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The strongest column : a matrix approach Kerr, Peter A. 1968

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THE STRONGEST COLUMN: A MATRIX APPROACH by PETER A. KERR B.A.Sc. ( 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 Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIEO SCIENCE i n the Department of CIVIL ENGINEERING UJe accept t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d The U n i v e r s i t y of B r i t i s h Columbia January 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 life representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l Engineering The University of Brit ish Columbia Vancouver 8, Canada Date January 12, 1968. In 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 o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study* 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 c o p y i n g o f 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 p o s e s may be g r a n t e d 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 u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f 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 no t be a l lou ied w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f C IVIL ENGINEERING The U n i v e r s i t y o f B r i t i s h C o l u m b i a , Vancouver 8 , Canada Date J a n u a r y 5 , 1968 - i -A b s t r a c t The s t r o n g e s t column problem i s defined, f o r t h i s t h e s i s , as the determination of the column shape which gives the maximum E u l e r b u c k l i n g load for a given length, volume of m a t e r i a l , type of c r o s s -s e c t i o n and type of t a p e r i n g . This t h e s i s presents a new method - the matrix method - f o r s o l v i n g some s t r o n g e s t column problems. In the matrix method a member i s approximated by a number of uniform sub-members. A s t r u c t u r e s t i f f n e s s matrix, with the e f f e c t of a x i a l force on d e f l e c t i o n s i n -cluded, i s generated from the sub-members. By s e t t i n g the determinant of t h i s matrix equal to zero the c r i t i c a l b uckling load and the buckled shape of the member i n the f i r s t mode are found. The s e c t i o n p r o p e r t i e s of the sub-members are then a l t e r e d , according to the constant s t r e s s c r i t e r i o n , so that the extreme f i b r e bending s t r e s s , determined from the f i r s t mode, i s the same i n each sub-member. The process i s repeated u n t i l the s t r e s s e s are s u f f i c i e n t l y c l o s e to being equal so that no f u r t h e r a l t e r a t i o n s are r e q u i r e d . The optimum shape i s taken from the l a s t i t e r a t i o n . The constant s t r e s s c r i t e r i o n i s based on the f a c t that, when c e r t a i n c o n d i t i o n s are s a t i s f i e d , the extreme f i b r e bending s t r e s s at any s e c t i o n i s constant along the length of the s t r o n g e s t column when i t i s buckled i n the f i r s t mode. The matrix method gives r e s u l t s i n very c l o s e agreement with those found by previous authors for cases where the constant s t r e s s c r i t e r i o n i s v a l i d . For the one example presented where the constant s t r e s s - i i -c r i t e r i o n was not v a l i d , the column b u c k l i n g under i t s own weight, the matrix method gave poor r e s u l t s . TABLE OF CONTENTS A b s t r a c t L i s t of Figures Acknowledgements Page 1. I n t r o d u c t i o n 1 2. Review of Previous I n v e s t i g a t i o n a) Previous I n v e s t i g a t o r s 4 b) Governing Equations 5 3. Extension to Previous I n v e s t i g a t i o n a) Necessary C o n d i t i o n f o r Constant S t r e s s C r i t e r i o n 14 b) E l a s t i c R e s t r a i n t Boundary Co n d i t i o n s 16 4. matrix method 21 5«, Examples Demonstrating the Accuracy of the matrix method 24 6. Conclusions 32 ( References 36 LIST OF FIGURES Page F i g . 1 8 F i g . 2 Optimum Shape for Fixed-Free Column of S i m i l a r S e c t i o n s 13 F i g . 3 17 F i g . 4 Optimum Shape f o r Fixed-Free Column of S i m i l a r S e c t i o n s 25 F i g . 5 27 F i g . 6 Volume of the Optimum Column of S i m i l a r Sections Buckling Under i t s Own Weight 29 F i g . 7 Shape Obtained by Matrix Method f o r Column of S i m i l a r Sections Buckling Under i t s Own Weight 31 ACKNOWLEDGEMENTS The author wishes to thank h i s s u p e r v i s o r , Dr. R.F. Hooley, and a l s o Dr. D.L. Anderson, f o r t h e i r encouragement and guidance during the development of t h i s work. Gratitude i s a l s o expressed to the Na t i o n a l Research Co u n c i l of Canada for some f i n a n c i a l support, and a l s o to the U.B.C. Computing Centre f o r the use of i t s computer. January, 1968 Vancouver, B r i t i s h Columbia. The Strongest Column; A matrix Approach Chapter 1. I n t r o d u c t i o n I t i s w e l l known that the Euler b u c k l i n g load of a column can be i n -creased by ta p e r i n g the s i d e s of the member along i t s l e n g t h . For example, a pin-ended column which i s roughly the shape of a c i g a r , w i l l have a greater c r i t i c a l load than a column of constant s e c t i o n which has the same length and volume of m a t e r i a l . The s t r o n g e s t column problem i s d e f i n e d f o r t h i s t h e s i s as the de-termi n a t i o n of the column shape which gives the maximum E u l e r buckling load f o r a given length, volume of m a t e r i a l , type of c r o s s - s e c t i o n and type of t a p e r i n g . The shape so found i s def i n e d as the optimum shape. In the a n a l y s i s to follow members ,are considered to buckle i n one plane o n l y . The method developed i n t h i s t h e s i s f o r the determination of the st r o n g e s t column i s a p p l i c a b l e only to those columns which s a t i s f y the f o l l o w i n g c o n d i t i o n s . The c e n t r o i d of every s e c t i o n must be on a s t r a i g h t l i n e which forms the l o n g i t u d i n a l a x i s of the member. In other words, s i n g l e mem-bers only are considered while curved members are not permitted. The major p r i n c i p a l a x i s of every cross s e c t i o n along the length of the member must l i e i n a plane which passes through the l o n g i t u d i n a l axis of the member. In other words, a member cannot be t w i s t e d . The f o l l o w i n g r e l a t i o n between the moment