 0.6780 Cantilever > Restrained L = 360 Properties as above 10 Segments @ 36" 15 Segments @ 24" S t rength of S t r e n g t h of M a t e r i a l M a t r i x Va lue M a t e r i a l M a t r i x Va lue F lange Moment F lange Moment F lange Moment - F lange Moment X (K ip inches) (K ip inches) (K ip inches) (K ip inches ) 0 6 9 . 8 69.795 6 9 . 8 69.807 36" 4 1 . 5 41.473 -72" 24.15 24.155 24.15 24.161 108" 13.26 13.244 -144" 5.86 5.846 5.36 5.87 180" 0 . 0 . - -F i g . 6 TEST RESULTS FOR K o 12. Vh T 1.0 .9 .8 .7 .6 . 5 .4 . 3 .2 .1 0 > X I =41.6in 4 E = 3 0 0 0 0 k / i n 2 " » T J =1.25 in 4 G =1000 k/ in 2 h= 10 L = I40' L = 3 ' L = 6 ' L = 35 ->x .IL . 2 L .3 L . 4 L . 5 L F i g . 7 EFFECT OF FLANGE WARPING I t can .be 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 c a r r y -i n g t o r s i o n f o r members of t h i s type i s c o n s i d e r a b l e and i n the case of shor t members, the 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 . Th is would i n d i c a t e tha t c a u t i o n shou ld 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 lements . However, a t h i r t y foo t beam of the same type as represented i n F i g . 7 was analyzed 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 s e r i o u s . The l i n e a r m a t r i x developed 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 , shou ld be used i n the 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 . Th is i s i m p o r t a n t , as f l a n g e warping may account f o r a l a r g e p a r t of the t o r s i o n a l s t r e n g t h of a wide f l a n g e s e c t i o n . 13. CHAPTER I I I DEVELOPMENT OF NON-LINEAR DIFFERENTIAL EQUATIONS The team element may be sub jec ted to moments, shears and a x i a l l oads i n the major p r i n c i p a l axes as shown i n F i g . 8 . L > M= M-~ + v x V <-F i g . 8 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 behav iour i s no longer l i n e a r , and a s t r u c t u r e composed of these elements may reach a c o n d i t i o n of i n s t a b i l i t y . 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 the 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 shear cente r of the s e c t i o n c o i n c i d e s w i t h the 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 . 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 ther than the c e n t r o i d are found from the r e l a -t i o n y i= y + <J>n. Due to the presence of P, M and V, the d i f f e r e n t i a l elements are 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 . 9 ( b ) , ( c ) , ( d ) , (e) where ' o = (P _ MQ) _ „ M and T = V (A I ) o ~ ~ A y I w 14. (a) (b) web (c) flange curature = z' =4>' p (e) flange F i g . 9 ELEMENTAL BEAM SECTIONS IN DISPLACED POSITION UNDER THE ACTION OF PRIMARY STRESSES The shear s t r e s s T i s assumed constant over the web and the bending 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 cons ide red as g e n e r a t i n g l a t e r a l p r e s -sures 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 ac t on the element where i i ? i (a - Mn )• 1 ' (a - MQ. ) (y + $> n ) t ( ° I ) y l • fc " ( 0 I ) (8) p = TtcJ. + 2xt £ ' T n — 2T(J) t (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 genera t ing v e r t i c a l p ressures 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 ) . " n n . ° ( ° 2 1 ) * p e ^ ^ By i n t e g r a t i n g these p ressures over t h e i r r e s p e c t i v e areas and d i v i d -i n g by dx , the f o r c e s and torques per u n i t l e n g t h can be o b t a i n e d . They a re g i v e n by + h/2 L a t e r a l fo rce/Length = . (p + p ) dn dx (11) dx J . 0 T -n <= - h/2 + h/2 + w/2 Torque/Length = 1 _ j ( p & + p^) ndn dx + 1_ pdp dx (12) dx dx n = - h/2 p = - w/2 •+ w/2 V e r t i c a l fo rce/Length = JL_ ^ . p ^ dp (13) dx p = - w/2 Now, the 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 eq . (4 ) to g i v e : +h/2 ^ E l y " " - ' | - ( o o - ^ ( y " + L + d r i (14) n = - h / 2 1 c V 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) to g i v e : 2 '" ' 11 + h/2 , , , , , E l h <ji CtJ) ' = ' t - ( a - MnHy + tj> n) n t + 2Xcp tn] dn 2 - ( ° I.) = - h / 2 y + w/2 + w/2 ( ° 2 i ) < f ' P e d P + f ( ° 2 1 ) $ P e d P ( 1 5 ) p = - w/2 y p = - w/2 y 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 r- w / 2 EI z / (a - Mh ) , . I (a + Mh ) , y } o -zj : <J> pedp + } o — ' $ pedp (16) P = - w/2 1 y} P = - w/2 1 ^V By m u l t i p l y i n g o u t , i n t e g r a t i n g , and u s i n g the symmetry p r o p e r t i e s of the s e c t i o n , eqs . ( 1 4 ) , ( 1 5 ) , (16) reduce to the governing d i f f e r e n t i a l e q u a t i o n of the s e c t i o n as f o l l o w s : M M • I I I I • f 2EIy = - Py + M<j> + 2cj>V (17) 2 M M t I I I I I E l h <f> - Cd> = - a I $ +My (18) 2 ° P EI z = 0 (19) y -where I = p o l a r moment of i n e r t i a about c e n t r o i d . P Equat ion (19) i s the e q u a t i o n govern ing the v e r t i c a l d e f l e c t i o n s of the s e c t i o n . I t s t a t e s tha 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 the 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 d i