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Stiffness matrix for twist bend buckling of narrow rectangular sections Zavitz, Bryant Allan 1968

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A STIFFNESS MATRIX FOR TWIST BEND BUCKLING OF NARROW RECTANGULAR SECTIONS BY BRYANT A. ZAVITZ B.Sc. (CIVIL ENG.) The U n i v e r s i t y of A l b e r t a , 1962 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 standard The U n i v e r s i t y of B r i t i s h Columbia May 1968 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and Study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by hits representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of Brit ish Columbia Vancouver 8 , Canada Date 3 /?<T<3 i ABSTRACT The s t i f f n e s s p r o p e r t i e s of a short narrow r e c t a n g u l a r beam as modi-f i e d by a primary bending moment and shear s t r e s s d i s t r i b u t i o n i n the major plane are presented. The beam i s a segment taken from a longer member the " s t r u c t u r e . " A d i s t r i b u t i o n of bending s t r e s s i s assumed over the beam segment length and i t s e f f e c t on the s t i f f n e s s p r o p e r t i e s i n l a t e r a l bending and t o r s i o n obtained. The s t i f f n e s s m a t r i x i s used to o b t a i n the c r i t i c a l v alue of load f o r a number of w e l l known examples of narrow r e c t a n g u l a r beams and the r e s u l t s are shown to be i n good agreement. The r e s u l t s of an energy s o l u t i o n , which produces a symmetrical m a t r i x , are presented. Comparison w i t h c l a s s i c a l examples shows accurate r e s u l t s w i t h the added b e n e f i t that the symmetrical m a t r i x lends i t s e l f much more r e a d i l y to more complicated problems. i i i TABLE OF CONTENTS Page Ab s t r a c t i Acknowledgements i i Table of Contents i i i Not a t i o n v CHAPTER I INTRODUCTION 1 CHAPTER I I BASIC EQUATIONS 3 2.1 General Remarks 2.2 D e r i v a t i o n CHAPTER I I I METHOD OF SOLUTION 7 3.1 General Remarks 3.2 D e r i v a t i o n of S t i f f n e s s M a t r i x 3.2.1 Column 1 3.2.2 Column 2 3.2.3 Column 3 CHAPTER IV ALTERNATE METHOD OF SOLUTION 17 4.1 General Remarks 4.2 D e r i v a t i o n of S t i f f n e s s M a t r i x 4.2.1 Column 4 4.2.2 Column 5 4.2.3 Column 6 4.3 Co n s i d e r a t i o n of E c c e n t r i c Load 4.4 Co n s i d e r a t i o n of Restrained Centre of Ro t a t i o n and A x i a l Loads TABLE OF CONTENTS Cont'd. CHAPTER V CHAPTER VI CHAPTER V I I DERIVATION USING THE ENERGY METHOD 5.1 General Remarks 5.2 B r i e f D e s c r i p t i o n of the Method APPLICATION TO STABILITY PROBLEM 6.1 General Remarks 6.2 Numerical Examples CONCLUSIONS Bi b l i o g r a p h y Appendix A . l D e s c r i p t i o n of Computer Program A.2 F o r t r a n IV L i s t i n g V NOTATION E = modulus of e l a s t i c i t y G - shear modulus of e l a s t i c i t y I = moment of i n e r t i a J = t o r s i o n constant L = length of " s t r u c t u r e " I = length of beam segment a^byO = c e n t r o d i a l , major and minor a x i s of beam element XsljiZ ~ f i x e d co-ordinate system e = e c c e n t r i c i t y of load a p p l i e d to " s t r u c t u r e " g = e c c e n t r i c i t y of r e s t r a i n t a p p l i e d to " s t r u c t u r e " 3 = t o r s i o n a l d e f l e c t i o n i n z d i r e c t i o n P = l a t e r a l d e f l e c t i o n i n x d i r e c t i o n Mx = moment about p o s i t i v e x a x i s My = moment about p o s i t i v e y a x i s M = moment about p o s i t i v e z a x i s F = f o r c e along p o s i t i v e x a x i s M = i n t e r n a l moment about a x i s b x M = i n t e r n a l moment about a x i s o T = i n t e r n a l moment about a x i s a V = i n t e r n a l shear along c Q = i n t e r n a l shear along b M = average value of M i n segment P = v e r t i c a l load on s t r u c t u r e AL, = f i x e d end moments F v i NOTATION Cont'd. H = a x i a l load i n member 6 = d e f l e c t i o n v e c t o r f o r beam segment f = f o r c e v e c t o r f o r beam segment 6 = d e f l e c t i o n v e c t o r f o r r e s t r a i n e d beam segment f = f o r c e v e c t o r f o r r e s t r a i n e d beam segment k = r e s t r a i n e d 6 x 6 s t i f f n e s s matrix k = r e s t r a i n e d 4 x 4 s t i f f n e s s m a t r i x KjKl = 6 x 6 s t i f f n e s s matrices f o r beam segment J p T = tra n s f o r m a t i o n matrices o = f i b r e s t r e s s at c r i t i c a l load Y = c r i t i c a l load c o e f f i c i e n t U = s t r a i n energy q - l a t e r a l load <|> = s t a b i l i t y f u n c t i o n c o - e f f i c i e n t X = s t a b i l i t y f u n c t i o n c o - e f f i c i e n t i i ACKNOWLEDGEMENTS The author wishes to express h i s g r a t i t u d e to Dr. R. F. Hooley f o r h i s i n v a l u a b l e guidance during the research and i n the p r e p a r a t i o n of t h i s t h e s i s . A l s o , the author i s g r a t e f u l 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 support i n the form of an a s s i s t a n t s h i p . May 1968. Vancouver, B r i t i s h Columbia A STIFFNESS MATRIX FOR TWIST BEND BUCKLING OF NARROW RECTANGULAR SECTIONS CHAPTER I INTRODUCTION The e f f e c t of primary bending on the s t a b i l i t y of narrow re c t a n g u l a r beams has been e x t e n s i v e l y i n v e s t i g a t e d by, Timoshenko and Gere ( 1 ) , Goodier ( 2 ) , B l e i c h ( 3 ) , Vlasov (4) and others. I n Hartmann (5) and B e l l (6) methods of determining c r i t i c a l loads f o r plane s t r u c t u r e s are presented. A l a r g e amount of work has a l s o been done on the e f f e c t of a x i a l load on the s t a b i l i t y of members and s