of i n e r t i a , l ( x ) , and the area, A(x) K x ) = * A ( x ) ~ 1 / n (1) - 2 -must hold, where a and n are constants. An example of a s e c t i o n which does not s a t i s f y ( l ) i s a wide-flange shape where the width and t h i c k -ness of the flanges are kept constant but the depth of the web v a r i e s along the length of the member. In t h i s case the exponent of the func-t i o n A(x) i n ( l ) would not be a constant but would also be a f u n c t i o n of x, the l o c a t i o n along the l o n g i t u d i n a l a x i s . The values of the constants a and n i n ( l ) are determined from the shape of the c r o s s - s e c t i o n of the member and the manner i n which the member i s tapered along i t s l e n g t h . To i l l u s t r a t e how the value of n i s determined, consider a column which has a s o l i d r e c t a n g u l a r s e c t i o n . I f the width of the column i s kept constant while the depth i s v a r i e d along the length, then n = -^/Z. This type of t a p e r i n g i s d e f i n e d , f o r t h i s t h e s i s , as s e c t i o n s of con-s t a n t width. I f both dimensions of the s e c t i o n are varied along the length of the member i n such a way that the aspect r a t i o between them i s constant, then n = -^/2. This type of t a p e r i n g defines s i m i l a r s e c t i o n s which may be v i s u a l i z e d as ones where every s e c t i o n i s a photo-graphic enlargement or re d u c t i o n of every other, A f u r t h e r c o n d i t i o n which must be s a t i s f i e d i s the c o n d i t i o n that the column has only one type of ta p e r i n g along i t s length. This means that a column could not have s e c t i o n s of constant width for one-half of i t s length, f o r i n s t a n c e , and s i m i l a r s e c t i o n s f o r the remainder. The l a s t c o n d i t i o n r e q u i r e s that the column be under a s i n g l e con-stant a x i a l load a p p l i e d at the ends. The weight of the member i t s e l f i s neglected. One case of s p e c i a l i n t e r e s t which does not s a t i s f y t h i s c o n d i t i o n i s presented i n Chapter 5 as a column buckling under i t s own - 3 -weight. Most previous i n v e s t i g a t o r s [1, 2, 4, 5, 6, 7 ] a have approached the st r o n g e s t column problem by means of v a r i a t i o n a l c a l c u l u s . A b r i e f review of the r e s u l t s and extent of t h e i r i n v e s t i g a t i o n s i s given i n Section (a) of Chapter 2. This t h e s i s presents a new approach to the st r o n g e s t column problem which w i l l be c a l l e d the matrix method. The method i s approximate and determines a s o l u t i o n by i t e r a t i o n using s t r u c t u r a l s t a b i l i t y matrices. The b a s i c c r i t e r i o n used i n the matrix method to determine the o p t i -mum shape, i s the c o n d i t i o n of constant s t r e s s , which r e q u i r e s the maxi-mum bending s t r e s s to be constant along the length of the member when buckled i n i t s fundamental mode. Chapter 3, Se c t i o n (a) develops the co n d i t i o n s t h a t must be s a t i s f i e d f o r the constant s t r e s s c r i t e r i o n to be v a l i d . The examples considered by previous i n v e s t i g a t o r s have a l l had end co n d i t i o n s which were some s t a b l e combination of f i x e d , pinned or f r e e . I t i s shown i n Chapter 3, Se c t i o n (b) that the strongest column can al s o be determined f o r some combinations of end c o n d i t i o n s which i n c l u d e springs or e l a s t i c r e s t r a i n t s a gainst t r a n s l a t i o n or r o t a t i o n . Numbers i n brackets designate References at the end of the t h e s i s . - 4 -C h a p t e r 2 . Review of P r e v i o u s I n v e s t i g a t i o n a) P r e v i o u s I n v e s t i g a t o r s In t h i s s e c t i o n a r e v i e w o f p r e v i o u s a u t h o r s and the e x t e n t of t h e i r i n v e s t i g a t i o n s i s g i v e n i n c h r o n o l o g i c a l o r d e r . The b e g i n n i n g s o f the s t r o n g e s t column problem can be t r a c e d to Lagrange [ 1 ] , i n 1773, who f i r s t c o n s i d e r e d the problem by means o f v a r i -a t i o n a l c a l c u l u s but came to an i n c o r r e c t s o l u t i o n . In 1 8 5 1 , C l a u s e n [ 2 ] c o r r e c t l y d e t e r m i n e d the s t r o n g e s t co lumn, by v a r i a t i o n a l c a l c u l u s , f o r any p i n - e n d e d column of s i m i l a r s e c t i o n s . F e i g e n [ 3 ] , i n 1 9 5 2 , d e t e r m i n e d the optimum shape f o r a p i n - e n d e d round t h i n - w a l l e d tube o f c o n s t a n t w a l l t h i c k n e s s . In t h i s example the v a l u e o f n i n e q u a t i o n ( l ) i s e q u a l to - ^ / 3 . Hence, w i t h r e g a r d to the type of t a p e r i n g , t h i s example i s e q u i v a l e n t to the s o l i d r e c t a n g u l a r column of c o n s t a n t w i d t h . However, F e i g e n ' s d e r i v a t i o n was based on the assumpt ion t h a t the c o n d i t i o n o f c o n s t a n t s t r e s s was v a l i d . A p r o o f , showing t h i s a s s u m p t i o n to be c o r r e c t , was not p r e s e n t e d u n t i l l a t e r by Tad jbakhsh and K e l l e r [ 5 ] , In 1 9 6 0 , K e l l e r [4 ] used v a r i a t i o n a l means to d e t e r m i n e the same r e s u l t as C l a u s e n [2] f o r p i n - e n d e d columns w i t h s i m i l a r s e c t i o n s . In 1 9 6 1 , Tad jbakhsh and K e l l e r [5 ] ex tended K e l l e r ' s p r e v i o u s work on columns o f s i m i l a r s e c t i o n s to de te rmine the optimum shapes c o r r e s p o n -d i n g to the boundary c o n d i t i o n s where one end i s f i x e d and the o t h e r i s f i x e d , p i n n e d or f r e e . T a d j b a k h s h and K e l l e r went on to f o r m u l a t e the s t r o n g e s t column problem i n a more g e n e r a l f o r m . They d e r i v e d c e r t a i n n e c e s s a r y c o n d i t i o n s wh ich must be s a t i s f i e d i n o r d e r f o r the s t r o n g e s t column to have the a b s o l u t e maximum b u c k l i n g l o a d . A summary o f t h e i r - 5 -r e s u l t s i s given i n g r e a t e r d e t a i l i n the next s e c t i o n . In 1966, K e l l e r and Niordson [6] considered a f i x e d - f r e e column of s i m i l a r s e c t i o n s b u c k l i n g under i t s own weight. This case w i l l be covered more thoroughly i n Chapter 5. By way of an energy