s p l a c e m e n t . I t should be noted though that there w i l l be some e f f e c t on the y , z and <f> d e f l e c t i o n s due to v e r t i c a l d e f l e c t i o n , but i n t h i s d e r i v a t i o n the v e r t i c a l d e f l e c t i o n s are assumed to be s m a l l and t h e i r e f f e c t i s taken as z e r o . The exact s o l u t i o n f o r the d i f f e r e n t i a l equat ions f o r the 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 to o b t a i n . Ins tead an i t e r a t i v e technique w i l l be deve loped. I f the beam i s represented by s e v e r a l e lements , these w i l l be much s h o r t e r than the s t r u c t u r e . Th is means the d e f l e c t i o n s of the element r e l a t i v e 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 than the s t r u c t u r e d e f l e c t i o n s and consequent ly the element w i l l be much s t i f f e r than the 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 than the 17. c r i t i c a l P , M and V f o r the element . Thus the P, M and V i n each element w i l l have on ly 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 ; consequent ly the l i n e a r shape, p r e v i o u s l y o b t a i n e d , w i l l be q u i t ^ c l o s e to the f i n a l d e f l e c t e d shape. By p l a c i n g the l i n e a r d e f l e c t i o n s , which were p r e v i o u s l y o b t a i n e d , i n t o the R . H . S . of eqs . (17 ) 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 which the e f f e c t of M, P and V w i l l be approx imate ly accounted f o r ; s o l v i n g these new equat ions f o r homogeneous boundary c o n d i t i o n s y i e l d s increments i n y and <j>. Th is p rocess can be repeated u s i n g the newly obta ined y and <}> to get a f u r t h e r re f inement on the l i n e a r y and cf>. Th is may be s imply w r i t t e n as 2 E I y n + i = - p y n " + M+n' + 2 V V <20> 2 " " ' ' ' ' " E l h ch - C| , , = - o H + M y (21) — — T n+1 o p n n n+i where n = 0 , 1 , 2 . . . and y^ and <j>Q r e p r e s e n t the l i n e a r d e f l e c t i o n s . S i n c e the boundary c o n d i t i o n s are s a t i s f i e d by the l i n e a r d e f l e c t i o n s y and <j> , the sequence of new s o l u t i o n s y and <h , n = l , 2 . . . a s remarked o Y o n Y n above must s a t i s f y homogeneous boundary c o n d i t i o n s . Upon t e r m i n a t i o n of the i t e r a t i o n p rocedure , the f i n a l r e s u l t s may be ob ta ined by summing the y^ and <j) f u n c t i o n s o b t a i n e d , as 'shown i n eqs . (22) y = y D + y i + y2 • • • y n • = +0 + *1 + * 2 ••• + n ( ? ' 2 ) By u s i n g t h i s t e c h n i q u e , the f i n a l y and <f> obta ined 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 the terms i n the 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 , the use of s e v e r a l elements to represent a s t r u c t u r e reduces the e f f e c t of M, P and V on the element d e f l e c -t i o n s . Indeed, t h i s e f f e c t can 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 e lements ; i n these c i r c u m s t a n c e s , t h e n , i t can be main ta ined tha t one i t e r a t i o n of eqs . (20) and (21) w i l l g i v e s u f f i c i e n t accuracy i n the r e s u l t s . 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 , i t •'o o on ly remains to f i n d the f o r c e s due to y j and qb .^ These f o r c e s w i l l be the n o n - l i n e a r terms of i n t e r e s t , and the m a t r i x obta ined from them w i l l be c a l l e d K j . 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 , due to a p r e v i o u s l y ob ta ined set of y and <j>, b e i n g a p p l i e d to the l i n e a r d i f f e r e n -t i a l e q u a t i o n s . t i i . I I I t shou ld be noted tha t the use of y and y to f i n d shears and moments i m p l i e s t h a t the c o - o r d i n a t e system i n which the f o r c e s on the beam are represented t r a n s l a t e s and r o t a t e s w i t h the member. In o ther words , the f o r c e s a re tangent and p e r p e n d i c u l a r to the f i n a l d e f l e c t e d beam shape. Th is means t h a t the f o r c e s on the 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 c o -o r d i n a t e system. S i n c e the f o r c e s found from the d i f f e r e n t i a l equat ion need i on ly be m o d i f i e d by the c o s i n e of ang les <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 the p r i n c i p a l f o r c e s M, P and V a re a l s o rep resented i n these a x e s , they must a l s o be t ransformed i n t o 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 the s i n e of cf> or y . S i n c e f o r s m a l l d e f l e c t i o n theory s i n e 0 = 8 , the components w i l l be the f o r c e s of i n t e r e s t 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 . The j o i n t f o r c e s must be s u i t a b l y ad jus ted to account f o r the presence of these 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 the m a t r i x due to t h e i r e f f e c t s w i l l be c a l l e d K 2 . S i n c e these f o r c e s a re due on ly to the l i n e a r end d e f l e c t i o n s of the e lement , they are 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 the n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n s . For convenience of r e f e r e n c e , the e f f e c t i v e d i s t r i b u t e d loads w i l l be known as loads of the f i r s t type and the p o i n t loads w i l l be known as loads of the second t ype . 