t r u c t u r e s , i n Gere and Weaver (7) and i n McMinn (8) methods of modifying member s t i f f n e s s matrices to show the e f f e c t of a x i a l load on l a t e r a l s t i f f n e s s are shown. This method r e a d i l y adapts i t s e l f through the use of e l e c t r o n i c computers to the p r e d i c t i o n of c r i t i c a l load by e v a l u a t i o n of that load which produces a zero determinate f o r the s t r u c t u r e s t i f f n e s s matrix. In t h i s t h e s i s a method of determining the e f f e c t of a primary bending s t r e s s d i s t r i b u t i o n on the s t i f f n e s s m a t r i x of a narrow r e c t a n g u l a r beam i s presented. The w e l l known d i f f e r e n t i a l equations governing the f l e x u r a l -t o r s i o n a l behavior are determined and solved subject to the v a r i o u s boundary c o n d i t i o n s as d i c t a t e d by the d e f i n i t i o n of the s t i f f n e s s matrix. The assump-t i o n that the beam segment i s r e l a t i v e l y short compared w i t h the " s t r u c t u r e " dimensions i s made to s i m p l i f y the s o l u t i o n of the equations. Comparison of c r i t i c a l loads f o r s t r u c t u r e s determined by using the d e r i v e d m a t r i x agree very w e l l w i t h c l a s s i c a l s o l u t i o n s a v a i l a b l e . * Numbers i n brackets r e f e r to the B i b l i o g r a p h y 2. An a l t e r n a t e procedure using the energy method f o r determining a ma t r i x i s presented. The r e s u l t i n g symmetrical m a t r i x which lends i t s e l f r e a d i l y to more complex problems shows r e s u l t s which are c o n s i s t a n t w i t h a v a i l a b l e c l a s s i c a l s o l u t i o n s . 3 . CHAPTER I I BASIC EQUATIONS 2.1 General Remarks In order to study the behavior of a beam segment under the i n f l u e n c e of primary bending and a s s o c i a t e d shears i t i s necessary to d e r i v e the a p p l i c a b l e d i f f e r e n t i a l equations l i n k i n g bending and t o r s i o n . For the purpose of t h i s d e r i v a t i o n an i n f i n i t e s i m a l element of the beam segment of length dz as shown i n F i g . 2.1 i s considered. The element i s shown i n i t s general deformed p o s i t i o n as defined by the d e f l e c t i o n s u and B r e l a t i v e to the axes x3y3z which are f i x e d i n space. Since the element i s par t of a beam of narrow re c t a n g u l a r cross s e c t i o n , d e f l e c t i o n s i n the y-z plane are considered small and are neglected. F i g . 2.1 a l s o shows the co-ordinate axes a3b3c of the element which are c o i n c i d e n t w i t h the c e n t r o i d a l , major and minor axes of the member and are i n a t r a n s l a t e d r o t a t e d p o s i t i o n to the axes x3y3z. The i n t e r n a l f o r c e s a c t i n g on the element cross s e c t i o n are of two types: (1) Primary forces M and V of l a r g e magnitude due to primary bending i n the y-z plane. (2) Secondary forces T3Q3M due to bending i n the x-z plane and t o r s i o n , as these fo r c e s form r e s u l t a n t s t r e s s e s p a r a l l e l and perpendicular to the cross s e c t i o n , they are shown i n the a3b3e axes system. 2.2 D e r i v a t i o n The f o l l o w i n g f o u r equations a r i s e by demanding the element be i n e q u i l i b r i u m i n i t s deformed shape as shown i n F i g . 2.1.a. Z Hx = 0 dM- - Vdz = 0 x on rearrangement V - V = 0 (1) ig. 2.1 ELEMENTAL LENGTH OF BEAM SEGMENT E FX = o - dq - Vd$ = o • on rearrangement 6 ' + F B ' = 0 ( 2 ) I « = 0 . dM + M d$ + dM d$ - Qdz = 0 n e g l e c t i n g terms of higher order and rearranging M ' + M B ' - Q = 0 (3) E Af = 0 dT - M v"dz - 6M \i"dz = 0 2 x x n e g l e c t i n g terms of higher order and rearranging 2" - M y" = 0 (4) CC From Hooke's Law of e l a s t i c behavior the f o l l o w i n g w e l l known r e l a t i o n s h i p s are obtained. Ely" = M (5) GJV = T (6) R e l a t i o n s h i p (1) i s a s t a t i c a l i d e n t i t y r e l a t i n g Af to F. The remaining, CC ( 2 ) to (6) form 5 equations i n the 5 unknowns M,Z,T, y a n d 3 -By d i f f e r e n t i a t i n g equation (5) once and s u b s t i t u t i n g i n equation (3) the r e l a t i o n s h i p £ V " + M B ' - Q = 0 (7) i s obtained. By d i f f e r e n t i a t i n g equation (7) once and s u b s t i t u t i n g i n equation ( 2 ) the r e l a t i o n s h i p Ely" + M B " + 27B ' = 0 ( 8 ) CC r e s u l t s . S i m i l a r l y the r e l a t i o n s h i p GJ$" - M \i" = 0 ( 9 ) i s obtained. 6. Solution of (8) and (9) for y and 3 i n closed form i s not possible i n general since M 3 V and EI are functions of z. The beam could be considered OC s p l i t into many small parts wherein M 3 V and EI might be considered as con-CC stant. However simultaneous solution of the equations would be inconvenient and for this reason a s t i f f n e s s procedure as outlined i n Chapter I I I i s preferred. 7. CHAPTER I I I METHOD OF SOLUTION 3.1 General Remarks The end of a beam has i n general s i x degrees of freedom. I n the pre-vious d e r i v a t i o n d e f l e c t i o n s i n the y-z plane were considered s m a l l and neglected and the forces M and V were assumed to be defined. This then CC e l i m i n a t e s two degrees of freedom which when combined w i t h the assumption that a x i a l forces and d e f l e c t i o n s are sm a l l leaves three at each end of the beam. 1 They are u , u and 3 a s s o c i a t e d w i t h the forces T3M3 and Q. A 6 x 6 s t i f f n e s s m a t r i x K w i l l then govern the behavior of the beam segment F i g . 3.1 such that K 6 = f (10) F i g . 