approach T a y l o r [ 7 ] , i n 1967, reproduced the governing equations f o r the strongest column problem that were determined p r e v i o u s l y by Tadjbakhsh and K e l l e r . b) Governing Equations In t h i s s e c t i o n the strongest column problem i s formulated and a method of s o l u t i o n , using v a r i a t i o n a l c a l c u l u s , i s i n d i c a t e d . The pro-cedure i s taken from tha t used by Tadjbakhsh and K e l l e r [ 5 ] . C e r t a i n a l t e r a t i o n s have been made to t h e i r procedure for the sake of concise pre-s e n t a t i o n i n t h i s t h e s i s . The f o l l o w i n g p r e s e n t a t i o n i s v a l i d only for columns with a s i n g l e a x i a l l o a d a p p l i e d at the ends and for boundary c o n d i t i o n s which can be expressed i n a s p e c i f i e d form to be i n t r o d u c e d l a t e r . The theory i n t h i s s e c t i o n i s presented f o r the sake of c o n t i n u i t y i n the development of the theory i n the f o l l o w i n g chapter p e r t a i n i n g to the c r i t e r i o n of constant s t r e s s and e l a s t i c a l l y r e s t r a i n e d end c o n d i t i o n s . The f u r t h e r development i s based on the same equations and uses the same symbols. The sequence followed i n the f o r m u l a t i o n and s o l u t i o n of the problem i s b r i e f l y o u t l i n e d below. a) F i r s t the equations necessary f o r the formulation are presented. - 6 -b) In order to solve these equations new v a r i a b l e s are introduced and i n s e r t e d i n t o the i n i t i a l equations. The problem i s then i n a more convenient form f o r s o l u t i o n . c) Next, c o n d i t i o n s are presented which are necessary f o r the s o l u t i o n to y i e l d the maximum c r i t i c a l l o a d . d) F i n a l l y , a method of determining a s o l u t i o n s a t i s f y i n g these nece-ss a r y c o n d i t i o n s , i s i n d i c a t e d . The equations necessary f o r the form u l a t i o n of the problem are given i n the f o l l o w i n g s u b - s e c t i o n s . a) The equation of l a t e r a l e q u i l i b r i u m of a column i s : ^ § + P ^ f = 0 (2) dx dx where P i s an a x i a l l o a d , and w(x) denotes the l a t e r a l d e f l e c t i o n of the column from i t s s t r a i g h t p o s i t i o n . b) The bending moment ffl(x) i s given by IY1 = E l ( x ) d 2 w (3) where E i s the modulus of e l a s t i c i t y and l ( x ) i s the moment of i n e r t i a of the column i n the plane of b u c k l i n g . c) As seen p r e v i o u s l y i n Chapter 1, the type of tapering and c r o s s -s e c t i o n a l shape of a column s p e c i f y the values of 0/ and n i n the r e l a t i o n 1/ I(x) = a-A(x)~ / P (1) - 7 -d) The c o n d i t i o n f o r constant volume of m a t e r i a l r e l a t e s the volume V , to the area, A(x), according to f v n A(x)dx = V (4) e) F i n a l l y , four homogeneous boundary c o n d i t i o n s , expressed i n terms of w(x) and i t s d e r i v a t i v e s , are needed to complete the set of equations r e q u i r e d f o r the f o r m u l a t i o n of the problem. When ( l ) and (3) are i n s e r t e d i n (2), i t becomes an ordinary f o u r t h order homogeneous d i f f e r e n t i a l equation i n UJ(X). A 2 ( x ) d 4m(x) + 4A(x) dA(x) d 3 x ( x ) + 2A(x) d 2A(x) d 2yj(x) 4 3 2 2 dx dx dx dx dx 2JdA(x ) 1 2 d 2 i u(x) P d 2m(x) , » + dx J 2 + r „ 2 dx to; dx I f UJ(X) i s re q u i r e d to s a t i s f y the homogeneous boundary c o n d i t i o n s then there w i l l be a n o n - t r i v i a l s o l u t i o n f o r w(x) only i f P i s an eigenvalue. The lowest eigenvalue represents the c r i t i c a l buckling l o a d . I t i s d e s i r e d to f i n d the f u n c t i o n A(x) which maximizes the buckling load among a l l the f u n c t i o n s s a t i s f y i n g the constant volume c o n s t r a i n t ( 4 ) . That f u n c t i o n determines the optimum shape. In order to s o l v e f o r A(x) i t i s convenient to i n t r o d u c e new v a r i a b l e s . ? = x/L (6) $ ( ? ) = L 2 A ( x ) ^ d 2w(x) (7) 2 dx - 8 -PL A ( 0 = A(x) (8) (9) Upon i n s e r t i n g the new v a r i a b l e s i n t o ( 5 ) , a second order d i f f e r -e n t i a l equation i n i s obtained. 2 H i l l + \ A ( | ) ^ *(0 = 0 - t (10) The constant volume c o n s t r a i n t now becomes i n terms of | ^ A(c)d^ = V/L (11) The boundary c o n d i t i o n s expressed i n terms of the new v a r i a b l e §(§) are a_$(l) - a 2d$(0) + a 4$(0) = 0 (12) (13) where the are chosen a p p r o p r i a t e l y for the p a r t i c u l a r example. To demonstrate how the a- are chosen, c o n s i d e r the f i x e d - f i x e d I column shown i n F i g . 1. f F i g . 1 - 9 -The boundary c o n d i t i o n s f o r t h i s example i n terms of w(x) and i t s d e r i v a t i v e s are at x = 0 ui(o) = 0 (14a) dw(0) = 0 (14b) dx and at x = L UJ(L) = 0 (15a) dw(l_) = 0 (15b) dx The boundary c o n d i t i o n s (14) and (15) can be expressed i n terms °f <£•(§) by i n t e g r a t i n g (10) once with respect to | from | = 0 to | = _ 1 / n 2 2 f i r s t r e p l a c i n g A ( ^ ) * ( | ) by L d UJ(X) according to (7) and (9), and then i n s e r t i n g (14b) to obtain dw(x) = i _ l ^ * ( 0 ) _ ( 1 6 ) dx \ L d | dl? Now i n t e g r a t e (16) with respect to £ from | = 0 to g = | and use (14a) to o b t a i n Having used the boundary c o n d i t i o n s (14) at x = 0 to obtain (16) and (17), the boundary c o n d i t i o n s i n terms of §(|) can now be found from the remaining boundary c o n d i t i o n s (15) at x = L . S e t t i n g (16) i n t o (15b) and (17) i n t o (15a) the boundary c o n d i t i o n s i n terms of $ ( 5 ) are obtained. d£(0) _ d*(l) = Q ( 1 8 ) d$(0) _ 6(1) + $ ( 0 ) = 0 (19) - 10 -By s o l v i n g (19) for d_(0) and s u b s t i t u t i n g i n t o (18), a more convenient form of the boundary c o n d i t i o n s can be obta i n e d . $(0) + d_>(l) - 5(1) = 0 (20) $(1) - d_(0) - *(0) = 0 (21) d | Comparing (20) with (12) and (21) with (13), i t can be seen th a t the f o r the f i x e d - f i x e d column are «1 = 1 = 1 c>3 = -1 of, - -1 The c o n d i t i o n s r e q u i r e d f o r the formulation of the strongest column problem have now been expressed i n terms of the new v a r i a b l e s . The object now i s to f i n d a s o l u t i o n of the second order system represented by the equation (10) and the boundary c o n d i t i o n s (12) and (13), which a l s o s