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 „ . to which must be added the l i n e a r m a t r i x K . . 1 l- o A l though the n u m e r i c a l techn ique as d e s c r i b e d s i m p l i f i e s 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 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 order d i f f e r e n t i a l equat ion 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 . I t i s t h e r e f o r e p r o -posed to overcome t h i s work w i t h a f u r t h e r approx imat ion or s i m p l i f i c a t i o n to be d e s c r i b e d i n the next c h a p t e r . 20. CHAPTER IV METHODS OF APPROXIMATIONS. Before p r e s e n t i n g the next approx imat ion 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 prove 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 -mat ion a p p l i e d to a s i m p l e r and more f a m i l i a r problem. In the 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 reatment e n t a i l s d i v i d i n g the beam i n t o s e v e r a l segments by i n t r o -duc ing new j o i n t s a long the member as i n F i g . 1 0 ( a ) . w ± * v v v v v v w ' „ > ' - v „ i/ „ </ - I - ^  m L C 3 UJ ' — A w wL 2 D-12 wL 2 wL< 12 ->< Or „ L wL wL wL wL wL wL I 12 j I 1 <- 5 ^ wL -5<f-F i g . 10 REPLACING DISTRIBUTED LOADS BY FIXED ENp REACTIONS 21. The d i s t r i b u t e d load 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 these are p l a c e d on the beam as j o i n t l o a d s , the moments c a n c e l at the i n t e r i o r j o i n t s and a re present on ly a t the end j o i n t s , as i n F i g . 1 0 ( c ) . These are u s u a l l y i gnored as they have n e g l i g i b l e e f f e c t on the 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 are 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 , the end moments w i l l 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 . However, the end shears are p r o p o r t i o n a l to the l e n g t h of the e lement , whereas the end moments are p r o -p o r t i o n a l to the l e n g t h squared . Th is means tha t as the l e n g t h L goes to z e r o , the end moments decrease f a s t e r than the end s h e a r s . T h e r e f o r e , by d e c r e a s i n g L, which means i n c r e a s i n g the number of e lements , the moments approach z e r o . 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 loads on ad jacent members to approach each o ther i n v a l u e , the end moments not on ly approach z e r o , they approach each o ther but w i t h a s i g n d i f f e r e n c e . A good approx imat ion i s then ob ta ined by u s i n g s e v e r a l e lements , 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 on ly the end s h e a r s . Th is 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 r e a c t i o n s . Th is l e a d s to the next a p p r o x i m a t i o n 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 f o r l a t e r a l t o r s i o n a l b u c k l i n g : Apply the type one loads of the R . H . S . of eqs . (20 ) and (21) to a s imply supported element r a t h e r than a f i x e d e lement . Th is 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 due to loads of type one. The type two loads e x i s t unchanged by t h i s a p p r o x i m a t i o n . 22. CHAPTER V ILLUSTRATION OF METHODS OF APPROXIMATIONS As an i l l u s t r a t i o n of the above method and a p p r o x i m a t i o n , the buck -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 loads a c t i n g on a f i x e d element and then on a s imp ly supported element. Each n o n - l i n e a r 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 p a r t due to type one loads and a second p a r t due to type two l o a d s . The exact m a t r i x i s g i v e n by Gere and Weaver [ 2 ] . For a beam element under constant a x i a l l o a d , the d i f f e r e n t i a l equa-t i o n i s E l y = - P.y ' - (23) f o r the element of F i g . 1 1 ( a ) . 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 ended element f i r s t , u t i l i z i n g the s i g n convent ion of F i g . 1(b) f o r s h e a r , moment and l o a d . Propert ies - E,I A y " = M / E I 4 6 - 6 L.2 • i _ - 6 ^ I2x y ~ 2 + ~~3 x M / E I - 4 L y = — L + 2_ L ^ r r n . 6x x F i g . 