3.1 DEFINED FORCES AND DEFLECTIONS Where / and 6 are 6 - component vectors which represent the forces and d e f l e c t i o n s of F i g . 3.1. The f i r s t column of K represents / when 6^ = 1 and a l l other displacements are zero. S i m i l a r l y the second column represents the forces / when 6 ^  ™ 1 a n <i a H others are zero. ) The d e f l e c t i o n s f i w i l l be 6 1 = y a t Z = 0 6 2 = 6 at Z = 0 6 3 = y' at Z = 0 6 4 = y at Z = Z 6 5 = B at Z = I 6 6 = y' at Z = Z The forces / w i l l be f± = -£Ty'M - MB' - 73 at Z = 0 f„ =-Gy"g' + Af y' at Z = 0 f 3 = -2?Ty" - Af^P at Z = 0 /. = E l y 1 1 ' + Af 3 * - at Z = Z f_ = GJB' - M y' at Z = Z f6 = £Ty" + Af^B at Z == Z These forces are merely the components of the s t r e s s r e s u l t a n t s T, Af, Q, and V i n the x,y',z d i r e c t i o n s . There i s a f o r c e i n and that i s not obvious namely Af B 1 which a r i s e s because Af acts on a t w i s t e d cross s e c t i o n . x x In order to f i n d any column of K i t appears necessary to sol v e (8) and (9) subject to the boundary c o n d i t i o n s on 6 f o r that column. A method of successive approximations i s chosen to s a t i s f y (8) and (9). F i r s t they are r e w r i t t e n as: Ely"" = - Af 3" - 2FB' (11) and Gy"B" = M^y" (12) 9. I f the beam segment i s small enough such that the average M i s much less CO than the c r i t i c a l moment for the length Z, a s u i t a b l e i n i t i a l guess i s the l i n e a r d e f l e c t i o n curve f o r the column of K being considered. This guess i s then substituted into the r i g h t hand side of ( l l ) or (12) and the d e f l e c t i o n curve p or M s found. One such cycle i s s u f f i c i e n t i f the segment i s short enough. Before i l l u s t r a t i n g the method i n d e t a i l i t i s necessary to define M as x Mx = Q4 + ^ - ) - V(l - z) (13) where M represents the average moment over the length I, or the center l i n e moment. 3.2 Derivation of S t i f f n e s s Matrix 3.2.1 Column 1 By the d e f i n i t i o n of K, 6^ = 1 and 6 ^  ••• g = 0 t n e boundary conditions on (11) and (12) are therefore: - 1 at Z = 0 0 at Z = 0 0 at z = 0 0 at z = I 0 at z = I 0 at z = I The s o l u t i o n by successive approximations w i l l take the form: y = y l + v 2 + ' * * + yn B = B 1 + 6 2 + ... + 3 n Assume that i s the l i n e a r d e f l e c t i o n shape for column 1 of K and i s given by 2 3 y , = - 1 + ^ 7 - ^ (14) 1 r z J 10. This assumption w i l l s a t i s f y equation (11) i f the r i g h t hand s i d e i s zero and t h e r e f o r e i f f u r t h e r accuracy i s r e q u i r e d would be estab-l i s h e d to p a r t l y s a t i s f y equation (11). S u b s t i t u t i n g y 1' i n t o (12), where , , 6 12z y l - o - 3 (15) l l I gives G J ^ = [(Af + ) - V(l - 'a).] [-^ - - ] - (16) I n t e g r a t i n g (16) twice and i n t r o d u c i n g the boundary c o n d i t i o n s above gives GJQ, = Af [7* + ^  " ^  ] " ^  " ^  1 * z z z z 3 4 + V [ f + ^ " * J ] ( 1 7 ) z z Equations v(14) and (17) f o r y^ and B^ s a t i s f y equation (12) e x a c t l y , however s i n c e y^ was assumed to be the l i n e a r d e f l e c t i o n curve equation (11) i s not f u l l y s a t i s f i e d . The normal procedure i n t h i s method of s o l u t i o n would be to s u b s t i t u t e values of B^' and B^ ' 1 i n t o the r i g h t hand s i d e of (11) and s o l v e again f o r y , e s t a b l i s h i n g On the other hand the e s t a b l i s h e d values of y^ and B-^  may define the d e f l e c t e d shape of the beam segement to a s u f f i c i e n t degree of accuracy. To e s t a b l i s h a measure of the accuracy of y^ and B^ a comparison i s made of the values of end moments produced; f i r s t l y by the requirements of the boundary c o n d i t i o n s of the s t i f f n e s s m a t r i x a c t i n g on equation (11) w i t h the r i g h t hand s i d e zero, and secondly those produced by a l a t e r a l l o a d of magnitude represented by the r i g h t hand s i d e of (11) w i t h B = B r I n the f i r s t case the value of end moments w i l l be of the order represented by *, - ^ (18) 11. In the second case the value of the l a t e r a l load must be f i r s t e s t a b l i s h e d . To s i m p l i f y t h i s , values o f , 3 , ' , 3 '' and M w i l l be assumed as JL JL CC constant and of value r e p r e s e n t i n g t h e i r order of magnitude only. These assumed values are then — VI Mx = M + 2 ( 1 9 ) V = m+ h (20) GJ12 GJl These values of M , 3, 1 and 3., ' ' w i l l then d e f i n e the l a t e r a l X L 1 load on the beam segment as < = - ^ + F " J i r + ^ r i - 2 V ^ik + h J ^ 2 2 ) End moments corresponding to t h i s load would be of the order F q 12 S u b s t i t u t i n g q from equation (22) the end moments become M = - M2 _ 7MVI - bV1!2 (24) 12Gy 24C7J" 24Gy The values of M i n equations (18) and (24) cannot be compared h d i r e c t l y s i n c e equation (24) contains terms w i t h M and V. These terms may be reduced to a more convenient form by c o n s i d e r i n g the intended use of the matrix. I t i s intended to p r e d i c t c r i t i c a l loads f o r s t r u c t u r e s comprised of s e v e r a l beam segments. The maximum values of M and V that w i l l occur i n a segment of the s t r u c t u r e i s then the moment and shear corresponding t o the c r i t i c a l s t r u c t u r e l o a d i n g . For the purposes of t h i s accuracy check, consider a simply supported beam w i t h a poi n t load at the center l i n e . The w e l l known value of c r i t i c a l load i s 12. Where L i s the length of beam or " s t r u c t u r e . " By l e t t i n g L take the value L = a I (26) P - ^ f - (27) or 2 -,2 x 7 The maximum