a t i s f i e s the constant volume c o n s t r a i n t ( l l ) . The s o l u t i o n w i l l be composed of an eigenvalue \ q , and corresponding functions A Q ( § ) and § 0 ( ? ) . By means of v a r i a t i o n a l c a l c u l u s Tadjbakhsh and K e l l e r d e r ive the expressions * 2 ( § ) = A ( § ) ( n " " l ) / n (22) and n < 1 (23) - 11 -which must be s a t i s f i e d as necessary c o n d i t i o n s i n order for the s o l u t i o n to y i e l d the eigenvalue that i s the maximum of the lowest eigenvalues. This means that i f the c o n d i t i o n s (22) and (23) are s a t i s f i e d by a s o l u t i o n to (10) - (13) then that s o l u t i o n i s the only s o l u t i o n and the eigenvalue so found i s the maximum of the lowest eigenvalues. This s o l u -t i o n w i l l y i e l d the gre a t e s t c r i t i c a l load and the corresponding optimum shape f o r a column of a given l e n g t h , volume and end c o n d i t i o n s . A l l the equations and c o n d i t i o n s necessary f o r the development i n l a t e r parts of t h i s t h e s i s , have now been derived or presented. Thus the purpose i n presenting the procedure taken from Tadjbakhsh and K e l l e r has been s a t i s f i e d . However, f o r the sake of completeness, the remainder of Tadjbakhsh and K e l l e r ' s method of determining a s o l u t i o n i s b r i e f l y o u t l i n e d i n the f o l l o w i n g three s u b - s e c t i o n s . a) A second order n o n - l i n e a r d i f f e r e n t i a l equation i n $(^) i s obtained by using (22) to e l i m i n a t e A(|) i n (10) n+l d 2a(g) + X$(^) n" 1 = 0 b) The above n o n - l i n e a r equation, the boundary c o n d i t i o n s (12) and (13), and the c o n s t r a i n t of constant volume ( l l ) , are a l l determined i n terms of a new v a r i a b l e i n order to f a c i l i t a t e the s o l u t i o n of t h i s s e t of equations. c) This s et of equations, now expressed i n terms of the new v a r i a b l e , i s solved and from the s o l u t i o n the optimum shape, A(x), and the maximum c r i t i c a l buckling load, P, are obtained. - 12 -The optimum shape of a f i x e d - f r e e column of s i m i l a r s e c t i o n s , which was obtained by Tadjbakhsh and K e l l e r ' s method, i s shown i n F i g . 2. For r e f e r e n c e , the uniform column having the same c r i t i c a l load and c r o s s -s e c t i o n a l shape as the optimum column i s a l s o p l o t t e d . In t h i s case the volume of the optimum column i s 0.866 times the volume of the uniform column of equal s t r e n g t h . - 13 -OPTIMUM SHAPE FOR FIXED-FREE COLUMN OF SIMILAR SECTIONS A ( x 1 0 ) I? FIG. 2 - 14 -Chapter 3. Extension to Previous I n v e s t i g a t i o n a) Necessary Conditions f o r the Constant Stress C r i t e r i o n I t i s proposed i n t h i s t h e s i s to s o l v e the s t r o n g e s t column problem by the matrix method using the c r i t e r i o n of constant s t r e s s . The object now i s to determine under what c o n d i t i o n s the constant s t r e s s c r i t e r i o n w i l l y i e l d the s t r o n g e s t column. In the l a s t chapter i t was s t a t e d that i f the c o n d i t i o n s a ( ? ) 2 = fl(?)^ ( 2 2 ) and n < 1 (23) are s a t i s f i e d by a s o l u t i o n to (10) - (13) then that s o l u t i o n i s the only s o l u t i o n and the eigenvalue so found i s the maximum of the lowest eigen-values. I t w i l l be shown that an expression r e p r e s e n t i n g the constant s t r e s s c o n d i t i o n can be derived from (22). I f t h i s expression i s s a t i s f i e d then the constant s t r e s s c o n d i t i o n i s s a t i s f i e d . The f o l l o w i n g i d e n t i t i e s , which have been seen p r e v i o u s l y , are a l s o used. IKl(x) = E l ( x ) d 2w(x) (3) dx 2 _1 * ( | ) = L 2 A ( x ) - f r d__u(x) (7) dx 2 A(|) = A(x) (9) S e t t i n g (3) and (7) i n t o the l e f t - h a n d s i d e of (22), and (9) i n t o - 15 -the r i g h t - h a n d s i d e , e q u a t i o n .(22) becomes 2 n - 1 L 4 A ( x ) ~ > ( x ) I2 A(x) " [ E l ( x ) - ' T a k i n g the square r o o t and s o l v i n g f o r ITl(x) K x ) n+1 ffl(x) = E_ A C x ) 7 ^ (24) K x ) L 2 In o r d e r to proceed i t i s n e c e s s a r y to i n t r o d u c e a new e x p r e s s i o n w h i c h l i n k s the a r e a o f a member, A ( x ) , w i t h the d i s t a n c e from the c e n -t r o i d of the member to the ext reme f i b r e , c ( x ) , i n the p l a n e of b u c k l i n g , c ( x ) = (3A(x)m (25) where Q and m are c o n s t a n t s de te rmined by the c r o s s - s e c t i o n a l shape and type o f t a p e r i n g of the member. M u l t i p l y i n g both s i d e s of (24) by c ( x ) and i n s e r t i n g (25) i n t o the r i g h t - h a n d s i d e , (24) becomes m+(n+l) Hl(x)c(x) = REA(x) 2 n (26) K x ) L 2 R e c o g n i z i n g the l s f t - h a n d s i d e of (26) as the bending s t r e s s i n the extreme f i b r e s , i t i s o b v i o u s t h a t the exponent o f the A (x ) f u n c t i o n must e q u a l z e r o i f the s t r e s s i s t o be c o n s t a n t . T h e r e f o r e the e x p r e s s i o n m + (n+l ) = 0 (27) 2n - 16 -must be s a t i s f i e d i f the c o n d i t i o n of constant s t r e s s i s to be s a t i s f i e d . The expression (27) has been d e r i v e d from the necessary c o n d i t i o n f o r a maximum of the lowest eigenvalue, (22), and the expression, (25), which defines an a d d i t i o n a l property of the c r o s s - s e c t i o n , c ( x ) . There-fore the constant s t r e s s c o n d i t i o n w i l l be v a l i d f o r the column shape which i s found as a s o l u t i o n of (10) - (13) and s a t i s f i e s (22) and (23), provided that (27) i s a l s o s a t i s f i e d . This shape i s the optimum shape. In other words the c o n d i t i o n of constant s t r e s s can be regarded as a c h a r a c t e r i s t i c of the optimum shape of the stro n g e s t column when (27) i s s a t i s f i e d . Since there i s only one optimum shape s o l u t i o n when (22), (23) and a l l the previous c o n d i t i o n s are s a t i s f i e d , then a column shape f o r which the constant s t r e s s c o n d i t i o n i s v a l i d w i l l be the optimum shape i f (27) i s s a t i s f i e d a l s o . Thus i t can be concluded that the c r i t e r i o n of constant s t r e s s w i l l y i e l d the st r o n g e s t column when (23), (27) and the c o n d i t i o n s mentioned p r e v i o u s l y i n Chapter 1 are s a t i s f i e d . I t may r e a d i l y be seen that the optimum shape can be found by way of the constant s t r e s s c r i t e r i o n f o r columns of constant width (m = 1, n = -^/3) and s i m i l a r s e c t i o n s (m = ^"/2, n = ~^/2) by s e t t i n g t h e i r r e s p e c t i v e values of m and n i n t o (27). b) E l a s t i c R e s t r a i n t Boundary C o n d i t i o n s I t was shown p r e v i o u s l y i n Chapter 2, Section (b) that the boundary c o n d i t i o n s , expressed i n terms of f > ( | ) , are a.