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UBC Theses and Dissertations

Buckling of a ring in an elastic foundation Choukalos, William 1964

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B U C K L I N G O F A R I N G I N A N E L A S T I C F O U N D A T I O N by WILLIAM CHOUKALOS 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, 1950 A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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 UNIVERSITY OF BRITISH COLUMBIA May, 1964 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 reference and study* I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that,- copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written' permission*. Department of CIVIL ENGINEERING The U n i v e r s i t y .of B r i t i s h Columbia, Vancouver 8 . Canada ABSTRACT This t h e s i s d e r i v e s the d i f f e r e n t i a l equation f o r the i n plane e x t e n t i o n l e s s b u c k l i n g of a r i n g i n an e l a s t i c f o u n d a t i o n . The e l a s t i c f oundation can exert f o r c e s p r o p o r t i o n a l to r a d i a l and t a n g e n t i a l displacements and a couple p r o p o r t i o n a l to r o t a t i o n . The e x t e r n a l uniform load i s always d i r e c t e d toward a f i x e d point on the i n i t i a l r a d i u s . A s p e c i a l case of t h i s i s h y d r o s t a t i c pressure where the load remains normal to the r i n g . In other cases the load may remain p a r a l l e l to the i n i t i a l r a d i u s or remain d i r e c t e d t o -wards the i n i t i a l c e n t r e . Complete s o l u t i o n s are presented f o r a f u l l r i n g and s e v e r a l graphs of c r i t i c a l pressure versus f o u n d a t i o n s t i f f n e s s i n d i c a t i n g the general behaviour. A f u l l s o l u t i o n i s given f o r a p a r t i a l r i n g with hinged sup-ports along with a number of graphs. A method i s pre-sented but no s o l u t i o n s are given f o r a p a r t i a l r i n g with f i x e d supports. F i n a l l y , a s o l u t i o n i s given f o r the case of a f u l l r i n g under h y d r o s t a t i c load with r a d i a l e l a s t i c sup-ports having a d i f f e r e n t s p r i n g constant f o r inward and outward displacements. This s o l u t i o n , presented g r a p h i c a l l y f o r a l l combination of s p r i n g constants, can r e p r e s e n t the b u c k l i n g of a c u l v e r t under a high f i l l . ACKNOWLEDGEMENTS The author wishes to express h i s indebtedness to h i s a d v i s o r , Dr. R.F. Hooley, f o r h i s i n v a l u a b l e a s s i s -tance i n the f o r m u l a t i o n of the b a s i c ideas and f o r the guidance given during the r e s e a r c h and prepara-t i o n of t h i s t h e s i s . He a l s o wishes to acknowledge the use of the d i g i t a l computer at the U n i v e r s i t y of B r i t i s h Columbia without which the l a s t part of t h i s t h e s i s could not have been attempted. Wm.C . May 7, 1964 Vancouver, B r i t i s h Columbia i i i TABLE OF CONTENTS Page CHAPTER I INTRODUCTION 1 CHAPTER I I DERIVATION OF THE DIFFERENTIAL EQUATION 3 CHAPTER I I I SOLUTION OF FULL RING CASE 11 CHAPTER IV SOLUTION OF PARTIAL RING CASE 1. Hinged Supports 18 2. F i x e d Supports 20 CHAPTER V NON-LINEAR FOUNDATION MODULUS 23 CHAPTER VI CONCLUSIONS 33 APPENDIX 1. FORTRAN LISTING AND BLOCK DIAGRAM FOR COMPUTER PROGRAM - - 34 APPENDIX 2. NUMERICAL EXAMPLES 43 BIBLIOGRAPHY 47 TABLE OF SYMBOLS Geometry r = r a d i u s of r i n g w = contained angle between supports i = poi n t on the r i n g 8 = angular measurement 3 = length c o e f f i c i e n t f o r r a d i a l s p r i n g o = length c o e f f i c i e n t f o r t a n g e n t i a l s p r i n g a = length c o e f f i c i e n t e s t a b l i s h i n g d i r e c t i o n of a p p l i c a t i o n of c r i t i c a l l o a d i n g n = number of modes of the buckled shape of the r i n g Deformations u = r a d i a l displacements v = t a n g e n t i a l displacements cp = angular r o t a t i o n e = t a n g e n t i a l s t r a i n A = change of curvature S e c t i o n Proper t i e s cross s e c t i o n a l area moment of i n e r t i a of of r i n g cross s e c t i o n of r i n g V F or ces P = a p p l i e d e x t e r n a l uniform pressure p = net e x t e r n a l uniform pressure p c r = c r i t i c a l net e x t e r n a l uniform pressure q r = r a d i a l e l a s t i c f oundation r e a c t i o n M = bending moment V = shear f o r c e C = compressive f o r c e E l a s t i c P r o p e r t i e s E = Youngs modulus u = Poisson's r a t i o Foundation Constants K r = r a d i a l s p r i n g constant of e l a s t i c f oundation K t = t a n g e n t i a l s p r i n g constant of e l a s t i c f o u n d a t i o n = t o r s i o n a l s p r i n g constant of e l a s t i c f oundation Mis ce 1laneous x = c o e f f i c i e n t of d i f f e r e n t i a l equation 6 = c o e f f i c i e n t of d i f f e r e n t i a l equation m = r o o t s of c h a r a c t e r i s t i c equation B = c o e f f i c i e n t a = c o e f f i c i e n t b = c o e f f i c i e n t p = r a t i o of c o e f f i c i e n t s 4_6 v = foun d a t i o n modulus term BUCKLING OF A RING IN AN ELASTIC FOUNDATION CHAPTER I INTRODUCTION Problems r e q u i r i n g the dete r m i n a t i o n of the c r i t i c a l e x t e r n a l uniform l o a d i n g f o r a c i r c u l a r r i n g occur f r e q u e n t l y i n the f i e l d s of s t r u c t u r a l a n a l y s i s or en g i n e e r i n g mechanics and as a consequence have been e x t e n s i v e l y t r e a t e d i n standard r e f e r e n c e books. The treatment i n t e c h n i c a l l i t e r a t u r e f o r the b u c k l i n g of a r i n g i n an independent e l a s t i c foundation when loaded by a uniform e x t e r n a l pressure has been l i m i t e d to s p e c i a l l o a d i n g c o n d i t i o n s or has been r e s t r i c t i v e i n the form of the independent e l a s t i c f o u n d a t i o n . We d e f i n e an independent e l a s t i c foundation as one i n which an i n f l u e n c e on the e l a s t i c f o u n d a t i o n at one point produces no dependent i n f l u e n c e on the e l a s t i c f o u n d a t i o n at another p o i n t . T h i s t h e s i s t r e a t s v a r i o u s aspects of the b u c k l i n g of a r i n g i n an inde-pendent e l a s t i c f o u n d a t i o n . 