values of moment and shear are then I . * S E ( 2 8 , • T - S ^ L (29) a I S u b s t i t u t i n g these values i n t o equation (24) the end moment becomes: = - 4_£T_ - 2 8 J J - WEI_ UF „ 2 72 3 72 , 4 72 U U ; 3a Z. 3a Z- 3a Z-2 Considering then equation (18) and (30) and e l i m i n a t i n g the common f a c t o r EI/^ the requirement of f u r t h e r approximations to the d e f l e c t i o n curve namely ... depends on the magnitude of equation (30) as compared to the value 6. However s i n c e s e v e r a l approximations were i n v o l v e d i n o b t a i n i n g equation (30) a 2 more l o g i c a l approach would be to consider the value of 1/a as compared to u n i t y . Obviously a choice of 10 elements would le a d to r e l a t i v e magnitudes of 1:.01. The determination of the r a t i o that w i l l l e a d to s a t i s f a c t o r y r e s u l t s i s best determined by numerical t r i a l s s i n c e an i n c r e a s i n g number of segments w i l l l i k e l y lead to b e t t e r accuracy. Numerical t r i a l s presented i n Chapter V confirm that f o r c e r t a i n s i t u a t i o n s 10 segments i s very s a t i s f a c t o r y . With the assumption that f u r t h e r approximations are unnecessary and that equation (14) and (17) adequately d e f i n e the d e f l e c t e d shape of the member the f o r c e s f. to fr are evaluated by s u b s t i t u t i o n of Af , and u. i n t o the J 1 J 6 x 1 1 equations of Pa r t 3.1. The for c e s are as f o l l o w s : _ 12gJ M2 _ MV V2l T l t3 GJl GJ liGJ 13. I n the above expression the l a s t three terms as compared to the f i r s t term are small w i t h i n c r e a s i n g number of segments by the same reasons as presented p r e v i o u s l y . For t h i s reason they w i l l be neglected and the f o r c e taken as f = 1 2 E I  1 = I3 S i m i l a r l y the remaining fo r c e s are: •F = ^ - -^2 1 2 fk - _ 12ET 3 f* - E Z r which c o n s t i t u t e the f i r s t column of K F i g . 3.2. 3.2.2 Column 2 By the d e f i n i t i o n of %, 8^ = 1 w i t h the remaining d e f l e c t i o n s zero. The boundary c o n d i t i o n s on (11) and (12) are t h e r e f o r e : y = 0 at Z = 0 6 = 1 at Z = 0 y ' = 0 at Z = 0 y = 0 at Z = I B = 0 at Z = I y ' = 0 at Z = I The s o l u t i o n by s u c c e s s i v e approximations w i l l take the form 6 = 6 , +60 + . . . +B and y = y , + y . + . . . + y Assume that B n i s the l i n e a r d e f l e c t i o n curve f o r Column 2 of % f t h e r e f o r e : $1 = 1 - f <30> 14. This assumption w i l l s a t i s f y equation (12) i f the r i g h t hand s i d e i s zero. Pro-ceeding as before by d i f f e r e n t i a t i n g equation (30) and s u b s t i t u t i n g i n t o equation (11) EIV™-=Z£ (31) I n t e g r a t i n g four times and i n t r o d u c i n g the boundary c o n d i t i o n s above gives 4 3 2 EIvi = I2T - — + ~VT ( 3 2 ) The accuracy of y^ and 3-^  may be e s t a b l i s h e d as i n Case 1 and t h e r e f o r e f u r t h e r approximations ... y^ and 3^ ••• $n a r e neglected. With equations (30) and (32) the d e f l e c t e d shape of the member i s adequately defined and the forces to may be evaluated. Proceeding as i n Case 1 the fo r c e s are: •F ~E_ z T l ~ l 2 * - QL 72~ I f = -E + l J 4 1 2 * - QL T5 " I -F = Yk T6 6 where a d d i t i o n s to / and are neglected. These values c o n s t i t u t e the second column of K F i g . 3.2. 3.2.3 Column 3 By the d e f i n i t i o n of K3 6^ = 1 w i t h the other d e f l e c t i o n s equal to zero. The boundary c o n d i t i o n s on (11) and (12) are t h e r e f o r e : y = 0 at Z = 0 3 = 0 at Z = 0 y' = 0 at Z = 0 15. y = 0 .at Z = I 6 = 0 at Z = I y' = 0 at Z = Z Taking the s o l u t i o n i n the form of successive approximations as before the d e f l e c t e d shapes are: y. = a + 4 - — (33) and GJ^ = M [z + ^ - ^ 4 ' i -Y-\ iT~:2r + 2 i ] ( 3 4 ) On e v a l u a t i o n of the forces to the t h i r d column of K F i g . 3.2 i s obtained. Columns 4, 5 and 6 of K may be derived i n a s i m i l a r manner and the complete matr i x i s shown i n F i g . 3.2. Before d i s c u s s i o n of the use and accuracy of K an a l t e r n a t e method of d e r i v a t i o n f o r K i s presented i n the f o l l o w i n g s e c t i o n . 125T I3 i M V I 2 - 6EI I2 - 12EI I3 - Af V I 2 -6EI I2 M V I 2 GJ I - VI 6 - M V I + 2 - GJ I + VI 6 - 6EI I2 4ST I 6EI I2 VI 6 2EI I - 12EI I3 - M V l + 2 6EI I2 12EI I3 Af V I + 2 6EI I2 - Af V I 2 - GJ I VI 6 A7 v l + 2 GJ I VI 6 - 6EI I2 VI 6 2£T I 6EI I2 Af + — mi i F i g . 3.2 STIFFNESS MATRIX K 17. CHAPTER IV ALTERNATE METHOD OF SOLUTION 4.1 General Remarks A more d i r e c t method of ob t a i n i n g the s t i f f n e s s m a t r i x K i s to con-s i d e r a c a n t i l e v e r beam as shown i n F i g . 4.1. The end Z = I i s considered f r e e f o r d e f l e c t i o n s y and 3 and d e f l e c t i o n s i n the y - z plane are considered n e g l i g i b l e as before. (b) F i g . 