*(o) + <* 9da(i) + aMi) = 0 (12) ds ajSd) - cv 2d£(0) + a 4 § ( 0 ) = 0 (13) d § - 17 -where the ct^ depend on the p a r t i c u l a r example. To demonstrate how the were chosen, an example was given - a column f i x e d at both ends. The cases of end c o n d i t i o n s , considered by previous authors, which s a t i s f y (12) and (13), are: f i x e d - f i x e d , fixed-pinned, f i x e d - f r e e and pinned-pinned. I t can a l s o be shown that the case where one end i s f i x e d and the other i s f r e e i n t r a n s l a t i o n normal to the a x i s of the member, but f i x e d a g a i n s t r o t a t i o n , s a t i s f i e s (12) and (13). A type of end c o n d i t i o n which has not been e x p l i c i t l y considered before i s the spring or e l a s t i c r e s t r a i n t against t r a n s l a t i o n or r o t a t i o n . I f i t can be shown that a set of end c o n d i t i o n s which i n c l u d e a s p r i n g s a t i s f y the form of the boundary c o n d i t i o n s , (12) and (13), for a p a r t i c u l a r s e t of Q^, then a l l that has been s a i d p r e v i o u s l y concerning columns with f i x e d , pinned or free end c o n d i t i o n s i s e q u a l l y a p p l i c a b l e to the column with the e l a s t i c a l l y r e s t r a i n e d end c o n d i t i o n . The object now, then, i s to show that a column with a set of end c o n d i t i o n s which i n c l u d e a s p r i n g , s a t i s f i e s ! (12) and (13). Consider the example i l l u s t r a t e d i n F i g . 3, which has an e l a s t i c r e s t r a i n t against r o t a t i o n where k i s the s p r i n g constant. F i g . 3 - 18 -The bounday c o n d i t i o n s a r e a t x = 0 w(0) = 0 (28a) dju(O) = 0 (28b) dx and a t x = L w(l) = 0 (29a) BI(L) = - k d u ) ( L ) (29b) dx I t i s u s e f u l to note the s i m i l a r i t y between t h i s example , the f i x e d - s p r i n g c o l u m n , and the f i x e d - f i x e d column c o n s i d e r e d as an example i n C h a p t e r 2 , S e c t i o n ( b ) . N o t i n g t h a t the boundary c o n d i t i o n s (14) and (28) a t the end x = 0 , a re the same f o r both e x a m p l e s , i t can be seen from the d e r i v a t i o n g i v e n i n C h a p t e r 2 , S e c t i o n ( b ) , t h a t the e q u a t i o n s du)(x) = _ l [ d £ ( 0 ) - d*(g ) ] ( 1 6 ) dx XL dg d | and w(x) = i[e;d$(o) - $ ( | ) * * ( 0 ) ] X d | a l s o r e p r e s e n t the s l o p e and d e f l e c t i o n o f the f i x e d - s p r i n g column i n terms of $ ( § ) . One boundary c o n d i t i o n can be found d i r e c t l y by i n s e r t i n g " (29a) i n t o (17) t o o b t a i n « ( 1 ) - d § ( 0 ) - * (0) = 0 (30) The o t h e r can be found by o p e r a t i n g on (29b) through the use of the - 19 -f o l l o w i n g equations. m(x) = E l ( x ) d 2 w ( x ) (3) 2 dx $(0 = L 2 A " l / n ( x ) d _ _ i ( x ) (7) S u b s t i t u t i n g ffl(L) from ( 3 ) , (29b) becomes d 2w(L) = _ _k d_(L) ( 3 1 ) dx 2 EI(L) dx 2 S u b s t i t u t i n g d ui(L) from (7) and dw(l_) from (16), (31) becomes dx 2 dx (32) $(1) _ _ k ["__.(0) _ d a d ) ] L 2 A " l / n ( L ) EI(L)XL l d ? d | S o l v i n g (30) for d__(lO) and s u b s t i t u t i n g i n (32) EI ( L ) \ L . $(1) = - $ ( l ) + *(0) + d _ ( l ) k L 2 A " 1 / r t L ) d | and r e a r r a n g i n g $ ( • ) + d__(l) - P i + P_"fe(l) = 0 (33) d | L k J I t i s now apparent that (33) i s of the form (12) and (30) i s the form (13) when the are chosen as - 20 -I t i s i n t e r e s t i n g to compare these Q?^  with those obtained f o r the f i x e d - f i x e d column. I f the s p r i n g i n the f i x e d - s p r i n g column becomes i n f i n i t e l y s t i f f , or the s p r i n g constant, k, i n f i n i t e l y l a r g e , then the 0/3 term above approaches the value of -1 uihich i s p r e c i s e l y the value obtained f o r o/g i n the f i x e d - f i x e d case. Thus i t can be seen that the f i x e d - f i x e d column can be considered as a p a r t i c u l a r case, mith regard to end c o n d i t i o n s , of the somewhat more general f i x e d s p r i n g column presented here. The author has not proved that a l l p o s s i b l e combinations of end c o n d i t i o n s i n v o l v i n g springs s a t i s f y the general form of the boundary c o n d i t i o n s (12) and (13). However, the two other examples that the author has i n v e s t i g a t e d have both s a t i s f i e d (12) and (13). One example was a column which was completely f i x e d at one end, and at the other end was f i x e d against r o t a t i o n but e l a s t i c a l l y r e s t r a i n e d against t r a n s -l a t i o n . The other example was a column pinned at both ends with an e l a s t i c r e s t r a i n t a g a i n s t r o t a t i o n at one end. - 21 -Chapter 4. Matrix Method The matrix method mas developed as an a l t e r n a t i v e to the v a r i a -t i o n a l c a l c u l u s approach to provide a more convenient means of s o l v i n g s t r o n g e s t column problems. With the known a d a p t a b i l i t y of matrix methods to the d i g i t a l computer i n general, i t was a n t i c i p a t e d that the strongest column con-cept could be extended to s t r u c t u r e s more complex than the simple column, such as frames. The author has c a r r i e d i n v e s t i g a t i o n i n t o t h i s area, but with l i m i t e d success, due to problems encountered with p o i n t s of i n f l e c t i o n and s t a t i c a l indeterminacy i n c e r t a i n s t r u c t u r e s . However, a thorough d i s c u s s i o n of these subjects i s outside the scope of t h i s t h e s i s . In the matrix method a co n t i n u o u s l y tapered member i s approxi-mated by a number of sub-members. Each sub-member i s uniform along i t s own length but has d i f f e r e n t c r o s s - s e c t i o n a l p r o p e r t i e s from the adjacent sub-members. In t h i s manner