1 2 In the f i r s t part of the t h e s i s , a s i x t h order d i f -f e r e n t i a l equation i s d e r i v e d from which a general s o l u t i o n f o r the b u c k l i n g load of a r i n g i n an e l a s t i c f o u n d a t i o n i s obtained f o r a f u l l r i n g . A l s o i n c l u d e d i s a general s o l u t i o n f o r the uniform c r i t i -c a l load f o r p a r t i a l r i n g s with hinged supports. A f i f t h order d i f f e r e n t i a l equation, i n terms of the t a n g e n t i a l displacements, i s presented from which r e s t r i c t e d s o l u t i o n s can be obtained f o r the b u c k l i n g load i n a f i x e d ended p a r t i a l r i n g supported i n a r a d i a l e l a s t i c f o undation. The l a s t part of the t h e s i s deals with the b u c k l i n g of a r i n g supported by a non l i n e a r e l a s t i c founda-t i o n . A t h i n gauge s t e e l c u l v e r t under a high earth f i l l i s a p r a c t i c a l example of a r i n g which could have a d i f f e r e n t value of the foundation modulus for an outward or inward displacement of the r i n g . T h i s t h e s i s w i l l not deal with the d e t e r m i n a t i o n of the proper r a t i o s or values of the f o u n d a t i o n m o d u l i i . This i s a s u b j e c t best l e f t to experts i n the f i e l d of s o i l mechanics. This t h e s i s w i l l present s o l u t i o n s f o r v a r i o u s r a t i o s of the foundation m o d u l i i from which q u a n t i t a t i v e s o l u t i o n s can be determined f o r r a t i o s other than those given. 3 CHAPTER I I DERIVATION OF THE DIFFERENTIAL EQUATION The d e r i v a t i o n of an equation g e n e r a l l y n e c e s s i t a t e s a c a r e f u l assessment and statement of the assumptions used i n the d e r i v a t i o n . The l i m i t a t i o n s of the t r e a t -ment of the p a r t i c u l a r problem and the measure of the accuracy of the s o l u t i o n s to the problem are i n d i c a t e d by the b a s i c assumptions. The f o l l o w i n g assumptions were made i n the d e r i v a t i o n of the d i f f e r e n t i a l equation f o r the b u c k l i n g of a r i n g i n an e l a s t i c f o u n d a t i o n : (a) Hookes Law a p p l i e s . (b) C r o s s - s e c t i o n s of the r i n g remain plane during bending. (c) S t r e s s e s vary only i n the d i r e c t i o n of cu r v a t u r e . (d) Only e x t e n s i o n l e s s forms of b u c k l i n g occur, i . e . , the a x i a l deformations of the r i n g under l o a d i n g can be n e g l e c t e d and the a x i a l compression f o r c e remains unchanged during b u c k l i n g of the r i n g . (e) The th i c k n e s s of the r i n g i s smal l i n comparison to the r i n g ' s r a d i u s . ( f ) The r i n g buckles only i n i t s plane. 4 I t i s worth n o t i n g that the above assumptions apply e q u a l l y w e l l to the b u c k l i n g of a c y l i n d e r with the e x c e p t i o n that the s t r e s s i s m o d i f i e d by the f a c t o r , 1 - u*- , where u «. Poisson's r a t i o f o r the m a t e r i a l i n the c y l i n d e r . Consider a r i n g of constant c r o s s - s e c t i o n i n an e l a s t i c f o u n d a t i o n and loaded by a uniform e x t e r n a l pr e s s u r e , ( F i g u r e l a ) . At any given time the r i n g i s under the a c t i o n of an a p p l i e d e x t e r n a l uniform l o a d i n g P and a r e a c t i o n from the r a d i a l e l a s t i c f o u n d a t i o n q r . The net ex-t e r n a l uniform pressure p which causes b u c k l i n g i s given by P s p + q r and i s dependent upon the e l a s t i c f o u n d a t i o n con-d i t i o n s at the commencement of l o a d i n g . F i g u r e lb shows the c o n d i t i o n s at the commencement of l o a d i n g ( r a d i u s « r^) and at the p o i n t of b u c k l i n g ( r a d i u s • r ) f o r the case where the e l a s t i c f o u n d a t i o n r e a c t i o n i s zero at r ^ . The r i n g has been compressed and as a r e s u l t has had a uniform r a d i a l displacement of u 0 , t h e r e f o r e the e l a s t i c f o u n d a t i o n r e a c t i o n i s q r s K r u 0 where K r i s the r a d i a l f o u n d a t i o n modulus. Since 2 u Q s- pr *• EA where E s Youngs Modulus and A • area of the c r o s s -To f o l l o w page F i g u r e lb s e c t i o n of the r i n g , the proceeding equation reduces to 1r " K r p r 2 EA Then, f o r t h i s case the r e l a t i o n s h i p between the a p p l i e d e x t e r n a l pressure P and the net e x t e r n a l pressure p i s found upon s u b s t i t u t i o n to be P = p ( l + K r r 2 ) EA For the case where the r a d i a l f o u n d a t i o n modulus K r i s zero when the r i n g i s at the poin t of b u c k l i n g the simple r e l a t i o n s h i p P = p r e s u l t s . R e l a t i o n s h i p s f o r other c o n d i t i o n s can be r e a d i l y de-r i v e d , t h e r e f o r e f o r the remainder of the t h e s i s we w i l l d e al only with the net e x t e r n a l pressure p. We proceed to d e r i v e the d i f f e r e n t i a l equation by a p p l y i n g the laws of kine m a t i c s , s t a t i c s and Hooke. Consider a sma l l element XY of the r i n g j u s t before b u c k l i n g and t h i s same element X'Y' j u s t a f t e r b u c k l i n g occurs, (See F i g u r e 2 ) . Displacements i n the r a d i a l d i r e c t i o n are designated u and i n the t a n g e n t i a l d i r e c t i o n v as shown i n F i g u r e 2. F i r s t from the kinematics of the element X'Y' we can w r i t e that the u n i t t a n g e n t i a l s t r a i n of the element i s To f o l l o w page 5 u i i ^ v P o s i t i v e d i r e c t i o n s of displacements and that the angular r o t a t i o n at the end X of the element i n moving from X to X' i s given by eg = _1 (vr - in ' ) r The change of curvature A 9i i s the r a t e of change of the angular r o t a t i o n cp over the length of the element rdi9 , i . e . A'X = dicp rdl9 from which t% = l _ ( v ' - u") 2 r Next consider the s t a t i c s of the loadings on the d i s p l a c e d element X'Y' i n F i g u r e 3a. At end X' the r i n g i s sub j e c t e d to a bending moment M, a shear f o r c e V and a compressive f o r c e C. At end Y' the r i n g i s su b j e c t e d to the same f o r c e s as at X 1 and i n a d d i t i o n i n c r e a s e s i n the bending moment, shear f o r c e and compressive f o r c e equal to dM, dV and dC r e s p e c t i v e l y . We d e f i n e the shear f o r c e V as a c t i n g p e r p e n d i c u l a r to the r i n g when i n p o s i t i o n X'Y'. The compressive f o r c e C i s d e f i n e d as remaining normal to the o r i g i n a l r a d i u s and as such w i l l c o n t r i b u t e to the r o t a t i o n a l e q u i l i b r i u m of the element. The l o a d i n g on the element X'Y' i s the d i s t r i -buted t a n g e n t i a l e l a s t i c f oundation l o a d i n g K tv, the d i s t r i b u t e d r a d i a l e l a s t i c f o u n d a t i o n To f o l l o w page 6 F i g u r e 3b 7 l o a d i n g K ru, the d i s t r i b u t e d t o r s i o n a l e l a s t i c foun-d a t i o n Krf (u'-v) and a u n i f o r m l y d i s t r i b u t e d e x t e r n a l r l o a d i n g p. F i g u r e 3b i s a p h y s i c a l r e p r e s e n t a t i o n of the e l a s t i c f o u n d a t i o n . I t w i l l be noted that the ends of the spr i n g s which represent the r a d i a l and the t a n g e n t i a l e l a s t i c foundations are pinned at a poi n t Br from the centre of the r i n g i n the case of the r a d i a l s p r i n g and a point dr along the tangent to the r i n g i n the case of the t a n g e n t i a l s p r i n g . I t i s assumed i n the d e r i v a t i o n that 8 never approaches u n i t y and that S never approaches