4.1 FORCES FOR COLUMN 4 of K The method of ob t a i n i n g K i s to place the f r e e end 3 - I i n t o a d e f l e c t e d p o s i t i o n as d i c t a t e d by the boundary c o n d i t i o n s f o r columns 4 to 6 of K. The d e f l e c t e d shape of the beam i s assumed to be the l i n e a r d e f l e c t i o n f o r the p a r t i c u l a r column of K being considered. This assumption corresponds to the one i n Chapter I I I where no c o r r e c t i o n s to the f i r s t approximation 18. (the l i n e a r d e f l e c t i o n ) w e r e made. Equations f o r moments M and T at Z = Z from a f r e e body of the segment from Z = Z to Z = Z i n i t s d e f l e c t e d shape are then obtained. The moments T are r e q u i r e d i n columns 4 and 6 and M i n column 5 s i n c e the b a s i c shapes f o r these columns are r e s p e c t i v e l y ones of l a t e r a l d e f l e c t i o n and t o r s i o n . For t h i s d i s t r i b u t i o n of M or T the l i n e a r equations Ely " = M (35) and GJ8 ' = T (36) are solved and give d e f l e c t i o n s y or 3 at Z = Z. End forc e s are then a p p l i e d to provide equal and opposite end d e f l e c t i o n s to s a t i s f y the d e f l e c t i o n con-d i t i o n s f o r the column being considered. Forces 4 - 6 are then these c o r r e c -t i o n f o r c e s p l u s the o r i g i n a l l i n e a r f o r c e s along w i t h those forces due to realignment of the primary forces at Z = Z due to d e f l e c t i o n s . The forces 1 - 3 are obtained from e q u i l i b r i u m of the element. 4.2 D e r i v a t i o n of S t i f f n e s s M a t r i x 4.2.1 Column 4 Assume a l i n e a r d e f l e c t e d shape as shown i n F i g . 4.1 (a) such that 2 3 U = - •-3jV + '-?£r (37) r Z and y . . - M + ' i a l OS) l z J The us u a l l i n e a r forces as shown are r e q u i r e d to maint a i n the end Z = Z i n t h i s p o s i t i o n . Considering the f r e e body from Z = Z to Z = Z the equation f o r T at Z i s — VI T = V (1 - y) + "[(Af + ^ | ) - V (Z - s ) ] y' (39) 19. S u b s t i t u t i n g M and v' from equations (37) and (38) and i n t r o d u c i n g equation (36) 2 3 2 c W = F [ l + ^ - - ^ + ^ 3 1 + « f ^ f + ^ 3 ] (40) I? is I* is I n t e g r a t i n g once and s u b s t i t u t i n g the boundary c o n d i t i o n , 3 = 0 at Z = 0 the d e f l e c t i o n at Z = Z- i s : In order that the d e f l e c t i o n 3(Z) be zero the end torque M =} + f (42) Z ^ Is as shown i n F i g . 4.1.(b) i s a p p l i e d . The forces to are then the sum of the f o r c e s shown i n F i g . 4(a) and (b) along the r e s p e c t i v e d i r e c t i o n s . The fo r c e s to are obtained from e q u i l i b r i u m of the beam. These forces then represent column 4 of K F i g . 3.2. 4.2.2 Column 5 Assume a l i n e a r d e f l e c t e d shape as shown i n F i g . 4.2(a) such that 3 = f (43) and 3' = \ (44) The usu a l l i n e a r f o r c e as shown i s r e q u i r e d to maintain the end z = Z- i n t h i s p o s i t i o n . F i g . 4.2 FORCES-FOR COLUMN 5 OF K Considering the free body from Z = Z to Z = Z and employing equation (43), (44) and (35) Ely" = M (1 - j ) Integrating twice and introducing the coundary conditions, y _ i _ the d e f l e c t i o n s at Z = Z are: and y'(Z) = y(Z) = A/Z 7Z' 2EZ 12 (45) 0 at Z = 0, (46) VI' 3EI 12 In order that the d e f l e c t i o n s y'(Z) and y(Z) be zero the end forces and M V F = y + -7T x I 2 M = f y 6 (47) (48) (49) as shown i n F i g . 4.2.(b) are applied. The forces f^ to are then the sum of the forces shown i n Figs. 4.2.(a) and (b) along the respective d i r e c t i o n s noting the re-alignment of the primary bending moment and shear force due to the d e f l e c t i o n g . Forces to are obtained from equilibrium. These forces then represent column 5 of K F i g . 3.2. 21. 4.2.3 Column 6 Assume a l i n e a r deflected shape as shown i n F i g . 4.3.(a) such that and y -i _ z I Iz + • 3z' 3 + (50) (51) F i g . 4.3 FORCES FOR COLUMN 6 OF K The usual l i n e a r forces as shown are required. Considering the free body Z = Z to Z = I, introducing equations (50), (51) and (36), and integrating^ the d e f l e c t i o n g (Z-) i s . 2 — Vl Ml f r 0 s p a ) = " 3GJ ~ Gl ( 5 2 ) The correction force required (53) M =¥\ +M . z - 3 as shown i n F i g . 4.3.(b). The forces to f6 and f± to f3 are evaluated i n the previous manner. These forces are then those of column 6 as shown i n F i g . 3.2. 22. The methods presented i n Chapters I I I and IV g i v e i d e n t i c a l r e s u l t s ; the l a t t e r however lends i t s e l f to a b e t t e r p h y s i c a l understanding of the p r o -b lem. In both methods b e t t e r v a l u e s of u and g cou ld be o b t a i n e d , i n the f i r s t by making f u r t h e r approx imat ions and i n the second by c o n s i d e r i n g p r o -g r e s s i v e m o d i f i c a t i o n s to assumed shapes. These are i n f a c t the same procedure , however, as was shown i n Chapter I I I the n e c e s s i t y of f u r t h e r approx imat ions does not appear war ran ted . Numer ica l r e s u l t s presented i n Chapter VI c o n f i r m t h i s assumpt ion . 4.3 C o n s i d e r a t i o n of E c c e n t r i c Load The s t i f f n e s s m a t r i x as d e r i v e d i n Chapter I I I and IV show the e f f e c t of a pr imary bending moment and shear f o r c e a c t i n g a t the c e n t r o i d a l axes of the member. Th is pr imary f o r c e system i s assumed to remain constant throughout the b u c k l i n g d e f o r m a t i o n . Th is assumption i s v a l i d i f the deformat ions are s m a l l and the l o a d i s a p p l i e d at the c e n t r o i d of the s e c t i o n . However, i f loads are a p p l i e d above or below as shown i n F i g . 4.5 i t i s ev ident t h a t b u c k l i n g deformat ions c r e a t e t o r s i o n a l moments about the c e n -t r o i d a l axes of the member. In order to determine the e f f e c t of p l a c i n g a l o a d i n these p o s i t i o n s the s t r u c t u r e s t i f f n e s s m a t r i x must be m o d i f i e d a c c o r d i n g l y . (a) F i g . 