1 1 ORDINARY BEAM STIFFNESS DEFLECTIONS, FORCES AND CURVATURES 23. The e q u a t i o n of f o r 61 equals u n i t y i s 1 1 y = - 6 , 12x yo - 7 +•-—3 (24) L L S u b s t i t u t i o n of eq . (24) i n t o the R . H . S . of eq . (23) g ives E I y i "= " P [ - 6 _ , 12x] ( . L L I n t e g r a t i o n g i v e s E l y ' i " = " P [ - 6 x 6x^ ] r 2 + 3 + A ] L L J (26) E l y / = - P [ - 3 x ! + 2x1 + A x + B J ( 2 7 )  [ L L ] » E l y i = - P [ - x _ + 2L. 3 + A x _ + B x + l ( 2 g ) L 2L J E i y i = - p [ - x ! _ + * ! _ + A | ! + B^! + C X + D ] ( 2 9 ) 1 4L 10L J I t i s to be remembered tha t the y^ i n the above equat ion i s the y^ due to loads of type one. For a f i x e d e lement , the end c o n d i t i o n s a r e : @x = 0 y j = 0 @x = L . y1 = 0 y[ = 0 y[ = 0 Us ing these end c o n d i t i o n s and s o l v i n g f o r the cons tants g i v e s A = 6/5L B = - 1/10 C = 0 D = 0 24. E q s . ( 2 6 ) and (27) then g i v e end shears and moments as i i I I @x = 0 M = E l y ! = P/10 @x = L M = EIy1 = - P/10 V = E l y J " = - 6P/5L V = EIy[ " = - 6P/5L 11 For 6 2 = 1 , y d =. - 4 + 6x ( 3 0 ) L L S u b s t i t u t i o n of eqs . (30) i n t o eqs . (23) g i v e s E l y i = - P ( - 4 + 6x ) ( 3 1 ) ( L L 2 ) I n t e g r a t i o n g i v e s I I I 2 E l y j = - P ( - 4x 3x_ ) ( 3 2 ) ( L 2 1 ) ^ i ' = " P ["|-X- + 4+A1x + B 1 ) ) (33) i 3 4 2 • " ' • • ' i ' ^ ^ T 1 ^ * 1 " ^ ) ( 3 5 ) The end c o n d i t i o n s f o r a f i x e d element a re @x = 0 y ! = 0 x = L y x = 0 y[ = 0 y i = 0 S o l v i n g f o r the cons tants y i e l d s A : = 11/10 B : = - 2 L / 1 5 Ci = 0 Dl = 0 S u b s t i t u t i o n of A j , B j , C'1 and Dj into eqs. (32) and (33) y i e l d s the shears and moments of i n t e r e s t . i t t t (?;•; = 0 M = E l y = 2PL/15 @x = L M = Ely = PL/30 V = E l y / ' = - 11 P/10 V = E l y / ' = .- P/10 The ead forces for ,$3 and SL, equal to one can be found using s i m i l a r c a l c u l a t i o n s to the ones above. When the r e s u l t s are placed i n matrix form, they give the por t i o n of the non-linear matrix due to loads of the f i r s t type acting on a fi x e d ended element. The matrix i s given i n F i g . 12. 6P " 5L IIP 10 P 10 P 10 2PL 15 + ^ 30 5L + JL 10 6P 5L + 1 1 P 10 P ~ 10 + ^ 30 2PL 15 F i g . 12 NON-LINEAR BEAM COLUMN MATRIX FOR TYPE 1 LOADS,' FIXED END CONDITIONS For the p o r t i o n of the non-linear matrix due to loads of the second type, the value of the l i n e a r end displacement need be considered. See Fig.13. F i g . 13 NON-LINEAR BEAM COLUMN TERMS FOR TYPE 2 LOADS 26. For 6^ = 1 , as i n F i g . 1 3 ( a ) , the re are no components of P a c t i n g i n the shear d i r e c t i o n , so there are no c o n t r i b u t i o n s i n the 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 n F i g . 1 3 ( b ) , the load P which the j o i n t must p rov ide has a component i n the 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 at the L .H .S of the s t r u c t u r e . Th is v a l u e must be entered i n the shear f o r c e p o s i t i o n of the m a t r i x f o r 6 2 = 1 . The R.H.S has no component. By t r e a t i n g the o ther d e f l e c t i o n s the same way, the 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 . 14. 0 +p 0 0 0 0 0. 0 0 0 0 - p 0 0 0 0 F i g . 14 NON-LINEAR BEAM COLUMN MATRIX FOR TYPE 2 LOADS When the m a t r i c e s presented i n F i g s . 12 and 14 are added, the r e s u l t i s the complete second order m a t r i x as shown i n F i g . 15. 6P 5L . P 10 + f P 10 P 10 2PL 15 + •£ + & 5L 6P 5L P 10 2PL 15 F i g . 15 COMPLETE NON-LINEAR BEAM COLUMN MATRIX, FIXED END CONDITIONS 27. In summary, t h i s matrix wat- found by using the l i n e a r deflected shape to generate a load to use i n eq.'23). This load was applied to an element f i x e d at the ends. I t i s of i n t e r e s t to note at t h i s point the r e l a t i o n between the exact matrix f o r buckling, containing sine and cosine functions, and the approx-imate matrix i n F i g . 16. I f the se r i e s expansions for the sine and cosine are substituted into the exact matrix, the second term of t h i s expansion gives the approximate matrix derived above. Equation (23) w i l l now be applied to a pin ended element. F i g . 12 s t i l l represents the l i n e a r deflected shapes and the type one loads. To solve f o r the forces, i t i s only necessary to integrate eq.(23) t i twice since the known end conditions of y = 0 at the ends w i l l solve the two constants of 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 give the end shears. However, because of the simple type one loads f o r these cases, s t a t i c s can be used to determine the end shears, as i n F i g . 16. 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 , the end forces f o r 63 and 64 equal to one can be obtained. Combining the end forces into matrix form gives the portion of the non-linear matrix due to loads of the f i r s t type acting on a pinned element. See F i g . 17. p L - p ., I L 0 0 0 0 0 0 P L + P 0 0 0 0 F i g . 17 NON-LINEAR BEAM COLUMN MATRIX FOR TYPE 1 LOADS, PINNED END CONDITIONS The p o r t i o n of the matrix due to loads of the second type remains un-changed. Therefore, the complete matrix i s found by adding the matrix of Fig.17 and F i g . 14. This y i e l d s a r e l a t i v e l y simple matrix, as shown i n F i g . 18. p 0 P 0 + -L L 0 0 0 0 P 0 P 0 + T L L 0 0 0 0 F i g . 