zero. The uniform e x t e r n a l l o a d i n g p i s assumed to remain d i r e c t e d t o -wards a point on the o r i g i n a l r a d i u s . This p o i n t on the o r i g i n a l r a d i u s i s l o c a t e d from the centre of the r i n g by the dimension ar (see F i g u r e 3a). The equation f o r the r a d i a l e q u i l i b r i u m , when w r i t t e n for the element i n i t s f i n a l p o s i t i o n X'Y', to the f i r s t order i s dV + prd@ + K rurd6 - CdQ = 0 which upon d i v i d i n g through by d9 and r e a r r a n g i n g reduces to V = C - pr - K r r u ... (3) The equation f o r the t a n g e n t i a l e q u i l i b r i u m , when w r i t t e n f o r the element i n i t s f i n a l p o s i t i o n X'Y', to the f i r s t order i s Vdi9 + dC + K.vz-die + pu,1 di9 t + pv ( a ) dl9 = 0 1 - a vrhich upon d i v i d i n g through by dl9 and r e a r r a n g i n g reduces to V - _C' -K rv/ -p(u' + a v) X 1 - a The equation f o r r o t a t i o n a l e q u i l i b r i u m to the f i r s t order when w r i t t e n about poin t Y' i s dM - K«*u'd9 + Cdu - V,rdl9 +. Krfvde = 0 which upon d i v i d i n g through by rdl9 and r e a r r a n g i n g reduces to V s M* + C u' - K^(u'-V) r r r r F i n a l l y from Hookes Law and the assumption of ex-t e n s i o n l e s s b u c k l i n g we can w r i t e that the tangen-t i a l s t r a i n £ = 0 and f u r t h e r that the change i n curva t u r e i s r e l a t e d to the bending moment as f o l l o w s AOC = - M_ EI where E e Youngs Modulus and I s Moment of I n e r t i a of the r i n g s c r o s s -s e c t i o n . We note that the preceeding 7 independent equations c o n t a i n 7 unknowns, namely u, v, e, , M, C and V, t h e r e f o r e we can. proceed to so l v e f o r one unknown by u s i n g the 7 independent equations. I t i s convenient i n the m a n i p u l a t i o n of the above equations to f i r s t e l i m i n a t e the r a d i a l shear term i n equation (3) by d i f f e r e n t i a t i n g equation (5) and s u b s t i t u t i n g . Then equation (3) becomes M" + C u" - K«5 u" s C - pr - K r r u K^v' r r r r and c o l l e c t i n g terms i n C we o b t a i n C ( 1 - u") = M" + pr + K r r u - Kj<(u"-v' > r r r r Upon d i v i d i n g through by 1 - _u" and n o t i n g that f o r r i n f i n i t e s i m a l s 1 = 1 + u" 1 - u." r r . the above equation reduces to C - M" + pr + pu" + K r r u - K^(u"^v') s r r r by n e g l e c t i n g a l l products of the 7 unknowns. In the d i f f e r e n t i a t i o n of equation (5) i t was assumed that C remained a constant and t h e r e f o r e no cross product term C'-ii 1 r e s u l t e d thus making r the problem non l i n e a r . I f the above e x p r e s s i o n f o r G was d i f f e r e n t i a t e d and m u l t i p l i e d by _u' r only terms of the second order r e s u l t s and there-f o r e our assumption of C being a constant when d i f f e r e n t i a t i n g equation (5) i s reasonable. 10 We next s u b s t i t u t e f o r V, C and C 1 i n equation (4) by using equations (5) and ( 8 ) . C o n s i d e r i n g only terms of the f i r s t order t h i s f o l l o w i n g equation r e s u l t s M1 + pu 1 - K«5(u' - v) = -M* ' ' -pu'' 1 - K r r u ' r r r r + K«*(uV ' ' -v' ') - K t r v - p(u' + q v) r r 1 - a ...(9) The above equation s t i l l contains as unknowns M, v and u. From equations (1) and (6) we can w r i t e that v 1 s - u t h e r e f o r e equation (2) becomes upon s u b s t i t u t i o n for v' A X s-l__ ( - u" - u) r 2 which upon s u b s t i t u t i o n i n t o equation (7) produces M s EI (u" + u) r 2 D i f f e r e n t i a t i n g equation ( 9 ) , s u b s t i t u t i n g f o r the above expressions f o r v ! and M and r e a r r a n g i n g produces the f o l l o w i n g s i x t h order l i n e a r homogeneous d i f f e r e n t i a l equation to the f i r s t order i n terms of the r a d i a l displacements u V I + (2 + pr£ - K«5 r£) u I V E I E I + (1 + 2 pr£ + K r r 4 - 2K«5 r 2 ) u" E I ~ E T ~ ET " ( a p r 3 + K t + Krf r£) u = 0 ...(10) 1 - q E I E I E I CHAPTER I I I SOLUTION OF FULL RING CASE The s o l u t i o n of any s i x t h order l i n e a r homogeneous d i f f e r e n t i a l equation can be a tedious task i f r e -course i s made to a s o l u t i o n by the a u x i l i a r y equation method. In the case of the complete r i n g only p e r i o d i c f u n c t i o n s are p o s s i b l e s o l u t i o n s be-cause the boundary c o n d i t i o n s are s a t i s f i e d by the requirements that a l l displacements as w e l l as these d e r i v a t i v e s are p e r i o d i c over 2Tf radians . Therefore as pointed out by Stevens *, a s o l u t i o n i s r e a d i l y obtained by t a k i n g u s u Q s i n nQ and s u b s t i t u t i n g i n t o the d i f f e r e n t i a l equation (10) to determine the c o n d i t i o n s under which u Q i s not zer o. Making the above s u b s t i t u t i o n and r e a r r a n g i n g gives the f o l l o w i n g equation from which the c r i t i c a l pressure can be determined: Per l l ( n 2 - 2 - 1 . q ) - ( n 2 - l ) 2 EI n2 1 - a + K r r ^ + K t r 4 + K«5 r£ ( a 2 - l ) 2 . . . (11) * Reference (1) In the above equation n d e f i n e s the p e r i o d i c i n t e r v a l or number of c y c l e s of the buckled form of the f u l l r i n g and t h e r e f o r e i t i s d e s i r a b l e at t h i s time to i n -v e s t i g a t e the a d m i s s i b l e values of n. F i g u r e 4 shows the p l o t of the s i n f u n c t i o n and the buckled form of the r i n g f o r value of n. = 0, 1 and 2. When n s 0, the r i n g s u f f e r s no displacement, t h e r e f o r e t h i s case rep-r e s e n t s the unbuckled form of the r i n g . The c o n d i t i o n n = 1 re p r e s e n t s a r i g i d body t r a n s l a t i o n and i s f o r our purposes a t r i v i a l case. The c o n d i t i o n n a 2 rep-r e s e n t s the lowest value of n which i s c o n s i s t e n t with the problem. Then a l l values of n ^ 2 are a d m i s s i b l e . A g r a p h i c a l r e p r e s e n t a t i o n of the c r i t i c a l pressure obtained from equation (11) i s p r o h i b i t i v e because of the l a r g e number of independent parameters, namely a, K r, Kt a n d Ktf. I t i s r e l a t i v e l y easy to p l o t a graph f o r a p a r t i c u l a r problem f o r which a l l but two of the parameters are zero or can be expressed as a r a t i o of the remaining parameters. The remainder of t h i s chapter w i l l deal with s p e c i a l s o l u t i o n s of the general equation. The f a c t o r q , which l o c a t e s the point on the 1 - a o r i g i n a l r a d i u s of the r i n g towards or from which the load remains d i r e c t e d a f t e r b u c k l i n g begins, provides To f o l l o w page F i g u r e 4 f o r the f u l l range of the d i r e c t i o n of a p p l i c a t i o n of the uniform l o a d i n g . This i s true f o r the case of a r i n g supported i n an e l a s t i c f o u n d a t i o n as w e l l as f o r the case of a r i n g not supported i n an e l a s t i c foun-d a t i o n . Consider f i r s t the case of the r i n g not supported i n an e l a s t i c foundation, i . e . K r s Kt = Ktf = 0. From equation (11) we can w r i t e Per = U 2 - I ) 2 EI n 2 - 2 - j _ ( q ) r 3 n 2 1 - a o r Per = f EI r 3 where f i s a c o e f f i c i e n t dependent on l y upon n and a Two modes of b u c k l i n g w i l l govern i n a l l ranges of a with the ex c e p t i o n of a > .813 < 1. To determine the p o i n t at which another mode of b u c k l i n g governs we set equal the c o e f f i c i e n t s f f o r the two modes under c o n s i d e r a t i o n , i . e . ( t i ! 