4.5 (b) STRUCTURE LOADED AT BOTTOM CHORD 23. Considering the structure in Fig. 4.5 the load on the bottom edge may be considered as a load . at the centroid plus a torsional moment applied along the displacement. This moment is proportional to the magnitude of the load and the value of the displacement. Obviously i t is identical in effect on the structure to the effect of a torsional spring of stiffness Pxe and may be treated by introducing a torsional restraint of that value in the diagonal element of the structure stiffness matrix corresponding to the torsional dis-placement at that location. It is also evident that i f the load were above the centroid of the section, introduction of a negative torsional restraint would have the correct effect on the structure. Numerical trials as presented in Chapter VI establish the accuracy of this procedure. 4.4 Consideration of Restrained Centre of Rotation and Axial Loads In the performance of tests on lateral buckling of beams the use of a restrained centre of rotation is convenient and naturally many building systems such as girders with decking display this type of restraint. Also the effect of axial loads, must be included in a buckling analysis. Firstly the effect of axial load may be included by introducing the stability function of Gere and Weaver (7). These functions, shown below, are shown in the stiffness matrix Fig. 4.6 where H is the axial load in the member. 5 1 = A 3 sin A/12<J> 5 2 = A (1-cos A ) /6<j) 5 3 = X (sin X - X cos A ) /4(j> 54 = * - s i n *)72<1> where A 2 = HL2/EI cfi = 2 - 2 cos A - A sin A I n a d d i t i o n , an e x t r a term appears because.of the e f f e c t of H on the t o r s i o n a l r i g i d i t y GJ. and i s represented by a modified t o r s i o n a l 'o_. A r i g i d i t y GJ = (GJ - ~ H) i 1 W I x S z 3 1 M V I 2 ' " 6 E 1 x s I2 2 - 1 2 B J x s Z 3 1 - M V I 2 - 6EI 2 * S 2 M V I 2 GJ I - VI 6 - Af V 1 2 - GJ I VL 6 ~ 6 E I x S I2 2 6 f f J x s z 2 2 VI 6 2EI _ Z 2 4 Z J 1 -Id - .V 1 2 6 Z ? J x S 2 2 12EI 3 X S l z J 1 Af V 1 2 6 Z ? I x S 2 2 z z z M V I ~ 2 - GJ • I. 6 I F Z + 2 HL I - FZ 6 - 6EI _ 2 * S 2 VI 6 2EX I X S4 6 Z ? J x S z 2 2 M + —2 4 f f J x s Z 3 F i g . 4.6 MATRIX WITH STABILITY FUNCTIONS 25. C o n s i d e r a t i o n of a r e s t r a i n e d centre of r o t a t i o n at a distance g from the c e n t r o i d as shown i n F i g . 4.7 has the e f f e c t of reducing the number of degrees of freedom from 6 to 4. F i g . 4.7 RESTRAINED CENTRE OF ROTATION With reference to the deformation and f o r c e system <5 and / as shown i n F i g . 4.8 and the system used i n Chapters I I I and IV F i g . 3.1 the f o l l o w i n g equivalences are noted. 6 2 = 6 2 f2 = f2-f19 6 3 = 6 3 / 3 = f3 « 4 = 6 4 + 3 6 5 7 4 = f h 6 5 = 6 5 ?5=f5-fkg 5 6 = 6 6 ^ 6 = f6 26. F i g . 4 . 8 RESTRAINED FORCE AND DEFLECTION SYSTEM These equivalences may be placed i n the matrix form •6 = T 6 and where y and are given by: T = 1 9 1 1 1 9 1 1 (54) (55) 1 -9 1 1 1 -9 1 1 F i g . 4 . 9 TRANSFORMATION MATRICES 27. n o t i n g that 2 1 - 1 = T^* from which (TJCT^-S = / (57) k = T KT * (58) A new 6 x 6 s t i f f n e s s m a t r i x k referenced to the d e f l e c t i o n s 6 and forc e s / i s t h e r e f o r e a v a i l a b l e by tran s f o r m a t i o n of K. However the d e f l e c t i o n s 6^ and <5^  are zero by v i r t u e of the r e s t r a i n t . This enables the removal of rows and columns 1 and 4 of Ic and the development of a new 4 x 4 matri x Ti. The matri x Ti i s not shown s i n c e the m u l t i p l i c a t i o n s i n equation (58) are c a r r i e d out n u m e r i c a l l y f o r each p a r t i c u l a r case. The r e d u c t i o n to k enables the a n a l y s i s of r e s t r a i n e d centre of r o t a t i o n problems. 28. CHAPTER V DERIVATION USING THE ENERGY METHOD 5.1 General Remarks D e r i v a t i o n of a s t i f f n e s s m a t r i x by d i f f e r e n t i a t i o n of the s t r a i n energy f u n c t i o n f o r the beam w i l l by d e f i n i t i o n produce a symmetrical matrix. The r e s u l t s of such a d e r i v a t i o n , as obtained i n p r i v a t e c o n s u l t a t i o n w i t h Anderson ( 9 ) , are presented i n t h i s s e c t i o n . For the purpose of t h i s d e r i v a t i o n the s i m p l i f y i n g assumption that moment on the segment was constant, and hence the shear equal to zero, was made. 5.2 B r i e f D e s c r i p t i o n of the Method B l e i c h (8) presents an energy expression f o r the beam as f o l l o w s U = 1/2 / (£Ty" 2 + GJ8'2 + 2M y " g ) dz (59) 0 X which corresponds to the a d d i t i o n a l s t r a i n energy of a beam subjected to an i n i -t i a l moment M and a twist-bend d e f l e c t i o n . x Assumptions as to the fu n c t i o n s y and 3 are the same as presented i n Chapter I I I page 9 where y = y l + y 2 + *"' + y n and B - e x + B 2 + . . . +en As before y^ and 6^ are taken as the l i n e a r v a l u e s , y^ and 6 2 c a l -c u l a t e d from the e f f e c t of the moment on the l i n e a r d e f l e c t e d shape and sub-sequent d e f l e c t i o n s neglected. The second d e r i v i t i v e s of U w i t h respect to the d e f l e c t i o n s 6 ... <S , which represent the components of the symmetrical 1 o ma t r i x Kl, are shown i n F i g . 5.1 12EX M - 6EI - 12EI - M - 6EI I3 I I2 I3 I I2 Af I GJ I Af - M I - GJ I 0 - 6EI I2 - AT 4 EI I 6EI I2 0 TEI I - V2EI - A7 6EI 12EI M 6EI I3 l I2 I3 I I2 - M I - GJ I 0 M I GJ I M - 6EI I2 0 2EI I 6EI I2 M kEI I F i g . 