18 COMPLETE NON-LINEAR BEAM COLUMN MATRIX, PINNED END CONDITIONS. I t i s of i n t e r e s t to note that t h i s i s the non-linear matrix for a pin-ended strut under a x i a l load. 29. The e f f e c t of the approx imat ions 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 inned column, a f i x e d column w i t h a p i n at the center and c. f i x e d column were ana lyzed u s i n g the m a t r i c e s presented i n F i g . 18 and F i g . . 5 . Each column was ana lyzed u s i n g a v a r y i n g number of e lements . The per cent er.rors i n the 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 the number of elements used f o r each column, as shown i n F i g . 19. I t shou ld be noted tha t the f o r c e s due to loads of the second type i n the m a t r i x are of no importance i n members w i t h cont inuous d e f l e c t i o n s . Th is a r i s e s from the f a c t tha t the a d j o i n i n g ends of elements i n a cont inuous s t r u c t u r e have the same d e f l e c t i o n s but the end f o r c e s are 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 of end f o r c e s to each j o i n t from each member then c a n c e l each other out . 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 , c o n t i n u i t y of s l o p e no longer e x i s t s and the end f o r c e components of ad jacent elements may not be s e l f c a n c e l l i n g . Th is i m p l i e s tha t the type one loads a re a l l that i s necessary to a n a l y z e cont inuous s t r u c t u r e s , but tha t they w i l l f a i l to ana lyze s t r u c t u r e s w i t h i n t e r i o r p i n s . 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 i n t e r i o r p inned 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 the e f f e c t s of loads of the second type bear t h i s ou t . In c o n c l u s i o n t h e n , i t i s proposed t o . c a l c u l a t e y 1 on the b a s i s of y 1 = y 1 = 0 at each end and j u s t use the c a l c u l a t e d end shears i n the m a t r i x i r a t h e r than u s i n g = 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 shears and moments i n the m a t r i x . The type two te rms , which are added to the above r e s u l t s , are the same f o r e i t h e r set of assumed end c o n -d i t i o n s . Accuracy P l o t s of f i x - f i x Matrices p i n - pin Fig. 19 PLOT OF % ERROR VS. NO. OF ELEMENTS FOR BEAM — COLUMN MATRICES FOR 3 COLUMN TYPES 31. CHAPTER VI • DEVELOPMENT OF LATERAL STABILITY MATRIX i A p p l i c a t i o n of Approximations Only one i t e r a t i o n w i l l be used to develop the matrix. The l i n e a r p o r t i o n of the matrix, corresponding to the d e f l e c t i o n s y^ and <j>o, has already been obtained. The non-linear terms of the matrix due to y and <J> w i l l now be J l 1 found by so l v i n g the d i f f e r e n t i a l equation for y^ and ^ using boundary condi-tions o f y 1 = y 1 =<f'1=<f>1 = 0 at each end. Shears and torques are then found from y^ and <f> . I t i s deduced from the previous section that i t i s not neces-i i sary to use y^ = y^ = 0 and = <J>j = 0 at each end as boundary conditions. Once again, the type two load terms are independent of whichever boundary con-d i t i o n s are used. However, the use of s t a t i c s to determine the end shears on a pinned element was discarded as the type one loads were more complex than those associated with the beam column. Instead, the governing d i f f e r e n t i a l equations, a f t e r 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 i o n s i n the R.H.S were integrated twice and then solved for the end conditions. Equations (20) and (21) become for n = 0, ti i t i t I I 2EIy 1 = - Py Q + M<f>o + 2V$Q (36) 2 "" I I i i i t Elh 4>, - C6, = - o I <j> + My (37) — 2 — 1 1 o p o Jo Integrating twice, eqs.(36) and (37) become I I I I 2EIy 1 = -'Py + ffWQ dxdx + 2V/<j>odx + Ax + B (38) 2 i i I I •Elh A - = - o I (j) + //My dxdx + Dx + E - (39) — - — y l 1 o p o J o 32. i t t i I n t e g r a t i n g fM§ dxdx by p a r t s , and remembering M = V and V = 0 , g i v e s i t JM<$> dxdx = JMd) dx - ft, Vdx (40) o o o I n t e g r a t i o n of /Mcfi^dx by p a r t s g i ves T /Mcf> dx = Mtt - V/(J> dx (41) o o o v J S u b s t i t u t i o n of (41) i n t o (40) g i v e s i i J7:McJ> dxdx = Mcf> - 2VJ"<j> dx (42) 0 o o A s i m i l a r i n t e g r a t i o n g i v e s i t //My Q dxdx = My Q - 2V/y Q dx (43) Now, s u b s t i t u t i o n of eqs . (43) and (42) i n t o eqs . (38) and (39) g i v e s 1 i i 2 E I y i = - P y Q + McJ>o + Ax + B (44) 2 i i E l h tt, - Ccf), = - a I <J> + My - 2V/y dx + Ex + D (45) — — 1 1 o p o o o I I I I The end c o n d i t i o n s of <f>j = cj)^  = y i = y i = u w e r e then a p p l i e d to equat ions (44) and (45) . These equat ions lend themselves to t h i s approach , as the L .H .S of both equat ions become zero at x = 0 , L , and the R.H.S of the equat ions c o n t a i n s on ly 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 at x = 0 , L, the i n t e g r a t i o n of y^ , and four unknown c o n s t a n t s . These cons tants A , B , C , D are e a s i l y found and the n o n - l i n e a r m a t r i x f o r c e s due to loads of type one are then obta ined by a p p l i c a t i o n of the d i f f e r e n t i a l equat ions (46) . I I I i i Shear = E I y x = - P y Q + M<J> + Vcj> + A (46) i 2 I I I i t Torque = C A , - E l h cb, = + a I <f> - M y + V y - E T l T i o p T o ' o 1 o 33. i i C a l c u l a t i o n of Non-linear S t i f f n e s s Matrix for Type One Loads The terms of the non-linear matrix w i l l be c a l c u l a t e d f i r s t f o r 67 ii for which y^ i s given by one h a l f of F i g . 