2 - D 2 = ( n 2 2 - I ) 2  n L 2 - 2 - 1 ( q_) n 2 2 - 2 - 1 ( q ) n j * 1 " a n 2 2 1 - a where n^ i s the mode of b u c k l i n g which w i l l no longer govern and n 2 i s the mode of b u c k l i n g which w i l l j u s t s t a r t to govern. From the preceeding equation we de-termine that one mode governs to q = .813, two modes govern from a = .813 to a = .97, and so on f o r higher modes of b u c k l i n g . 14 F i g u r e 5a i s the graph of f against a f o r the b u c k l i n g of a r i n g not supported i n an e l a s t i c foundation and F i g u r e 5b i s a p i c t o r i a l r e p r e s e n t a t i o n of graph i n F i g u r e 5a. F i g u r e 5b i l l u s t r a t e s that f o r values of a below 1, the load acts to r e s t o r e the r i n g to i t s unbuckled shape and f o r values of a above 1, the load acts to i n c r e a s e the buckled shape of the r i n g . The values of the c o e f f i c i e n t f f o r a's very c l o s e to 1 are not s t r i c t l y accurate because i t was assumed i n the d e r i v a t i o n of the d i f f e r e n t i a l equation that the angle of the l o a d i n g was s u f f i c i e n t l y small f o r i n f i n i t e s i m a l geometry to apply, i . e . s i n S • S and cos S » 1. The c l a s s i c a l treatment of b u c k l i n g of r i n g s has been r e s t r i c t e d to the case where the l o a d i n g remains d i r e c t e d towards the r i n g ' s centre ( a = G) f o r which i t was found that f o 4.5 and to the case where the l o a d i n g remains d i r e c t e d normal to the r i n g ( h y d r o s t a t i c loading) f o r which i t was found f a 3.0. We f i n d then that the c l a s -s i c a l treatments are but s p e c i a l cases of the more gene r a l theory i n t h i s t h e s i s . For the case where the l o a d i n g remains d i r e c t e d normal to the r i n g and where n = 2, we can deduce that a = 1.333 and that the l o a d i n g r o t a t e s through an angle of 3v r r a d i a n s from the unbuckled to the buckled shape. In g e n e r a l , f o r t h i s c o n d i t i o n of l o a d i n g , the term JL ( q ) n2 1 - q To f o l l o w page 14 F i g u r e 5a 15 must equal - 1 i n which case f z ( n 2 - l ) 2 = n 2 - 1 , n 2 - 1 a = n 2 , n 2 - 1 and the angle of r o t a t i o n of the l o a d i n g from the un-buckled to the buckled c o n d i t i o n i s ( n 2 - l ) v r i n r a d i a n s . I d e n t i c a l r e s u l t s were obtained f o r the case where the lo a d i n g remains normal to the r i n g by d e r i v i n g the d i f f e r e n t i a l equation only i n terms of the normal ap-p l i c a t i o n of the l o a d i n g . Consider now the more general case of the r i n g being supported by an e l a s t i c f o u n d a t i o n with only a r a d i a l s p r i n g , i . e . Kt = = 0. For t h i s case equation 11 reduces to Per r£ ( n 2 - 2 - 1 . a ) = ( n 2 - l ) 2 + K r £ ^ EI n2 1 - a EI The g r a p h i c a l p l o t of t h i s equation as a f u n c t i o n of the two dimensionless parameters p c r r 3 a g a i n s t I T . . .a 2) / K r _r^_ f o r v a r i o u s modes of the EI buckled form of the r i n g as shown i n F i g u r e 6. The diagrams c o n s i s t of a s e r i e s of curved s e c t i o n s cor-responding to the number of modes of b u c k l i n g f o r the v a r i o u s values of a« F i g u r e 6 16 For the case where the l o a d i n g remains normal to the r i n g , equation (12) reduces to Per l l = (n2 -1) + 1 .Kr r 4 EI n2 _ L EI ... (13) The p l o t of t h i s l o a d i n g c o n d i t i o n i s a l s o shown i n F i g u r e 6 and i s designated as the h y d r o s t a t i c l o a d i n g case. For values of / K r J£^_ greater than 3.0 the diagram f o r t h i s V E I l o a d i n g c o n d i t i o n can be more simply r e p r e s e n t e d by a s t r a i g h t l i n e which i s tangent to the f a m i l y of curved s e c t i o n s . I t i s apparent that the e r r o r of basing any c a l c u l a t i o n s on the s t r a i g h t l i n e p l o t i s n e g l i g i b l e and that the degree of e r r o r decreases as the governing number of the buckled modes of the r i n g i n c r e a s e s . The equation of the tangent l i n e i s P e r l l = 2 / K r l l W T V T? T EI v EI which can be f u r t h e r reduced to p c r r = 2JK~E1 where p c r r i s the c r i t i c a l a x i a l load i n the r i n g . A s o l u t i o n f o r the b u c k l i n g of an i n f i n i t e l y long column i n an e l a s t i c f o u n d a t i o n has been presented by Hetenyi * as f o l l o w s N c r = l j KEI where N c r a c r i t i c a l a x i a l load K r foundation modulus The s i m i l a r i t y of the two equations shows that when the dimensionless parameter /KR i s greater than 3.0, the V EI * Reference (2) e f f e c t of the curvature of the r i n g i s no longer a s i g n i f i c a n t item i n the b u c k l i n g of a r i n g i n an e l a s t i c f o u n d a t i o n . The s e v e r a l examples and f i g u r e s i n t h i s chapter show that the e l a s t i c f o u n d a t i o n p r i n c i p a l l y f o r c e s the r i n g to buckle i n higher modes with r e s u l t i n g greater b u c k l i n g s t r e n g t h of the r i n g . 18 CHAPTER IV SOLUTION OF PARTIAL RING CASE 1. HINGED SUPPORTS For the case of a p a r t i a l r i n g with hinged supports, a l l displacements and t h e i r d e r i v a t i v e s are c y c l i c over a p e r i o d equal to the contained angle between the supports. Then a s o l u t i o n to the s i x t h order d i f f e r e n t i a l equation i s u = u Q s i n N 8 where N = 2nTT u and u i s the contained angle between the supports Proceeding as f o r the f u l l r i n g case by s u b s t i t u t i n g f o r u i n the d i f f e r e n t i a l equation and determining the con-d i t i o n s f o r which uo i s not zero, r e s u l t s i n the general equation Pcr ~ 2 " a )= ( N 2 - l ) 2 EI N 2 1 - a + K r j r ^ + K t r 4 + ( N 2 - l ) 2 Krf E I FIT" N 2 EI ...(14) In the above equation f o r N, n d e f i n e s the number of c y c l e s of the buckled form of the r i n g w i t h i n the con-t a i n e d angle w . Since the supports are hinged and capable of r e s i s t i n g any r i g i d body t r a n s l a t i o n , a l l values of n ^ 1 are a d m i s s i b l e . 