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 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 n = n*"^ 1 term i n s e r i e s s o l u t i o n of /2) (EIh)= - E l h / 2 . The torque i due to pure t o r s i o n i s C . 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 -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 + h/2 fi4 = y - 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 - /h 'h/2) + C4> /h h/2) h/2) 'h/2) + ctf'/h «= - E I (y - h/2) - C /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 = 0 i 4- = 0 x = L = 1 • = 0 The s o l u t i o n of eq . (2 ) f o r t h i s case i s = 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 = 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 ' - 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 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 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 + 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 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 + 2cj>V (17) 2 M M t I I I I I E l h - 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 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 . 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 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 , the sequence of new s o l u t i o n s y and 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 , 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 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/ „ < Or „ L wL wL wL wL wL wL I 12 j I 1 <- 5 ^ wL -5o, has already been obtained. The non-linear terms of the matrix due to y and 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 =1 = 0 at each end. Shears and torques are then found from y^ and . 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 = 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 + Mo + 2V$Q (36) 2 "" I I i i i t Elh 4>, - C6, = - o I + My (37) — 2 — 1 1 o p o Jo Integrating twice, eqs.(36) and (37) become I I I I 2EIy 1 = -'Py + ffWQ dxdx + 2V/odx + Ax + B (38) 2 i i I I •Elh A - = - o I (j) + //My dxdx + Dx + E - (39) — - — y l 1 o p o J o 32. i t t i I n t e g r a t i n g fM§ dxdx by p a r t s , and remembering M = V and V = 0 , g i v e s i t JM<$> 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" 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 + 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 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 + Vcj> + A (46) i 2 I I I i t Torque = C A , - E l h cb, = + a I - 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) = 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 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 £ = £/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 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 ^ . 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