a continuously tapered member can be approximated i n a step-wise f a s h i o n by a number of sub-members. I t i s obvious that the gre a t e r the number of sub-members, the f i n e r the approximation. The c r i t i c a l b u c k l i n g load and fundamental mode of the member are determined by way of w e l l known matrix methods. A s t r u c t u r a l s t a b i l i t y matrix [8] i s generated from the sub-members to obtain a matrix equation of the form where K i s the l i n e a r s t r u c t u r e matrix, P K, i s the s t r u c t u r e s t a b i l i t y o ' o 1 matrix, D i s the unknown column displacement vector, and F i s the known - 22 -vector of a p p l i e d loads. The c r i t i c a l value of P q i s found from the homogeneous equation E Ko + P o K l 3 l°\ = 0 which can be rearranged i n t o a more standard form of the eigenvalue problem as - [ K J " 1 [ K J \0) = 1 ID] (34) P o Since only the lowest c r i t i c a l load and corresponding mode shape are r e q u i r e d , a v a r i a t i o n on the Stodola method of f i n d i n g eigenvalues and eigenvectors by matrix i t e r a t i o n [9] i s used. To s t a r t the i t e r a t i o n procedure, an i n i t i a l guess f o r the D vector i s i n s e r t e d i n t o the l e f t -hand side of (34). The l e f t - h a n d s i d e of (34) i s m u l t i p l i e d out to obtain a right-hand s i d e which i s i n s e r t e d back i n t o the D vector on the L.H.S. f o r the second i t e r a t i o n . This procedure w i l l converge to the highest eigenvalue, l / P Q t o r the c r i t i c a l load corresponding to the fundamental mode. The o b j e c t now i s to arrange the c r o s s - s e c t i o n a l p r o p e r t i e s of the sub-members r e l a t i v e to one another so that the maximum bending s t r e s s at the mid-length of each sub-member i s the same accordi n g to the constant s t r e s s c r i t e r i o n . From the eigenvector the moment at the mid-length of each sub-mBmber i s c a l c u l a t e d . The s e c t i o n p r o p e r t i e s of the sub-members are then modified so that the r a t i o of the moment to the s e c t i o n modulus (which i s equal to the s t r e s s ) i s the same f o r each sub-member. - 23 -Thi s completes one i t e r a t i o n . A new s t r u c t u r e s t a b i l i t y matrix i s c a l c u l a t e d from the newly modified sub-members and the process i s repeated. When the s t r e s s e s are s u f f i c i e n t l y c l o s e to being equal the optimum shape i s taken from the l a s t i t e r a t i o n . The author has not been able to develop a rigorous proof of the convergence of t h i s i t e r a t i o n system. However, every example which has been t r i e d that met the requirements f o r the constant s t r e s s c r i t e r i o n to y i e l d the optimum shape, has converged. These examples i n c l u d e every s t a b l e combination of end c o n d i t i o n s f i x e d , pinned and f r e e , f o r columns of both s i m i l a r s e c t i o n s and constant width. Also columns with springs a g a i n s t r o t a t i o n at one end and f i x e d or pinned at the other, have been t r i e d f o r v a r i o u s values of s p r i n g constants. - 24 -C h a p t e r 5 . Examples D e m o n s t r a t i n g the A c c u r a c y of the M a t r i x Method The purpose o f t h i s c h a p t e r i s to demonst ra te the a c c u r a c y of the m a t r i x method th rough the use of two e x a m p l e s . The r e s u l t s found by the m a t r i x method are compared w i t h the e x a c t s o l u t i o n s found by way o f v a r i a t i o n a l c a l c u l u s . The f i r s t example i s the s t r o n g e s t f i x e d - f r e e column of s i m i l a r s e c t i o n s b u c k l i n g under a s i n g l e c o n s t a n t a x i a l l o a d a p p l i e d a t the e n d s . T h i s example was seen p r e v i o u s l y i n C h a p t e r 2 , S e c t i o n ( b ) . The optimum shape found from T a d j b a k h s h and K e l l e r ' s e x a c t s o l u t i o n [5 ] i s r e p r o d u c e d i n F i g . 4 from F i g . 2 . The smooth c u r v e r e p r e s e n t s the e x a c t s o l u t i o n . The s t e p p e d column r e p r e s e n t s the r e s u l t s o b t a i n e d from the m a t r i x method a f t e r f i v e i t e r a t i o n s u s i n g ten sub -members . An i n s p e c t i o n o f the graph shows t h a t the a r e a o f each sub-member, a t i t s m i d - l e n g t h , i s very c l o s e to the a r e a o f the e x a c t shape a t t h a t p o i n t . The g r e a t e s t d e v i a t i o n of the m a t r i x method from the e x a c t shape a t any p o i n t i s l e s s than one p e r c e n t o f the t r u e v a l u e a t t h a t p o i n t . The a reas under the cu rves shown i n F i g . 4 r e p r e s e n t the volumes of the r e s p e c t i v e c o l u m n s . I t was s t a t e d p r e v i o u s l y t h a t the volume of the s t r o n g e s t column was found to be 0 . 8 6 6 t i m e s the volume of the u n i f o r m column of e q u a l s t r e n g t h . The volume of the s tepped column i s 0 . 8 7 5 t imes the volume o f the u n i f o r m co lumn, wh ich i s one p e r c e n t g r e a t e r than t h e t r u e vo lume. The second example c o n s i d e r e d i s the s t r o n g e s t f i x e d - f r e e column of s i m i l a r s e c t i o n s b u c k l i n g under i t s own w e i g h t . One of the c o n d i t i o n s p r e s e n t e d i n Chapte r 1 , wh ich must be s a t i s f i e d - 25 -OPTIMUM SHAPE FOR FIXED-FREE COLUMN OF SIMILAR SECTIONS A (x10"5) if FIG. 4 - 26 -i n order f o r the constant s t r e s s c r i t e r i o n to be v a l i d , i s not s a t i s f i e d by t h i s example. In t h i s case the loads are not a p p l i e d at the ends of the column but are d i s t r i b u t e d along the length of the member and vary with the shape of the member. K e l l e r and Niordson [6] have a l s o considered the s t r o n g e s t column problem f o r the column bu c k l i n g under i t s own weight. K e l l e r and Niordson considered the s t r o n g e s t column to be the t a l l e s t column which can be found, s t a b l e a g a i n s t buckling, f o r a given volume of m a t e r i a l and c r o s s - s e c t i o n a l shape. Dealing only with columns of s i m i l a r s e c t i o n s they found th a t the height of the s t r o n g e s t column was 2.034 times the height of a uniform column of the same volume of m a t e r i a l and the same c r o s s - s e c t i o n a l shape. They