19 Consider f i r s t the case where the r i n g i s not supported by an e l a s t i c f o u n d a t i o n , i . e . K r = Kt = Kj$ o 0. From equation (14) we can w r i t e Per = ( N 2 - l ) 2 EI. N 2 -2 - 1 . a r 3 " - ' N2" 1 - q or p c r s f EI T3" where f i s a c o e f f i c i e n t dependent upon n, a and w . F i g u r e 7 i s a p l o t of f f o r v a r i o u s contained angles and f o r v a r i o u s d i r e c t i o n s of a p p l i c a t i o n of the c r i t i c a l load a f t e r b u c k l i n g . The graph shows that where a i s 1.33 the r i n g i s no longer capable of c a r r y i n g a load when the contained angle w s 2TT . For values of a between 1 and 1.333 the contained angle at which the r i n g i s no longer capable of c a r r y i n g load w i l l be l e s s than 2 TT . The graph a l s o shows that more than one mode of b u c k l i n g can govern i n some r e g i o n s . Consider now the case of the r i n g being supported by an e l a s t i c foundation with only r a d i a l s p r i n g s , i . e . K t = s 0. Equation (14) reduces to Pcr r 3 ( N 2 - 2 - 1 . q ) = ( N 2 - l ) 2 + K r r 4 N 7 1 - q I T T h i s equation has been p l o t t e d as a f u n c t i o n of the two dimensionless parameters p c r r 3 & K r r 4 f o r EI IT s e v e r a l values of the contained angle « f o r the case where the c r i t i c a l l o a d i n g remains d i r e c t e d towards To f o l l o w page 19 F i g u r e 7 20 the r i n g ' s centre ( a = 0) and f o r the case where the c r i t i c a l l o a d i n g remains normal to the r i n g (hydro-s t a t i c c a s e ) . See F i g u r e s 8a and 8b. 2. FIXED SUPPORTS Rings with f i x e d supports do not lend themselves to a sim-p l i f i e d s o l u t i o n of the s i x t h order d i f f e r e n t i a l equation because a l l displacements and t h e i r d e r i v a t i v e s are not n e c e s s a r i l y c y c l i c over a p e r i o d equal to the contained angle between supports. A c l a s s i c a l s o l u t i o n of the s i x t h order d i f f e r e n t i a l equation i s t e d i o u s . I f r e s t r i c t i o n s are made as to the e l a s t i c f oundation and the d i r e c t i o n of a p p l i c a t i o n of the l o a d i n g a f t e r b u c k l i n g , a f i f t h order d i f f e r e n t i a l equation i n terms of the t a n g e n t i a l d i s p l a c e -ment can be d e r i v e d f o r which s o l u t i o n s are r e a d i l y ob-t a i n a b l e . Taking the case of an e l a s t i c f o u n d a t i o n with only r a d i a l s p r i n g s and a lo a d i n g which remains normal to the r i n g , the d i f f e r e n t i a l equation ?is: vV + T V * " +6v' = 0 ... (15) where x = 2 + p r 3 EI and 6 = 1 + p r 3 + K r r 4 EI . EI The f i v e r o o t s of the c h a r a c t e r i s t i c equation f o r the s o l u t i o n of the above d i f f e r e n t i a l equation are mi - 0 and ^2,3,4,5 = + J ~ 2 ± JZ J 1 ~ p2' where p 2 - 4 0 T2 T o f o l l o w page 20 F i g u r e 8 a To f o l l o w page 20 F i g u r e 8 b 21 Three cases e x i s t f o r the s o l u t i o n . p > 1 This i s the str o n g s p r i n g case and d e f i n e s 2 the c o n d i t i o n where K r > JL_ (p r ) . The s o l u t i o n EI 2 ( i i ) ( i i i ) to the d i f f e r e n t i a l equation f o r t h i s case i s i n terms of t r i g o n o m e t r i c and h y p e r b o l i c f u n c t i o n s of the r a d i a l angle. + B3 cos a 9 s i n h b 9 + B4 s i n a 9 si n h b 9 p = 1 This case d e f i n e s the boundary between the strong and weak s p r i n g cases. p 2 < 1 This i s the weak s p r i n g case where K r < JL_ ( p r ) 2 . The s o l u t i o n to the d i f f e r e n t i a l EI 2 equation f o r t h i s case i s i n terms of t r i g o n o -m e t r i c f u n c t i o n s of the r a d i a l angle, v = B ^  cos a 9 + B2 s i n a 9 + B3 cos b 9 + B 4 s i n b 9 + B 5 . . .(17) v s Bi cos a 9 cosh b 9 + B2 s i n a 9 cosh b 9 + B 5 .(16) and b The f i v e boundary c o n d i t i o n s to be a p p l i e d to equations (16) and (17) are v' » 0 Q = 0 v' = 0 v' * = 0 9 = 0 v' * = 0 8 r u> u » j ud9 s 0 o The f i v e boundary c o n d i t i o n s w i l l give a t r a n s c e n d e n t a l equation c o n t a i n i n g only f u n c t i o n s of the angles a GO and b u> from which the two dimensionless parameters ( p c r r3) and x/^r E!L c a n ^ e obtained. EI EI 23 CHAPTER V NON-LINEAR FOUNDATION MODULUS A completely general s o l u t i o n has been presented f o r the b u c k l i n g of a f u l l r i n g and the b u c k l i n g of a p a r t i a l r i n g with pinned supports when supported by an e l a s t i c f o u n d a t i o n . A l s o presented was the s o l u t i o n to the d i f -f e r e n t i a l equation and a procedure f o r determining the '. b u c k l i n g load f o r a p a r t i a l r i n g with f i x e d supports provided the e l a s t i c f o u n d a t i o n and the d i r e c t i o n s of loa d i n g a p p l i c a t i o n s were r e s t r i c t e d . These s o l u t i o n s assumed complete l i n e a r i t y of the e l a s t i c f oundation, that i s , that a given displacement i n e i t h e r a p o s i t i v e or negative d i r e c t i o n produces a r e a c t i o n of equal mag-nit u d e from the foun d a t i o n . T h i s chapter deals with the case where the s p r i n g constant of the e l a s t i c f oundation i s a d i f f e r e n t value f o r the outward and f o r the inward displacements of the r i n g . The term n o n - l i n e a r w i l l apply to t h i s case. The best procedure f o r s o l v i n g the d i f f e r e n t i a l equation f o r the b u c k l i n g of a r i n g i n a n o n - l i n e a r e l a s t i c foun-d a t i o n i s by f i n i t e d i f f e r e n c e equations. This i s par-t i c u l a r l y true when a d i g i t a l computer i s a v a i l a b l e . The s i x t h order d i f f e r e n t i a l equation d e r i v e d i n Chapter II could have been w r i t t e n as f i n i t e d i f f e r e n c e equations 24 and programmed f o r the d i g i t a l computer to give a com-p l e t e l y general s o l u t i o n f o r a r i n g i n a n o n - l i n e a r e l a s t i c f o u n d a t i o n . However, because a r a d i a l component of the e l a s t i c f o u n d a t i o n and the c r i t i c a l load remain-ing normal to the r i n g i s a reasonable approximation to the c o n d i t i o n s of a s t e e l c u l v e r t i n a high f i l l , i t was decided to use the f i f t h order d i f f e r e n t i a l equation presented i n Chapter IV, i . e . v v + T V 1 " + 5v' . 0 where x = p r£ + 2 EI and 6 = pr_3_ + 1.+ K r r 4 EI EI The c o e f f i c i e n t 6 can be r e w r i t t e n as 6 = p r_3 + 2 "+ K r r 4 - 1 EI EI or 6 = x + v where v = K r r j 4 - 1 EI which upon s u b s t i t u t i n g i n the d i f f e r e n t i a l equation produces VV + - r v " ' + T V ' + v V ' = 0 ...