5.1 STIFFNESS MATRIX Kl 30. CHAPTER VI APPLICATION TO STABILITY PROBLEM 6.1 General Remarks The s t i f f n e s s m a t r i x K as derived i n Chapter I I I and IV contains the e f f e c t on the u s u a l l i n e a r m a t r i x of bending moments and shears i n the plane of the beam element. This matrix may be employed to determine the c r i t i c a l value of load f o r narrow r e c t a n g u l a r beams. The procedure used i s to formulate a " s t r u c t u r e " composed of s e v e r a l s m a l l beam segments. With reference to the determination i n Chapter I I I of the r e q u i r e d number of segments, the r a t i o of segment to beam length of 1/10 was used. The member s t i f f n e s s matrices are b u i l t up using values of M and V from a l i n e a r a n a l y s i s of the s t r u c t u r e . These member s t i f f n e s s matrices are then entered i n t o a s t r u c t u r e s t i f f n e s s m a t r i x i n the usual manner. The a c t u a l determination of the c r i t i c a l load i s obtained by f i r s t assuming a value f o r the e x t e r n a l l o a d , performing the l i n e a r a n a l y s i s , b u i l d i n g the s t r u c t u r e s t i f f n e s s m a t r i x and e v a l u a t i n g the determinate. The e x t e r n a l load i s then incremented u n t i l the s t r u c t u r e s t i f f n e s s m a t r i x i s zero which by d e f i n i t i o n i s the c r i t i c a l value of loading f o r the s t r u c t u r e . 6.2 Numerical Examples For the purpose of comparing the r e s u l t s of c r i t i c a l load e v a l u a t i o n by the above mentioned method w i t h values as presented i n the l i t e r a t u r e the beam shown i n F i g . 6.1.a i s used. 31. M E M B E R DATA E = 30,000 k.s.i. J = 3.333 in 4 A = 10 i n 2 ..G = 10,0.0.0_k.s.i. 1= 0.833 in 4 / = 10 in F i g . 6.1 EXAMPLE BEAM The beam i s of narrow r e c t a n g u l a r cross s e c t i o n w i t h a length depth width r a t i o of 100:10:1. The " s t r u c t u r e " c o n s i s t s of 10 segments of length 10 and the s t r u c t u r e s t i f f n e s s m a t r i x i s of a s i z e as determined by the boundary co n d i t i o n s as d i c t a t e d by p a r t i c u l a r examples. S i x s t r a i g h t beams were i n v e s t i g a t e d f o r c r i t i c a l load and the r e s u l t s i n terms of y are shown i n F i g . 6.2. Case # End Cond. Loading Y * T l Y 2 1 S.S. Equal End Couples .1885 .1898 .1940 2 S.S. P o i n t at e.g. at .2545 .2750 .2600 3 S.S. Po i n t at Bot. at .2740 .2795 .2820 4 S.S. Po i n t at Top at .2350 .2360 .2370 5 S.S. Uniform .2130 .2145 .2160 6 C a n t i l e v e r P o i n t a t e.g. at End .2410 .2425 .2480 °ar = y/EIGJ/l *y From a v a i l a b l e theory **Y^ From t e s t using matrix K From t e s t using matrix K. F i g . 6.2 EXAMPLE RESULTS 32. Since the r e s u l t s agree w i t h those a v a i l a b l e from the l i t e r a t u r e to w i t h i n 3% i t i s assumed the method gives s u f f i c i e n t l y accurate r e s u l t s f o r t h i s simple type of s t r u c t u r e . 33. CHAPTER V I I CONCLUSIONS The author concludes that both matrices developed, namely K and X I , give s a t i s f a c t o r y r e s u l t s when a p p l i e d to c r i t i c a l load problems of simple s t r u c t u r e s . I n v e s t i g a t i o n i n t o the l i m i t s of the short length segment shows that a r e d u c t i o n i n the number of elements from 10 to 3 decreases the accuracy of the r e s u l t by approximately 5%. However due to the simple nature of the s t r u c t u r e s analysed, namely those w i t h simple primary bending s t r e s s d i s t r i -b utions i t would be i n a d v i s a b l e to p r e d i c t a d e f i n i t e number of segments f o r a s p e c i f i c . d e g r e e of accuracy. A more l o g i c a l procedure would be to p l o t c r i t i c a l l o a d vs number of segments and determine convergence f o r each s p e c i f i c case. The extension of the use of these matrices to more complex problems would be i n a d v i s a b l e u n t i l d e f i n i t e reasons are presented to e x p l a i n why two e n t i r e l y d i f f e r e n t matrices give c o n s i s t e n t r e s u l t s to the same problems. Obviously the m a t r i x Kl lends i t s e l f much more r e a d i l y to numerical a p p l i c a t i o n s i n c e i t s symmetrical nature may be f i t t e d i n t o e x i s t i n g s t r u c t u r e a n a l y s i s programs. When v a l i d explanations f o r the above mentioned problems are obtained other e f f e c t s such as the e f f e c t of a x i a l load on t o r s i o n a l r i g i d i t y and pre-v i o u s l y derived e f f e c t s of a x i a l load on l a t e r a l s t a b i l i t y may be added r e s u l t i n g i n a m a t r i x w i t h which complex problems of f l e x u r a l t o r s i o n a l stab-i l i t y under l a t e r a l and a x i a l loads could be analysed. BIBLIOGRAPHY 1. Timoshenko, S.P. and Gere, J.H., Theory of E l a s t i c S t a b i l i t y , Second E d i t i o n , McGraw-Hill, New York, New York, 1961, p. 252-277. 2. Goodier, J.N. , F l e x u r a l - T o r s i o n a l B u c k l i n g of Bars of Open Section  Under Bending, E c c e n t r i c Thrust, or Torsion Loads, C o r n e l l U n i v e r s i t y Engineering Experimental S t a t i o n , B u l l e t i n No.28, January 1962. 3. B l e i c h , F., Buckling Strength of Metal S t r u c t u r e s , McGraw-Hill, New York, New York, 1952, p. 109-166. 4. Vlasov, V.Z., Thin - Walled E l a s t i c Beams, Second E d i t i o n rev. and aug., Jerusalem, I s r a e l Program for S c i e n t i f i c T r a n s l a t i o n s , 1961, p.311-342. 5. Harmann, A. J . , The E l a s t i c F l e x u r a l - T o r s i o n a l Buckling of Planar  S t r u c t u r e s Ph.D. T h e s i s , U n i v e r s i t y of I l l i n o i s 1964. 6. B e l l , L.A., L a t e r a l S t a b i l i t y of Rectangular Beams M.Sc. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1966. 7. Gere, J.M., and Weaver, W., A n a l y s i s of Framed S t r u c t u r e s , D. Van Nostrand, Pr i n c e t o n 1965. 8. McMinn, S.J., M a t r i c e s For S t r u c t u r a l A n a l y s i s , Second E d i t i o n , E. & F. N. Spon L t d . , London, England, 1964, p. 170-201. 