did not, however, give an e x p l i c i t represen-t a t i o n of the optimum shape, e i t h e r by diagram or equation. They showed only t h a t an i t e r a t i o n method, p r e v i o u s l y developed by Niordson [10], could be adapted to t h i s purpose. K e l l e r and Niordson's length r a t i o of 2,034 can be converted to a volume r a t i o f o r columns of the same l e n g t h by means of a simple r e l a t i o n . This r e l a t i o n can be d e r i v e d from the w e l l known expression f o r a uniform column b u c k l i n g under a uniformly d i s t r i b u t e d a x i a l load [11]. ( q L ) c r = 7.837 E_I (35) For a uniform column of s i m i l a r s e c t i o n s , the area, volume and moment of - 27 -i n e r t i a can be expressed as A = C d V = C ] Ld 2L I = C 2 d 4 (36) (37) (38) where d i s a r e p r e s e n t a t i v e dimension of the c r o s s - s e c t i o n and C,, are constants. S u b s t i t u t i n g (36) - (38) i n t o (35) and s e t t i n g the load per u n i t l e n g t h , q, equal to the d e n s i t y of the m a t e r i a l , p, times the c r o s s - s e c t i o n a l area, A, a dimensionless r a t i o can be obtained EV (39) where C 3 i s a constant. Consider the three columns shown i n F i g . 5, a l l on the verge of buc k l i n g under t h e i r own weight. F i g . 5 Column Volume Length A V \ 1 rt I ft/ B V L C V L - 28 -These columns are af s i m i l a r s e c t i o n s , have the same c r o s s - s e c t i o n a l shape and are of the same m a t e r i a l . Column B i s the t a l l e s t column of volume V and columns A and C are uniform. K e l l e r and Niordson showed tha t ^ = 2.034 (40) Con s i d e r i n g columns A and C i t i s apparent from (39) that (41) U 4 ) S e t t i n g (40) i n t o ( 4 l ) gives V = V f i t = 0.0584 V (42) Applying (42) to columns A and B i t can be seen that, f o r columns of s i m i l a r s e c t i o n s , the same c r o s s - s e c t i o n a l shape and the same type of m a t e r i a l , the volume of the s t r o n g e s t column i s 0.0584 times the volume of the uniform column of the same height. S e v e r a l t r i a l s , with a d i f f e r e n t number of sub-members f o r each t r i a l , have been made using the matrix method to determine the optimum shape. The volume of the column, V, expressed as a f r a c t i o n of the volume, U q, of a uniform column of the same len g t h , has been p l o t t e d i n F i g . 6 versus the number of sub-members used f o r the t r i a l . Using 20 sub-members the volume of the column obtained by the matrix method i s 6Z% g r e a t e r than the value found by K e l l e r and Niordson. By e x t r a p o l a t i n g the curve, i t i s apparent that a c o n s i d e r a b l e discrepancy - 29 -VOLUME OF THE OPTIMUM COLUMN OF SIMILAR SECTIONS BUCKLING UNDER ITS OWN WEIGHT 25 k 20 15 (%) fO L 5 \-0 0 5.84 % 10 15 20 25 NUMBER OF S U B - M E M B E R S FIG. 6 - 30 -w i l l s t i l l e x i s t between the values obtained by the matrix method and the exact value f o r the str o n g e s t volumn even i f a much greater number of sub-members are used. In view of t h i s discrepancy, i t i s apparent that the a p p l i c a t i o n of the matrix method to columns which do not s a t i s f y a l l the c o n d i t i o n s r e q u i r e d f o r the v a l i d i t y of the constant s t r e s s c r i t e r i o n , can r e s u l t i n answers that are i n c o n s i d e r a b l e e r r o r . The shape obtained by the matrix method u s i n g 20 sub-members i s given i n F i g . 7 f o r the column of s i m i l a r s e c t i o n s buckling under i t s own weight. This shape i s not the optimum shape but i t i s the shape y i e l d i n g the l e a s t volume which the author has been able to ob t a i n using the matrix method. - 31 -SHAPE OBTAINED BY MATRIX METHOD FOR COLUMN OF SIMILAR SECTIONS BUCKLING UNDER ITS OWN WEIGHT d z D(X10"*) L FIG. 7 - 32 -Chapter 6 . Conclusions The s t r o n g e s t column problem i s d e f i n e d f o r t h i s t h e s i s as the determination of the column shape which gives the maximum E u l e r buck-l i n g load f o r a given l e n g t h , volume of m a t e r i a l , type of s e c t i o n and type of t a p e r i n g . This t h e s i s presents a new method - the Matrix Method - f o r s o l v i n g some stro n g e s t column problems. The i n i t i a l step i n the Matrix Method c o n s i s t s of approximating the member by a number of sub-members. A s t r u c t u r e s t i f f n e s s matrix, with the e f f e c t of a x i a l f o r c e on de-f l e c t i o n s i n c l u d e d , i s generated from the sub-members. By s e t t i n g the determinant of t h i s matrix equal to zero the eigenvalue r e p r e s e n t i n g the lowest c r i t i c a l b u c kling load i s found. The corresponding eigen-vector r e p r e s e n t s the buckled shape of the whole column i n the f i r s t mode. The s e c t i o n p r o p e r t i e s of the sub-members are then a l t e r e d accor-ding to the constant s t r e s s c r i t e r i o n (which w i l l be d i s c u s s e d more f u l l y l a t e r ) so that the extreme f i b r e bending s t r e s s generated from the f i r s t mode shape i s the same i n each sub-member. The process i s repeated u n t i l the s t r e s s e s are s u f f i c i e n t l y c l o s e to being equal so t h a t no f u r t h e r a l t e r a t i o n s are r e q u i r e d . The optimum shape i s taken from the l a s t i t e r -a t i o n . No convergence proof of the matrix method i s presented. However, for a l l the examples t r i e d , i n which the constant s t r e s s c r i t e r i o n was v a l i d , the matrix method converged to answers which were i n very c l o s e agreement with those found by previous authors. In the case presented, the f i x e d - f r e e column of s i m i l a r s e c t i o n s , the volume of the column found by the matrix method was one percent g r e a t e r than the volume found by - 33 -Tadjbakhsh and K e l l e r [ 5], For the example presented where the constant s t r e s s c r i t e r i o n was not v a l i d , the column b u c k l i n g under i t s own weight, the matrix method gave a volume 63% greater than the value found by K e l l e r and Niordson [ 6 ] . The matrix method i s only a p p l i c a b l e where the constant s t r e s s c r i t e r i o n i s v a l i d . This c r i t e r i o n was developed from the f a c t that the maximum bending s t r e s s at any s e c t i o n i s constant