3(a) or, y = 3 6x 0 ~z - — . L - L Integrating eq.(47) three times gives (47) 3 k fy dx = x_ - x 2 3 2LZ 4 L D (48) A constant should be added to eq.(48), but i t i s taken to be com-bined with D i n eq.(45). From F i g . 4, the values of the l i n e a r d e f l e c t i o n s at the ends of the element for 6 7 = 1 are: x = 0 y = 0 = 0 = 0 = 0 x = L 1/2 1/h 0 0 (49) The end conditions for s o l u t i o n of eqs. (44) and (45) are 11 11 y i = y i =+!=<!>! = 0 @x = O , L (50) Using eqs.(50), (49) with the d i f f e r e n t i a l equation (44) gives x = 0 2EI[0] = - P[0] + M[0] + A[0] + B x = L 2EI[0] = - P[l/2] + M[l/h] + A[L] + B (51) (52) Solving eqs. (51) and (52) for A and B gives 34. Using eqs.(49) and (50) with the d i f f e r e n t i a l equation (45) gives x = 0 Elh [0] - C[0] = - a I [0] + M[0] - 2V[0] + E[0] + D —z— o p (53) x = L Elh [0] - C[0] = - a I [1/h] + M[l/2] - 2V[L/4] + Ei L + D (54) o p From eqs.(53) and ( 5 4 ) , D = 0 VL E = c l (M + ^ ~ ) . „ o p - o 2 + V hL 2L 2 Substituting the values of A, B, D, E back into eqs.(46) and using the r e l a -tions i n (49) gives SHEAR @x = 0 Q = - P[0] + M[0] + V[0] + A = I (P - (M + ^ ) ) L ( 2 - f i - j - 2 - ) @x = L Q = - P[0] + M[0] + V[l/h] + A VL V + 1 (P - (M + -f^ )) h 1 ( 2 — T ) (55) TORQUE @x = 0 T = a I [0] - M[0] + V[0[ - E o p hL 2L (?x = L T = o I [0] - M[0] + V [ l / 2 ] - E o p hL 2L By using s u i t a b l e signs, and the r e l a t i o n Q = T/h, the column of the s t i f f n e s s matrix f or 67 = 1 was b u i l t from r e l a t i o n s (55) and i s shown i n F i g . 20. 35. jjr . VL VL 1 [P - o 2~] + 1 [V^ - M o + T + V ] 2L [2 h ] h [ hL 2L 2 ] M J L _ M . VL _1 [P. - o 2 ] - _1 [ o p - o 2 + y_ J 2L [2 h ] h [ hL 2L 2 ] 1 t l + I QL 2 [h L (2 vr _ L VL , VL M + — , , r , a I M + — , o 2 ) ] - ] . [ + o p - o 2 ] h ) ]- h [ hL 2L ] JL [V + 1 (P 2 [h L (2 VHjD ] + 1 f+ g o I p h ) ] h [ hL M + ^ o 2 ] 2L ] F i g . 20 TYPE 1 TERMS FOR NON-LINEAR MATRIX FOR 6 7 = 1 3 6 . 11 For §6 ='1) y i s given by one h a l f of F i g . 3(b) or y = - 1 + 3x L L 2 Integrating three times gives 3 . h fy = - x + x , From F i g . 4, the values of the l i n e a r d e f l e c t i o n at the ends of the element for 6g = 1 are: x = 0 y = 0 x = L y = o o •'o t . i y = 0 y = + 1/2 ^o J o 4>O = 0 - * D = 0 (57) <j> = 0 4> = + 1/h o o The end conditions for s o l u t i o n of the d i f f e r e n t i a l equation (44) and (45) are 11 I I s t i l l y i = y i (58) Using eqs.(57) and (58) with the d i f f e r e n t i a l equation (44) gives x = 0 2EI[0] = - P[0] + M[0] + A[0] + Bj (59) x = L 2EI[0] = - PfO] + M[0] + A : L + Bl (60) From eqs.(59) and (60), Bj and Aj are Aj = 0 Bj = 0 Using eqs.(57) and (58) with the d i f f e r e n t i a l equation (45) gives x = 0 EIh 2[0] - C[0] = - a I [0] + M[0] - 2V[0] + E,[0] + D, (61) i o n i * 2 [24] 2 ° P [ L 2 ] x = L Elh [0] - C[0] = - a I [0] + M[0] - 2V ~ + E, L + D, (62) .*"2— ° P 124 J l l From eqs.(61) and (62) T>1 = 0 E = - VL/12 37, Su b s t i t u t i o n of the values of A ^ . B j , C j ^ D j b a c k into eqs. (46) and using r e l a t i o n s (57) gives SHEAR x = 0 Q = - P[0] + M[0] + V[0] = 0 x = L Q = - P[l/2] + (M + ~ ) (1/h) + V[0] o z - - £ > < M o + f > (63) TORQUE x = 0 T = + a Q I p [ 0 ] - M[0] + V[0] - E1 = + VL 12 x = L T = + a l [1/h] + (M + ~ )[1/2] + V[0] - E, o p o 2 1 = + o I - (M + ? ) + VL o P o 2 — h x z By using the r e l a t i o n s i n (63) with s u i t a b l e signs, and the r e l a t i o n Q = T/h, the column of the matrix due to type one loads for 65 = 1 was b u i l t , as shown i n F i g . 21. S i m i l a r l y , the other s t i f f n e s s d e f l e c t i o n s were treated. The com-ple t e non-linear p o r t i o n of the matrix due to loads of type one i s and can be written as K, = Pk. + Mk, + Vk. , as i n F i g . 22. The portion of the 1 lp lm l v ^ matrix due to loads of type two must s t i l l be ca 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 - 1 t- M P + o + VL 2 ] 1 r o I . M [- o p + o + VL 2 - VL] 2 [ 2 h ] h [ h 2 12] 8 - 1 [- M P + o + VL 2 ] + 1 r a I M L- o p + o + VL 2 - VL] 2 [ 2 h ] h [ h 2 12] F i g . 21 TYPE 1 TERMS FOR NON-LINEAR MATRIX FOR 6 6 = 1 Pk . = P Pi Mk = M(f ) 1 2 3 4 5 6 7 . 8 1 2 3 4 5 6 7 8 X 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - a + b - a - b -(a+b)/L (-a+b)/L 0 0 (a+b)/L (a-b)/L - a - b - a - b (-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 a - b a + b -(a+b)/L (-a+b)/L 0 0 (a-b)/L (a+b)/L a + b a - b (-a+b)/L (-a-b)/L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +L +1 0 0 0 -1 0 -L 0 0 -1 0 0 0 +1. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -L +1 0 0 0 0 +1 +L 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 -5 ri -6/L -1 -1 0 +6/L +5 -1 +6/L 0 +1 +1 -6/L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 -6/1 +1 -5 0 +6/L +1 +1 +6/L 0 +5 -1 -6/L 0 a = 1/4 b = I /Ah P F i g . 