9. Anderson, D.L., A s s i s t a n t Professor of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, 1968. APPENDIX A A . l . D e s c r i p t i o n of Computer Program In a b r i e f manner the.sequence of operations and the p r e s e n t a t i o n of data r e q u i r e d i s as f o l l o w s : 1. A card i s read which bears the s t r u c t u r e data. (a) number of members, NM (b) e l a s t i c and shear modulus, E.G (c) moment of i n e r t i a and t o r s i o n constant, TT, T. (d) length of each segment, AL (e) number of degrees of freedom, NU 2. The next cards, one f o r each member, bear the average moment FM (I) and average shear VM (I) as determined by an independent e l a s t i c a n a l y s i s under an assumed load. 3 . Next, a s e r i e s of cards, one f o r each member, w i t h i t s code number, NCODE ( I I , J ) . 4 . The above member data and moment and shear values are used to determine member s t i f f n e s s matrices SM(I,J,K). 5. The above member s t i f f n e s s matrices are then used i n conj u n c t i o n w i t h the code numbers to b u i l d the s t r u c t u r e s t i f f n e s s m a t r i x , SM ( I , J ) . 6. A subroutine, "INVERT," i s c a l l e d from which the value of the deter-minate, "DETERM," i s obtained. 7 . The value of the determinate i s p r i n t e d out. I f i t i s greater than zero the values of FMC(I) and VM(I) are incremented by a f a c t o r , FA, and steps 4 t h r u 7 repeated. 8. Termination of c a l c u l a t i o n i s governed by o b t a i n i n g a determinate value of zero or l e s s . A.2 FORTRAN SOURCE LIST f ^ K T H W ' S O U R C E L I S T ISN SOURCE STATEMENT 0 $ IBFTC BAZ 1 DIMENSION N C O D E ( 2 5 , 6 ) , S M I 6 0 , 6 0 ) , S ( 2 5 . , 6 , 6 ) , F M ( 2 5 ) , V M ( 2 5 ) c. THIS MATRIX IS THE COMPLETE SHEAR 2 * 99 CONMNUE 3 R E A D ( 5 , 3 3 3 3 ) NST 5 V 3 3 3 3 FORMAT(110) ' 6 * WRITE*6 ,3 334) NST • 7 * 3 3 3 4 F 0 R M A T ' 1 X , 2 2 H THIS I S STRUCTURE NO. , 1 1 0 , / / / ) 10 JL TT R E A D { 5 , 1 0 0 ) N M , E » G , T T , T , A L , N U 13 151 F O R M A T ( I FiO.2 ) 14 * WR I-TE ( 6 , 1 0 0 ) NM, E , G , TT , T , AL , NU 15 V 100 F O R M A T ( 1 1 1 0 , 5 F 1 0 . l t 1 1 1 0 ) 16 * R E A D ( 5 , 1 0 1 ) ( F M ( I ) , 1 = 1 , 2 4 ) 2 3 W R I T E ( 6 , 1 0 3 5 ( F M { I ) , 1 = 1 , 2 4 ) 30 R E A D ( 5 , 1 0 1 ) ( V M ( I J , 1 = 1 , 2 4 ) 35 W R I T E ( 6 , 1 0 3 ) ( V M ( I ) , 1 = 1 , 2 4 ) 42 R E A D ( 5 , 1 5 1 ) T S " 43 W R I T E ! 6 , 1 5 1 ) T S 44 103 FORMAT ( 1 X . , 6 F 1 0 . 1) 45 101 F O R M A T ! 6 F 1 0 . 1 ) 46 DO 1 5 0 11=1,NM 47 R E A D { 5 , 1 0 2) (NCODE( I I , J ) , J = l , 6 ) 54 * W R I T E ( 6 , 1 0 4 ) { N C O D E ( I I , J ) , J = l , 6 ) 61 104 FORMAT ( I X , 61 10) ' ' " " 62 102 F 0 R M A T ( 6 I 1 0 ) 6 3 * 150 CONTINUE 65 B = G * T 66 A=E*TT 67 * C=A*B 70 F A = 1 . 0 7 1 * DO 8 0 0 K K = 1 , 1 0 72 * 200 CONTINUE 73 D 0 3 0 0 1=1,NM 74 A A = ( F M ( I ) * * 2 ) / C 75 * S( I , 1 , 1 ) = ( 1 2 . 0 * A / ( A L * * 3 ) ) 76 V S ( I , 1 , 2 ) = ( F M ( I ) / A L ) + V M ( I ) / 2 . 77 S( I, 1 , 3 ) = - { 6 „ 0 * A / ( AL**2) ) 100 S( 1 , 1 , 4 ) = - S ( 1 , 1 , 1 ) 101 S ( I , 1 , 5 ) = - F M ( I ) / A L + V M ( I ) / 2 . 102 S ( I , 1 , 6 ) = S ( I , 1 , 3 ) 1 0 3 * S ( 1 , 2 , 1 ) = ( F M ( I ) / A L ) + ( V M ( I J / 2 . 0 ) 1 0 4 S ( I , 2 , 2 ) = B / A l 105 * S ( I » 2 , 3 ) = ( V M ( I ) * A L / 6 . 0 ) 106 * S( 1 , 2 , 4 ) = - F M ( I ) / A L - V M { I ) / 2 . 107 * S ( 1 , 2 , 5 ) = - S ( 1 , 2 , 2 ) 110 S ( I » 2 * 6 ) = - ( V M ( I ) * A L / 6 . 0 ) 111 S ( I , 3 , 1 ) = S ( I , 1 , 3 ) 1 1 2 St I , 3 , 2 ) = - F M ( I ) - V M ( I ) * A L / 3 . 113 S ( I , 3 , 3 ) = ( 4 . 0 * A / A L ) 114 S ( I , 3 , 4 ) = - S ( 1 , 1 , 3 ) 1 1 5 + SI I , 3 , 5 ) = - V M ( I ) * A L / 6 . 116 S ( I , 3 , 6 ) = ( 2 . 0 * A / A L ) 117 S( 1 , 4 , 1) = - -S( 1 , 1 , 1) 120 S ( I , 4 , 2 ) = - F M ( I J / A L - V M C I ) / 2 . 121 * S( 1 , 4 , 3 )=S( I , 3 , 4 ) FORTRAN SOURCEJLISX FORTRAN SOURCE LIST BAZ ISN SOURCE STATEMENT 122 S U , 4 , 4 ) = S ( I , 1 , 1) 123 S<1,4,5)=FM(I)/AL-VM(I)/2* 124 * S ( I , 4 , 6 ) = S ( 1 , 4 , 3 ) 125 S ( I , 5 , i ) = - ( F M U ) / A L ) + ( V M ( I ) / 2 . 0 ) 126 S U , 5 , 2 ) = - S ( I , 2 , 2 ) 127 S ( I , 5 , 3 ) = S ( I , 2 , 6 ) 130 S<I» 5 » 4 ) - ( F M { I ) /A L)-{V M( I ) / 2 • 0 ) 131 * S( 1,5,5)=S( 1,2,2 )• • 132 * S I I , 5 , 6 ) = S ( I , 2 , 3 ) 133 S( 1,6,1) = S( 1,3,1) 134 * S(I,6,2)=-VM{I) *AL/6. 135' * S U ,6, 3) = S( I , 3,6) 136 S ( I , 6 , 4 ) = S ( I , 4 , 6 ) 137 * S ( I , 6 , 5 ) = F M ( I ) - V M ( I ) * A L / 3 „ 140 * S U ,6, 6) = S( I ,3,3) 141 * 300 CONTINUE 143 * DO 400' J=1,NU 144" DO 400 1=1,NU 145 SM(J,I )=0.0 146 4-T 400 CONTINUE 151 DO 500 1=1,NM 152 DO 500 K=l,6 153 IF(NCODE{I,K))420,500,420 154 M=NCODE(I,K) 155 * DO 502 J = l , 6 156 * I F ( N C O D E ( I , J ) ) 4 3 0 , 5 0 2 , 4 3 0 157 * 430 N = N C O D E U » J ) 160 * SM(M,N)=SM{M,N)+S(I, K, J ) 161 502 CONTINUE 163 500" CONTINUE 166 * SM(15,15 )=SM{15,15)+TS*5.0 167 * 501 F0RMAT16F20.6) 170 00 600 JJJ=1,NU 171 DO 600 111=1,NU 172 * S M ( J J J , II I ) = S M ( J J J , I I I ) / 1 0 0 . 0 173 "600™' CONTINUE' 176 CALL INVERT (SM,NU,60,DETERM,COND) 177 WRITE(6,650)FA,DETERM 200 650 FORM AT(13H LOAD FACTOR=,IF 10.3,13H DETERMINATE=,1E20.6) 201 * W R I T E ( 6 , 1 0 3 ) ( F M ( I ) , 1 = 1 , 1 2 ) 206 WR I T E ( 6 ,10 3 ) ( V M ( I ) , I = 1,1 2 ) 213 * WRITE(6,151)TS 214 IF(DETERM) 701,701, 700 215 * 700 FA=FA+0.05 216 TS=TS*FA 217 * DO 6801=1,24 220 * FM( I ) = FM(I )*FA 221 VM( I ) =VM(I) *F A 222 * 680 CONTINUE 2 24 800 CONTINUE 226 701 CONTINUE 227 * GO TO 99 230 702 _ CONTINUE 231 STOP 232 * END 

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