(18) At any point on the r i n g i the d e r i v a t i v e s of the above equation may be approximated by the f o l l o w i n g f i n i t e d i f f e r e n c e equations 2 5 v i - 3 + 4 v i - 2 - 3 v i - l + 3 v i + l " 4 v i + 2 + v i + 3 v i - 2 + 2 v i - l " 2 v i + l + V i + 2 J . . . ( 1 9 ) v i - l + v i + l ) where V i _ 3 , vi_2> v i + 1 , v i + 2 , v i + 3 are the t a n g e n t i a l displacements at p o i n t s on the r i n g sepa-r a t e d by a d i s t a n c e h. Since the d i s t a n c e h should be s m a l l before reasonable accuracy can be expected i n the above equations, i t i s d e s i r a b l e to l i m i t the length of the r i n g f o r which f i n i t e d i f f e r e n c e s are w r i t t e n i n order to l i m i t the number of the unknown v o r d i n a t e s . Chapter I I I showed that a p e r i o d i c curve s a t i s f i e s the boundary c o n d i t i o n s f o r d i s p l a c e -ments and t h e i r d e r i v a t i v e s , t h e r e f o r e a h a l f wave of the buckled r i n g i s the l e a s t length we need consider for the f i n i t e d i f f e r e n c e equations. The length of a h a l f wave of the buckled r i n g i s TT where n i s the n number of modes of b u c k l i n g . D i v i d i n g t h i s l ength in x, equal p a r t s gives f o r the i n t e r v a l of the f i n i t e d i f f e r e n c e equations x n F i g u r e 9 shows a curve f o r the t a n g e n t i a l displacements v f o r a h a l f wave of the buckled r i n g when the number of equal i n t e r v a l s x i s 1 5 . I t w i l l be n o t i c e d that w r i t i n g f i n i t e d i f f e r e n c e equations f o r p o i n t s at the 26 ends of the curve i n v o l v e p o i n t s on the adjacent h a l f waves of the buckled r i n g . The requirements of com-p l e t e c o m p a t i b i l i t y of the ends of adjacent h a l f waves of the buckled r i n g enables us to ad j u s t the d i f f e r e n c e equations near the supports by adding the negative value of the o r d i n a t e i n the adjacent h a l f wave to the same o r d i n a t e i n the h a l f wave under c o n s i d e r a t i o n . F i g u r e 9 shows the c o r r e c t e d f i n i t e d i f f e r e n c e coef-f i c i e n t s . W r i t i n g the f i n i t e d i f f e r e n c e equations f o r the d i f -f e r e n t i a l equation i n matrix form w i l l give the f o l -lowing matrix equation 1 Pri! (y) + _VL [ S K ] (y) . - _ T _ r T 2 l (y) - _x_[T3l<y) 2 h 5 2h L J 2 h 3 L A - 2 h L where [ T l J i s a square matrix (g x g) of the c o e f f i c i e n t s of the f i n i t e d i f f e r e n c e equations f o r v^ (see F i g u r e 10), [JSKJ and [^ T3J are square matrices (g x g) of the c o e f f i c i e n t s of the f i n i t e d i f f e r e n c e equations f o r v' (see F i g u r e 11), jjT2j i s a square matrix (g x g) of the c o e f f i c i e n t s of the f i n i t e d i f f e r e n c e equations f o r v 1 1 ' (see F i g u r e 12), (y) i s a v e c t o r of the t a n g e n t i a l displacements v and h i s the i n t e r v a l between the o r d i n a t e s of the curve f o r the t a n g e n t i a l displacement v. .(20) To f o l l o w page 26 > T r > in in i o i m o m i l-l r-I i m 1 1 in- -o _ m — tn bo- tn i o - o rn i r-4 I I N -on I CM I CM CM I O CM CM I O r - l CM CM CM t CM r - l i -4 o 4J - d - CO CM r-4 I-I 1-4 CO r-4 <2 •a c3 u #-« r-4 CM CO ITfl t-4 CO VI CO CU CO •U u u 4J a a 1-1 C •rl •rl co •rl -o o o o o Cu cu Cu •rl CU o. u 4J 4-1 >> 4J < < < H CM I CM Tr O ; CM CO r - l i - l o o u 4-1 CO CO CM r - l CO CO *s 4J OS 4J C C! CM •rl 1-4 •rl o o CO Cu CO Cu 4J 4-J C r-4 C! t-4 •rl CO •rl <B o o o O cu •rl a. •rl CU Cu u >> 4-1 <c H < H CO a c c <U > •rl O O J 3 •rl CO 4-1 4-1 CO CU 4J O d O - r l O a) cu o a 4J <D to r l 0) > 4-4 4-1 4-1 •rl O T l CO cu cu 4-> > •rl * r l a 4-i •rl CO 4-4 > •H •d u cu cu 44 T J o CU V4 H o i - i 4-1 o CJ F i g u r e 9 To f o l l o w page 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 -4 6 -4 1 2 -4 0 5 -4 1 3 4 -5 0 5 -4 1 4 -1 4 -5 0 5 -4 1 5 -1 4 -5 0 5 -4 1 6 -1 4 -5 0 5 -4 1 7 -1 4 -5 0 5 -4 1 8 -1 4 -5 0 5 -4 1 9 -1 4 -5 0 5 -4 1 10 -1 4 -5 0 5 -4 1 11 -1 4 -5 0 5 -4 1 12 -1 4 -5 0 5 -4 13 -1 4 -5 0 4 14 -1 4 -6 4 M a t r i x of c o e f f i c i e n t s of f i n i t e d i f f e r e n c e equations f o r F i g u r e 10 \ To f o l l o w page 26 1 2 10 11 12 13 14 0 •1 1 0 -1 1 0 • 1 1 0 • 1 1, 0 • 1 1 0 •1 1 0 •1 1 0 • 1 1 0 • 1 1 0 -1 1 0 •1 1 0 • 1 1 0 1 -1 0 M a t r i x of c o e f f i c i e n t s of f i n i t e d i f f e r e n c e equations f o r v' F igure 11 To f o l l o w page 26 9 10 11 12 13 14 1 2 •1 -2 1 0 -2 2 -1 0 2 -1 1 •2 . 0 2 •1 1 •2 0 2 •1 1 •2 0 2 •1 1 •2 1 0 -2 2 •1 0 2 -1 1 -2 0 2 -1 1 •2 0 2 •1 1 •2 0 2 •1 1 -2 0 -2 2 0 -1 1 -2 -1 M a t r i x of c o e f f i c i e n t s of f i n i t e d i f f e r e n c e equations f o r v ' 1 1 F i g u r e 12 27 M u l t i p l y i n g through by 2h , l e t t i n g i> = v h 4 2 and a o T h reduces the matrix equation to [ T l ] (y) + [rf S K ] (y) s - O - [ T 2 ] (y) - a [h 2T3] ( y ) _ The matrix equation upon a d d i t i o n of m a t r i c e s reduces [ A ] (y) = - a [ B ] (y) M u l t i p l y i n g both s i d e s by the i n v e r t e d A matrix produces the equation y i = - » [ A " 1 ] [ B ] yo or y i : -o[cJ y Q . . .(21) Equation (21) i s i n a form f o r a p p l y i n g an i t e r a t i o n process f o r the d e t e r m i n a t i o n of Cf from which i n turn the c r i t i c a l l o a d i n g can be obtained. T h i s i n v o l v e s assuming a curve yg, c a l c u l a t i n g y^ from equation (21), r e p l a c i n g yQ by y'i,' and by repeated i t e r a t i o n s c l o s e on a rvalue of o such that equation (21) i s s a t i s f i e d at each p o i n t of the curve. The computer program f o r s o l v i n g equation (18) was w r i t t e n i n the F o r t r a n I I language f o r the IBM 1620 d i g i t a l computer at the U n i v e r s i t y of B r i t i s h Columbia. A block diagram and a complete l i s t i n g of the program i s given i n Appendix 1. The program length exceeded the a v a i l a b l e 30,000 d i g i t c a p a c i t y of the computer, t h e r e f o r e the program was d i v i d e d i n t o three parts? Link (0), Link (1) and Link ( 2 ) . A d e s c r i p t i o n of the program f o l l o w s . A l l input data i s read i n t o the computer i n Link (0) and c o n s i s t s of the f o l l o w i n g : DN Number of modes of b u c k l i n g i n the f u l l r i n g FK I n i t i a l input value f o r the foundation modulus term v DFK The increment i n FK WC R a t i o of the foundation modulus terms *0* where s foundation modulus f o r inward r a d i a l displacements of the r i n g and VQ = foundation modulus f o r outward r a d i a l displacements of the r i n g NC The number of c y c l e s of i t e r a t i o n on equation (21) Y0(I) The or d i n a t e s of the i n i t i a l l y assumed curve f o r t a n g e n t i a l displacements The remainder of Link (0) i s devoted to ensuring, f i r s t l y , that the storage f o r Ma t r i x T l i s zero and, secondly, to making up M a t r i x T l i n accordance with F i g u r e (10). The f i r s t o p e r a t i o n i n Link (1) i s the c a l c u l a t i o n of the value of the i n t e r v a l i n the f i n i t e d i f f e r e n c e equations (designated as DIM i n the program). The 29 second and f o u r t h powers of the i n t e r v a l are a l s o c a l -c u l a t e d and designated DUM and DOM r e s p e c t i v e l y . The n o n - l i n e a r i t y of e l a s t i c f oundation i s next i n t r o -duced. The value of the found a t i o n modulus to apply to the SK Matrix i s dependent upon the s i g n of