22 NON-LINEAR MATRIX FOR TYPE 1 LOADS 40. i i i C a l c u l a t i o n of Non-linear Matrix f o r Type Two Loads The e f f e c t of the type two loads can best be calculated 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 into 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, the component l i n e a r d e f l e c t i o n s are given i n F i g . 23. (a) Ay F i g . 23 COMPONENT DEFLECTIONS FOR 6 ? = 1 By applying M, P and V and taking t h e i r components about the R.H.S, of the sec t i o n as shown i n F i g . 24, gives F i g . 24 FORCE COMPONENTS DUE TO END DEFLECTIONS FOR 6 = 1 41. In F i g . 24(a), there are i;o components acting i n any of the allowable j o i n t d e f l e c t i o n s . Therefore: Shear = 0 Moment = 0 Torque = 0 For F i g . 24(b), the components acting on the cross section which must be supplied by the j o i n t are Shear = + V/h Moment = - (M + )/h o 2 Torque = 0 The L.H.S. of the deflected shapre for 6 7 = 1 has no components, as a l l the d e f l e c t i o n s are zero. The column i n the matrix f o r loads of the second type f o r <J7 = 1 i s given i n F i g . 25. ° For <$6 = 1, the component l i n e a r d e f l e c t i o n s are given i n F i g . 26. (a) - (b) F i g . 26 COMPONENT DEFLECTIONS FOR 6 6 = 1 Applying M, P and V, and taking t h e i r components along the allowed d e f l e c t i o n s gives: ( i ) For the L.H.S. - no components, no forces. ( i i ) For the R.H.S. - see F i g . 27. 42. 5" 2h O 2 2h + £_ 2h 2h F i g . 25 TYPE 2 TERMS FOR NON-LINEAR MATRIX FOR S 7 = 1 43. F i g . 27 FORCE COMPONENTS DUE TO END DEFLECTIONS FOR 6 6 = 1 For F i g . 27(a) components acting along the allowed d e f l e c t i o n s are M/2, P/2. In s t i f f n e s s matrix sign convention, Shear = - P/2 Moment = 0 Torque = (M + VL/2)/2 For F i g . 27(b) the slope changes continuously from top to bottom. Therefore i n t e g r a t i o n along the cross section must be used. Consider the e f f e c t of a at z on an element of area tdz, where o = + P - M z = a - M z T r o r A I I z z A force =aztdz slope @ z i n x d i r e c t i o n i s z/h Therefore, the component of force at z i s oztdz h 44. Integrating with respect to z to get the shear and torque gives + h/2 Shear . (a zt z = - h/2 ( 2 M z -t) . r v cz I h { z ) , M M + VL/2 + r = o Torque + h/2 2  1 z = - h/2 ( M z t) , r ( dz I h " z ) a I o z Because the flanges s u f f e r the same angular displacements as the sect i o n as a whole, the end slopes of the flanges must be taken into account. See F i g . 28. h 1 = e F i g . 28 VERTICAL. FLANGE DEFLECTIONS AS FUNCTIONS OF <j> The slope of the 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 flange i s t £<j> = £/h at point £. The d i f f e r e n t i a l force at point £ i s A force = oedE, The v e r t i c a l shear component i s + w/2 (a e£ M h£e) r ") d? w/2 ( 21 = 0 z ) 45. The torque i s + w/2 2 2 3 + w/2 . (a e£ - M h£ e) o I M £ e ;^ r^  21 d ? - - ~ + i^~ - - w/2 ( 2 > Z - w/2 However, the bottom flange has the same configuration with a moment stress on i t of opposite sign. Therefore the net torque c o n t r i b u t i o n of the flanges under moment stress i s zero. However, the a x i a l , l o a d c o n t r i b u t i o n i s the same for the bottom flange as f o r the top. Therefore, the t o t a l c o n t r i b u t i o n of the shape i n F i g . 27(b) i s given 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 Af t e r combining these forces, the column i n the matrix f o r 65 = 1 due to type two loads i s given i n F i g . 29. The remaining s i x columns can be found from s i m i l a r c a l c u l a t i o n s . This matrix, c a l l e d K 2, can be written as K 9 = Pk + Mk + Vk ^ pz m2 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 given i n F i g . 31 and i s obtained by adding the matrices and K 2. i v Numerical Examples The matrix i n F i g . 31 was used to c a l c u l a t e the c r i t i c a l loads of several structures, and the r e s u l t s were compared to the t h e o r e t i c a l s o l u t i o n s . A determinant p l o t method of s o l u t i o n was used. That i s , the deter-minant of the structure matrix f o r increasing values of M, P and V were 46. P 0 1 M + — • M . + — J? - o p + o 2 + o 2 4 , 2 2h 2h p + a i M + f M + — i _ + o P + o 2 - o 2 4 , 2 2h 2h F i g . 29 TYPE 2 TERMS IN NON-LINEAR MATRIX FOR 6 6 = 1 Pk = P ' P2 Mk •« M C{7T) ni2 o hL V k v 2 " V<T2h> 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 a + b a - b 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 7 0 0 0 0 - a + b - a - b 0 0 8 0 0 0 0 - a - b - a + b 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 4 + L 0 0 0 0 '. 0 0 0 5 0 0 0 0 0 0 - L/2 + L/2 6 0 0 0 0 0 0 - L/2 + L/2 7 0 0 0 0 0 + L 0 0 8 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 - 6/L + 6/L 0 0 0 0 4 - 6 0 - 6/L .  + 6/L 0 0 0 0 5 0 0 0 0 0 0 - 3 + 3 6 0 0 0 0 0 0 - 3 + 3 7 0 0 0 0 0 + 6 + 6/L - 6/L 8 0 0 0 0 - 6 0 + 6/L - 6/L a =.1/4 b = I /Ah P Fi g . 