along the le n g t h of the s t r o n g e s t column when i t i s buckled i n the fundamental mode when c e r t a i n c o n d i t i o n s are s a t i s f i e d . These c o n d i t i o n s are: a) The member buckles i n one plane. b) The member must be s t r a i g h t , not twisted or curved. c) The member must have only one type of t a p e r i n g along i t s l e n g t h ( i . e . not constant width half-way and s i m i l a r s e c t i o n s half-way). d) The member must have only one type of c r o s s - s e c t i o n along i t s length ( i . e . not square half-way and c i r c u l a r half-way). e) The member must be under a s i n g l e constant a x i a l load a p p l i e d at the end» f ) The f o l l o w i n g r e l a t i o n between the moment of i n e r t i a , I, and the area, A, must hold K x ) = o- A ( x f l / n where a and n are constants and g) n < 1 h) The f o l l o w i n g r e l a t i o n between, c, the d i s t a n c e from the c e n t r o i d of the member to the extreme f i b r e and the area must hold. c(x) = p A ( x ) m where p and m are constants. - 34 -i ) The r e l a t i o n m + (n+l) = 0 2n must a l s o hold. j ) The boundary c o n d i t i o n s must s a t i s f y the f o l l o w i n g general form. 0 ^ ( 0 ) + a 2d$(l) + » 3$(1) = 0 a n * ( l ) - a 2d#(D) + a 4 $ ( 0 ) = 0 d S Previous authors have shown that the boundary c o n d i t i o n s of f i x e d -f i x e d , f i xed-pinned, f i x e d - f r e e and pinned-pinned s a t i s f y the above general form. The author has f u r t h e r shown that three cases of boundary c o n d i t i o n s with e l a s t i c r e s t r a i n t s a l s o s a t i s f y t h i s general form. These cases are: i ) f i x e d at one end and pinned at the other with a r o t a t i o n a l s p r i n g at the pinned end, i i ) pinned at both ends with a s p r i n g a gainst r o t a t i o n at one end. i i i ) f i x e d at one end with the other f i x e d a g a i n s t r o t a t i o n but e l a s t i c a l l y r e s t r a i n e d a g a i n s t t r a n s l a t i o n . A l l the above c o n d i t i o n s ( a - j ) must be s a t i s f i e d i n order f o r the constant s t r e s s c r i t e r i o n to y i e l d the stro n g e s t column. Having d e s c r i b e d the matrix method, i t s accuracy, and i t s r e s t r i c t -ing c o n d i t i o n s , two questions a r i s e . How general i s i t ? UJhat savings i n weight are achieved through the use of the matrix method? In answer to the f i r s t q u e s tion, an examination of the r e s t r i c t i n g c o n d i t i o n s shows that there are many p o s s i b l e cases of columns where the - 35 -loading and boundary c o n d i t i o n s do not s a t i s f y the necessary requirements. I t i s evident then, that the matrix method can only be used on a l i m i t e d range of problems. More work needs to be done i n t h i s area i n order to obtain a general method f o r f i n d i n g the column of l e a s t weight for the t r a n s f e r of load between two or more p o i n t s . In answer to the second question, the weight savings can be i l l u s -t r a t e d by an example. I t was p r e v i o u s l y found that the volume of the s t r o n g e s t f i x e d - f r e e column of s i m i l a r s e c t i o n s was 0,866 times the volume of the uniform column of equal s t r e n g t h . This means that i n the i d e a l case f o r t h i s example there i s a maximum 13.4$ r e d u c t i o n i n weight. Thi s maximum weight r e d u c t i o n cannot be r e a l i z e d i n a p r a c t i c a l a p p l i c a t i o n because, according to the theory presented i n t h i s t h e s i s , the optimum shape i s determined only from the bending s t r e s s e s due to buckling without regard f o r a x i a l s t r e s s e s . This means that where there i s zero bending moment the s t r o n g e s t column w i l l have zero area. Conse-quently i f a column i s to have s u f f i c i e n t area everywhere f o r a x i a l s t r e s s requirements then there must be l e s s than 13.4$ r e d u c t i o n i n weight. Thus i t may be concluded that, while the matrix method provides an a l t e r n a t i v e s approach to e x i s t i n g methods of s o l u t i o n , i t does not provide the answer to the general problem of determining the column of l e a s t weight r e g a r d l e s s of the type of s e c t i o n , t a p e r i n g or l o a d i n g . - 36 -References 1. Lagrange, J.L., "Sure l a f i g u r e des colonnes", M i s c e l l a n e a  T o u r i n e n s i a (Royal S o c i e t y of T u r i n ) , Tomus V, p. 123, 1770-1773. Summarized i n I. Todhunter and K, Pearson, "A History of the E l a s t i c i t y and Strength of M a t e r i a l s " , Cambridge U n i v e r s i t y Press, Cambridge, England, V o l . 1, pp. 66-67, 1886. 2. Clausen, T., "Uber die Form a r c h i t e k t o n i s c h e r Saulen", B u l l e t i n Physico-mathematiques et Astronomigues, Tome 1, pp. 279-294, 1849-1853. Summarized i n I. Todhunter and K. Pearson, Cambridge, England, V o l . 2, pp. 325-329, 1893. 3. Feigen, M., "Minimum Weight of Tapered Round Thin-Walled Columns", J o u r n a l of Applied Mechanics. Trans. ASME, V o l . 74, pp. 375-380, 1952. 4. K e l l e r , J.B., "The Shape of the Strongest Column", Archive  f o r R a t i o n a l Mechanics and A n a l y s i s , V o l . 5, pp. 275-285, 1960. 5. Tadjbakhsh, I., and K e l l e r , J.B., "The Strongest Column and I s o p e r i m e t r i c I n e q u a l i t i e s f o r Eigenvalues", J o u r n a l of  App l i e d Mechanics, V o l . 29, No. 1, Trans. ASME, V o l . 84, S e r i e s E, pp. 159-164, Mar. 1962. 6. K e l l e r , J.B., and Niordson, F.I., "The T a l l e s t Column", J o u r n a l of Mathematics and Mechanics, V o l . 16, No. 5, pp. 433-452, 1966. 7. T a y l o r , J.E., "The Strongest Column: An Energy Approach", J o u r n a l of Applied Mechanics, Trans. ASME, V o l . 34, No. 2, pp. 486-487, June 1967. 8. Hartz, B.J., "Matrix Formulation of S t r u c t u r a l S t a b i l i t y Problems", J o u r n a l of the S t r u c t u r a l D i v i s i o n , ASCE, V o l . 91, No. ST6, pp. 141-157, Dec. 1965. 9. Karman, T. and B i o t , M., "Mathematical Methods i n Engineering McGrauj-Hill Inc., New York, N.Y. pp. 313, 1940. 10. Niordson, F.I., "On the Optimal Design of a V i b r a t i n g Beam", Quart. App. Math., V o l . 23, pp. 47-53, 1965. 11. 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. 103, 1961. 

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