the r a d i a l displacement at any point i on the r i n g . Since the i n i t i a l input curve i s f o r t a n g e n t i a l displacements, i t i s necessary to determine the s i g n of the r a d i a l displacement at point i i n terms of the t a n g e n t i a l d i s -placements. Chapter I I showed that the slope of the t a n g e n t i a l displacement curve i s equal to the negative value of the r a d i a l displacement at the same point i on the r i n g , i . e . v 1 = - u. The slope of the curve of the t a n g e n t i a l displacement at each p o i n t i i s evaluated from which the r a d i a l displacements ( d e s i g -nated U(I) i n the program) are computed. The program, i n making up the A M a t r i x , takes the T l Matrix from Link (0) and then i n t r o d u c e s an IF s t a t e -ment to check the s i g n of the r a d i a l displacement at each point i . I f the s i g n of U(I) i s p o s i t i v e , the program adds to the T l M a t r i x the value of vQ h 4 (designated FK*D0M i n the program) m u l t i p l i e d by the c o e f f i c i e n t s of the SK Ma t r i x (see F i g u r e 11 f o r these c o e f f i c i e n t s ) . I f the s i g n of U(I) i s negat i v e , the program adds to the T l Matrix the value of v ^  h 4 ( d e s i g n a t e d F K * D O M * W C i n t h e p r o g r a m ) m u l t i p l i e d b y t h e c o e f f i c i e n t s o f t h e S K M a t r i x . T h e A M a t r i x , w h e n c o m p l e t e d , i s i n v e r t e d b y a s u b r o u t i n e e s -p e c i a l l y s u i t e d t o h a n d l i n g m a t r i c e s w i t h z e r o d i a g o n a I s . M a t r i x B i s m a d e u p b y t a k i n g t h e c o e f f i c i e n t o f t h e f i n i t e d i f f e r e n c e e q u a t i o n s f o r v ' 1 ' ( s e e F i g u r e 12 f o r t h e s e c o e f f i c i e n t s ) a n d a d d i n g t h e p r o d u c t s o f h 2 a n d t h e c o e f f i c i e n t s o f t h e f i n i t e d i f f e r e n c e e q u a t i o n s f o r v ' ( s e e F i g u r e 11 f o r t h e s e c o e f f i c i e n t s ) . T h i s c o m p l e t e s L i n k (1). L i n k (2) i s a m a j o r DO l o o p f o r c a r r y i n g o u t t h e i t e r a t i o n p r o c e d u r e t o s o l v e e q u a t i o n (21). T h e n u m b e r o f i t e r a t i o n c y c l e s a r e s e t b y t h e i n p u t d a t a ( d e s i g n a t e d N C ) . V a l u e s o f y ^ , ( d e s i g n a t e d Y1 ( I ) i n t h e p r o g r a m ) a t e a c h p o i n t i a r e c o m -p u t e d u s i n g e q u a t i o n (21) a n d t h e n m u l t i p l i e d b y -1 t o a c c o u n t f o r t h e n e g a t i v e s i g n i n t h e e q u a t i o n . T h e i n i t i a l c u r v e yQ i s d i v i d e d b y y-^ t o c o m p u t e t h e v a l u e o f 0" ( d e s i g n a t e d R H ( I ) i n t h e p r o g r a m ) a t e a c h p o i n t i o f t h e r i n g . T o e n s u r e a g a i n s t a d i v i s i o n b y a v e r y s m a l l n u m b e r , e a c h v a l u e o f y\ i s f i r s t checked against the mean square root value of the sum of the squares of the y^'s. I f the absolute value of yi at any point i i s le s s than 17. of the mean square root value, that value of y^ i s r e p l a c e d by 17. of the mean square root v a l u e . A f t e r the computation of the a's, the o r d i n a t e s of the yg : c u r v e are r e p l a c e d by the o r d i n a t e s of the y^ curve. The value of a at each point i i s compared with the mean square root value of the sum of the squares of the a's. I f the absolute value of CT at each poin t i i s w i t h i n 27. of the mean square root value during any of the i t e r a t i o n c y c l e s NC, the computer c a l c u -l a t e s and types out the values of the dimensionless foundation and load constants (designated i n the pro-gram as FKX and FX r e s p e c t i v e l y ) . The computer i n -crements the foundation modulus term FK by DFK and re t u r n s to the s t a r t of Link ( 1 ) . If a l l the values of a do not converge during the NC i t e r a t i o n c y c l e s , the computer r e t u r n s to the s t a r t of Link (1) r e t a i n i n g the same value of the foundation modulus term. The r e c a l c u l a t i o n of the r a d i a l displacements ensures that any s h i f t i n the zero of the r a d i a l displacement curve w i l l be accounted f o r i n the make up of the new FK M a t r i x . During the program run, i t was seldom necessary to r e t u r n to Link (1) more than twice to ensure con-vergence of 0" . 32 The o p e r a t i o n of the program was checked by assuming complete l i n e a r i t y of the r a d i a l e l a s t i c f o u n d a t i o n (WC s 1.0). The values obtained were i d e n t i c a l to those given by equation (13) and the equation of the tangent to the f a m i l y of curves gave, as b e f o r e , the r e l a t i o n s h i p Pcr r - /ry ir i ? ( 2 2 ) T h i s formula, when c o r r e c t e d f o r c y l i n d e r s by d i v i d i n g E by 1 - u 2 , has been recommended f o r the determination of the c r i t i c a l load f o r s t e e l c u l v e r t s i n high f i l l s . * The tangents to the f a m i l y of curves f o r other r a t i o s of the m o d u l i i of the e l a s t i c f o u n d a t i o n are a l s o s t r a i g h t l i n e s . The equation of the tangent l i n e , f o r the case where the r a t i o of f o u n d a t i o n m o d u l i i equals %, (WC = .5), i s Per r = K r EI . . .(23) and, f o r the case where there i s no e l a s t i c foundation for inward displacements of the buckled r i n g (WC = 0), i s P c r r K r EI ...(24) The p l o t of the computer data f o r the three cases s t u d i e d i s shown i n F i g u r e (13). From the above i t can be seen that the c o e f f i c i e n t of the c r i t i c a l a x i a l load can vary fvomJ^T to J~l depending upon the degree of n o n - l i n e a r i t y of the e l a s t i c f o u n d a t i o n . * Reference (3) To f o l l o w page 32 F i g u r e 13 CHAPTER VI CONCLUSIONS The e f f e c t of an e l a s t i c f oundation i s to f o r c e the r i n g to buckle i n higher modes with a r e s u l t i n g i n c r e a s e i n the c r i t i c a l l o a d i n g . The theory presented i n t h i s t h e s i s shows that the d i r e c t i o n of a p p l i c a t i o n of the c r i t i c a l l o a d i n g has a marked i n f l u e n c e on the magnitude of the c r i t i c a l l o a d i n g i n the range of found a t i o n modu-l i i . where the f i r s t a d m i ssible mode of b u c k l i n g governs. F i x e d ended p a r t i a l r i n g s i n e l a s t i c foundations warrant f u r t h e r treatment and the author b e l i e v e s that the pro-gramming of the s i x t h order d i f f e r e n t i a l equation as f i n i t e d i f f e r e n c e s f o r s o l u t i o n by a d i g i t a l computer i s the most s u i t a b l e l i n e of f u r t h e r i n v e s t i g a t i o n . The n o n - l i n e a r i t y of the e l a s t i c f oundation, such as i s p o s s i b l e i n the case of s t e e l c u l v e r t s i n deep f i l l s , can r e s u l t i n values of the c r i t i c a l l o a d i n g lower than those p r e s e n t l y recommended f o r design. 