30 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 0 + 1 0 0 0 - 1 0 0 - 3 + 3 0 0 0 0 0 0 - 3 + 3 0 0 0 0 + 1 + 1 - 6/L 0 - 1 - 1 0 + 6/L - 1 - 1 0 + 6/L + 1 + 1 - 6/L 0 0 0 0 0 0 0 - 3 + 3 0 0 0 0 0 0 - 3 + 3 - 1 - 1 0 - 6/L + 1 + 1 + 6/L 0 + 1 + 1 + 6/L o - 1 - 1 0 - 6/L 2 a = 1/4 b = I. /Ah F i g . 31 THE COMPLETE NON-LINEAR MATRIX P FOR LOADS OF TYPE 1 AND 2 49. calculated and p l o t t e d . The value of M, P and V at which the determinant equalled zero was taken as the c r i t i c a l load. This technique was preferred to an eigenvalue approach since i t avoids the problem of converging to a r e a l eigenvalue which a r i s e s when an eigenvalue technique i s applied.to systems containing unsymmetric matrices. A table of the r e s u l t s i s given i n Figs.32 and 32(a). A p l o t of the number of elements used against the accuracy obtained for four cases i s given i n F i g . 33. The e f f e c t of applying v e r t i c a l loads at points other than the cen-"troid can be accounted for by modifying the s t i f f n e s s matrix of the structure. 50. 10 ELEMENTS USED IN EACH STRUCTURE -° pinned fixed -> PURE EULER BUCKLING I=9.7in4 L=20' E = 30,000k/in 2 STRUCTURE ERROR 1.2% P * P -o pinned fixed PURE TORSIONAL BUCKLING STRUCTURE ERROR -o < 2 % < 3.5% Ip^OOOin 4 l = 5 0 0 i r 4 J=5 in 4 E = 30,000 k/in 2 L = 20* G = 11,500 k/in A = 10 i n 2 h = 9.5 in Fig. 32 TABLE OF RESULTS FOR TEST STRUCTURES 51. 10 ELEMENTS USED IN EACH STRUCTURE <; 1 pinned ) £ fixed LATERAL TORSIONAL BUCKLING STRUCTURE M ERROR |M .5% i w i i i i i i i i r~r i i i i i i i i i w r w 1 % 2 . 5 % 4 % 1=21 i n 4 Ip = l 5 l i n 4 A = 7. l25in 2 J = . l 4 8 i n 4 E = 30,OOOk/in 2 G= II , 5 0 0 k / i n 2 Fig. 3 2 a TABLE OF RESULTS FOR TEST S T R U C T U R E S 5 2 . Fig. 33 PLOT OF ACCURACY VS. NUMBER OF E L E M E N T S USED CHAPTER VII CONCLUSIONS An 8 x 8 matrix for the exact l i n e a r treatment of doubly symmetric •wide flange beams under t o r s i o n and l a t e r a l displacement was developed. This matrix allows each flange at eit h e r end to assume tra n s l a t i o n s and rotations independent to the other flange. An approximate matrix accounting for the e f f e c t of p r i n c i p a l plane forces on the l a t e r a l d e f l e c t i o n s was developed. When added to the l i n e a r matrix, i t makes poss i b l e the determination of the l a t e r a l - t o r s i o n a l buckling loads of wide flanged beams. The non-linear matrix was based on small p r i n c i p a l plane d e f l e c t i o n s and no d i s t o r t i o n of the cross-section. Applied external loads must maintain 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 obtain the non-linear s t i f f n e s s matrix, d i f f e r e n t i a l equations were developed by considering a displaced element under the act i o n of the p r i n c i p a l forces. These were then solved to f i n d the end forces which were entered i n the matrix. To ease the s o l u t i o n of the equation, a numerical technique was developed. 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 deflected shape into the R.H.S. of the d i f f e r e n t i a l equation to produce a known load, and then so l v i n g the L.H.S. for the new y^  and <J> ^ . The new de f l e c t i o n s were y^ and ^ were then placed into the R.H.S. and the process repeated. The end conditions used were to be those of a fi x e d element. E f f e c t i v e end point loads acting due to the i n i t i a l l i n e a r end d e f l e c t i o n s gave a second load set. Two approximations were then u t i l i z e d to further s i m p l i f y the solu-t i o n of the equations. The f i r s t e n t a i l e d using only one cycle of the i t e r a -54. t i o n scheme. The second was to apply the e f f e c t i v e l a t e r a l loads to an element with no end moment r e s t r a i n t . Tests of the element against known solutions i n d i c a t e that good accuracies can be obtained, but depend on the number of elements used i i - the a n a l y s i s . Acceptable accuracies were obtained using ten elements, the larges t error encountered being 5%. The advantages of t h i s matrix are several. Cases i n v o l v i n g general load, support and end conditions of sections with varying s e c t i o n properties may now be solved by simply breaking the structure up into several elements and applying the presented matrix. LIST OF REFERENCES Timoshenko, S.P., "Strength of M a t e r i a l s , Part I I " D. Van Nostrand Company, Inc., New York, N.Y. pp. 255-265, 1956 Gere, J.M., and Weaver, W., "Analysis of Framed Structures" D. Van Nostrand Company, Inc., New York, N.Y. pp. 430, 1965 Timoshenko, S.P., and Gere, J.M. , "Theory of E l a s t i c S t a b i l i t y " McGraw-Hill, Inc., New York, N.Y. pp. 251-270, 1961 B l e i c h , F., "Buckling Strength of Metal Structures" McGraw-Hill, Inc., New York, N.Y. pp. 149-160, 1952 Timoshenko, S.P., "History of Strength of M a t e r i a l " McGraw-Hill, Inc., New York, N.Y. pp. 393, 1953 Goodier, J.N., "Some Observations on E l a s t i c S t a b i l i t y " Proceedings of the F i r s t National Congress of Applied Mechanics 

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