34 APPENDIX 1 » COMPUTER PROGRAM $ ID 2793 SFORTRAN C CHOUKALOS MARK II LINK(O) DIMENSION Y0<15)»T1(14,14),A(14,14),B(14,14),C(14,14),Y1(14), 1RH(14),U(14) COMMON YO»Ti,A*B»C»Yl,RH,U,DN»FK,DFK,WC,NC,DUM READ i»DN»FK,DFK»WC READ 2,NC 51 FORMAT (5E14.7) 1 FORMAT (7F10.3) 2 FORMAT (6110) DO 3 1=1,11,5 3 READ l»YOW>,YO(1+1)»Y0l1+2),YO(1+3),YO<1+4) C MAKE UP MATRIX T l DO 4 1=1,14 60 4 J=l,14 4 Ti(i»J)=o.o T1<1,1j=-4*0 T l ( l , 2 i = o.O Tl( l , 3 ) = - 4 . 0 T l ( l , 4 ) = 1.0 Tl(2»l)"4*0 t l ( 2 , 3 ) = 5i0 Tl(2,4)=-4i0 Tl(2»5)= 1.0 T l ( 3 , l J = 4.0 Tl(3,2)=^5*0 Tl(3»4)= 5.0 Tl(3»5)=-4.0 Tl(3,6;= 1.0 Tl(12,9)=-i.6 T l ( 1 2 , i 0 ) = 4*0 Tl(12,ll)=-5.6 Tl(12,13)= 5*6 Ti(12*14i= J a4i0 Tl(I3»10i=-i*6 f i (i3» i i i = 4*o tl(13,12)= i i5*6 tl(13,14)= 4.0 T l ( 1 4 i l l i = - 1 * 6 35 Tl(14,12 Yi<14,13 t l ( 1 4 » 1 4 DO 5 1=4 HiI»1-1 DO 6 1=4 Tl<I.1-2 DO 7 1=4 t l ( I , 1 - 3 56 8 1=4 T i < i » i + i 56 9 1=4 T K i » i + 2 = 4.0 = -6.0 = 4*6 11 5 = -5.6 11 6 =4*0 i i , =-i.6 i i 7 8 • 5.6 i i 9 = -4.0 56 16 I = 4 » l i io T i ( i » i + 3 j = i * 6 CALL t ~ i 5 M A t ( T l » i 4 i l 4 ) CALL LINK(l) E M D 36 SFORTRAN L I N K ( i i DIMENSION YO(15)>T1<14,14),A<14,14),B(14,14),C<14,14),Yll14), 1RHU4) ,UU4) COMMON YO»Tl,A,B,C,Yl,RH,U,DN,FK,DFK,WC,NC,DUM 100 PRINT 101 101 FORMAT (13HLINK 1 STARTS) DlM=3.14159/<15.0*DN) D(JM=D1M**2 D0M=D1M**4 CALL E15VEC(YO»l5) 11 U(l)=-YO(2j/(2.0*D1M) (J{ 14) =Y0< 13)/(2.0*D1M I DO 12 1=2.13 12 U<I)=(YO(I-l)-YO(1+1))/(2.6*D1M) CALL Ei5VEC(U.l4) C MAKE UP MATRIX A (Tl+SK) DO 13 I= l i l4 DO 13 J=1,14 13 A(I ,J )=T1(I ,J ) DO 14 1=2*13 I F t U U ) ) 15,14,16 15 A ( l » I - l ) = A ( I , I - l ) - F K * D O M * W C A( I, I.+ i ) = A( I, I+l)+FK*DOM*WC 60 TO 14 16 A ( I , I - l ) = A i l , I - i ) - F K # D O M A l l » I + l ) = A ( I » I + i ) + F K * D O M 14 CONTINUE IF(U(1))17,40.19 17 ki1.2)=A(1,2)+FK*DOM*WC 60 TO 40 19 A(l»2)=A(l»2j+FK»DOM 40 IF(U(14)120,18,21 20 A(14,13)=A(14,13)-FK*D0M*WC 60 TO 18 21 A(14,13)=A(14,13)-FK*D0M 18 CONTINUE CALL Ei5MAT(A,l4,14) C INVERT MATRIX A CALL INVERT(A,14,DET,C) IF(DET)22,100.22 22 CONTINUE MAKE UP MATRIX B (T2+T3) DO 23 1=1,14 DO 23 J=l» 1 4 B(I»J)=0.0 B('l,l j=1.0 B(l»2)=-2 .0+DUM B< 1 » 3 ) = 1 . 0 B ( 2 » 1 ) = 2 . 0 - D U M B(2,3)=^2.0+DUM B ( 2 » 4 ) = 1 . 0 Q{ 1 3»ll) = - l i 0 B(13,12)=2.0-DUM B (13,14)=-2.0+DUM B ( 1 4 , 1 2)=- l i0 B(14,13)=2.0-DUM B(14,14)=-1.0 DO 24 1=3,12 B( 1,1-1)=2.0-DUM DO 25 1=3,12 B(1,1-2)=-1.0 DO 26 1=3,12 B(I,I+l)=-2.0+DUM DO 27 1=3,12 B<I,I+2)=1.0 MAKE UP MATRIX C (A*B) CALL MULT(A,B,C,i4 ,14) CALL LINK(2) END 3 8 SFORTRAN L I N K ( 2 ) D IMENSION Y 0 ( 1 5 ) » T 1 ( 1 4 , 1 4 ) , A ( 1 4 , 1 4 ) ,B.( 1 4 , 1 4 ) , C ( 1 4 , 1 4 ) , Y l ( 1 4 ) , 1 R H ( 1 4 ) , U ( 1 4 ) COMMON Y O » T i , A » B , C , Y l » R H , U » D N » F K , D F K » W C , N C , D u M P R I N T 101 101 FORMAT ( 1 3 H L I N K 2 S T A R T S ) 5 0 FORMAT(/// ) DO 28 IBM=1»NC C A L L M A T V E C ( C » Y 0 » Y J . » 1 4 » 1 4 ) 66 29 1=1*14 2 9 Y 1 ( I ) = - Y 1 ( I ) C A L L E 1 5 V E U Y l » 1 4 ) C F I N D MEAN SQ.ROOT V A L U E OF Y l SQY=0.0 DO 30 1=1,14 3 0 SQY=SQY + Y 1 ( I ) * * 2 S = S Q R T F ( S Q Y / 1 4 . i S 1 = S / 1 0 0 . 0 C F I N D RHO ( R H ) , F I R S T C H E C K I N G FOR SMALL V A L U E S OF Y ! DO 33 1=1,14 31 I F ( A B S F ( Y l ( I ) ) - S I ) 3 2 » 32 » 33 32 Y l(I ) = S l 33 C O N T I N U E DO 34 1=1,14 34 R H ( I ) = Y 0 ( I ) / Y 1 ( I ) C S C A L E Yl DO 35 1=1,14 35 Y 1 ( I ) = Y 1 ( I J / S C R E P L A C E YO WITH NEW Y i DO 36 1=1,14 36 Y 0 U ) = Y 1 ( I ) C F I N D MEAN SQ. ROOT V A L U E OF RHO FOR CONV. T E S T SQR=0.0 DO 37 1=1,14 37 SQR=SQR + R H ( I )**2 R=SQRTF(SQR/14•) R l=R/50»0 P R I N T 80,R 6 0 FORMAT (5 H R H 0 = , E 1 2 . 3 ) C CHECK CONV• OF RHO DO 39 1=1,14 39 38 I F ( A B S F ( R - R H < I ) ) - R l j 3 9 , 3 9 , 2 8 3 9 C O N T I N U E FKX=SQRTF(FK+ i.O) PX=R/DUM - 2 . 0 P R I N T 81»DN,FK»WC 81 FORMAT (17HNO. CYC OF BUCK = » F 6 . 1 , 2 X » 1 3 H F D N MODULUS =,F9 . 1 , 2 X , 4 1 = » F 6 . 3 ) P R I N T 8 2 , F K X , P X 82 FORMAT (1QHFDN TERM = » E 1 4 . 7 » 2 X » 1 1 H L 0 A D TERM = » E 1 4 . 7 ) P R I N T 50 FK=FK+DFK C A L L L I N K ( l ) 28 C O N T I N U E C A L L L I N K U J END $DATA 2.0 8 9.0 15.0 0.0 - . 2 0 8 - • 9 5 1 - . 7 4 3 - . 4 0 7 - . 9 9 5 - . 5 8 8 - . 5 8 8 - . 9 9 5 - . 4 0 7 - . 7 4 3 - . 8 6 6 - . 9 5 1 - . 8 6 6 - . 2 0 8 1 0 0 0 0 0 . 0 0 0 40 S t a r t Zero Storage of T l M a t r i x Make up T l M a t rix LINK (0) 41 Compute DIM, DIM, DOM Compute P r i n t U(I) A(I,J) = T1(I,J) = 0 >o Add FK(DOM)(WC) to AliMatr i x Add FK(DOM) to A M a t r i x Invert A Matrix = 0 S igna 1 P r i n t S t a r t Link 1 Make up C M a t r i x (A) (B) i Make up B Matrix Zero Storage of B M a t r i x LINK (1) Compute Y1(I) Y 1 ( I ) S -Y1(I) P r i n t Y 1 ( I ) J Compute Mean Square Root Value of Y1(I) = S < NC >S1 <: SI Y1(I) = SI Compute RH(I) : Y0(I) Y1(I) > R l - > FK = FK+DFK Pr i n t DN,FK,WC, FKX,PX Compute FKX,PX < R l Compute Mean Square Root Value of RH(I) = R Scale Y1(I) = Y1(I) S Y0(I)=Y1(I) LINK ( 2 ) 43 APPENDIX 2 NUMERICAL EXAMPLES EXAMPLE 1 A 30 foot diameter c u l v e r t i s to be i n s t a l l e d under a 45 foot high f i l l of compact earth weighing 110 pounds per cubic f o o t . The b a c k f i l l and surrounding earth i s assumed to have an e l a s t i c f oundation modulus of 15 pounds per sq. i n . per i n c h . I t i s f u r t h e r assumed that the uniform l o a d i n g remains normal to the r i n g and that t h i s l o a d i n g i s equal to the average v e r t i c a l pressure at the top of the c u l v e r t . Determine the gauge of 6" x 2" corrugated s t e e l c u l v e r t to s a t i s f y s t r e s s and s t a b i l i t y requirements assuming a f a c t o r of s a f e t y of 2 against the y i e l d s t r e s s and 2.5 against s t a b i l i t y . Average v e r t i c a l pressure at top of c u l v e r t p = 110 ( 45 ) 144 34.4 p . s . i . Maximum a x i a l load pr s 34.4(180) a 6,200 pounds per i n c h of width Allowable a x i a l s t r e s s when y i e l d s t r e s s i s 40,000 p . s . i . 40,000 = 20,000 p . s . i . 2 Area of corrugated p l a t e r e q u i r e d per inch of width A m 6,200 - .31 sq. i n . 20,000 A No. 3 gauge corrugated p l a t e has an area of .305 sq. i n . per inch of width and t h e r e f o r e i s acceptable provided i t meets the s t a b i l i t y requirements. The moment of i n e r t i a , The seam s t r e n g t h of m u l t i p l a t e c u l v e r t s should be checked. From t a b l e s s u p p l i e d by manufacturers of c u l v e r t s , a No. 3 gauge p l a t e with a 6 b o l t connec-t i o n per foot of width has an u l t i m a t e c a p a c i t y of 150,000 l b s . This provides a f a c t o r of s a f e t y of In checking the s t a b i l i t y of the c i r c u l a r c u l v e r t , assume a l i n e a r r a d i a l e l a s t i c f o u n d a t i o n . For t h i s case the c r i t i c a l a x i a l load i s given by equation (22) I r .146 i n . ^ per inch of width 150,000 6,200(12) 2.02 16,200 pounds per inch of width F a c t o r of s a f e t y 3 16,200 = 2.61 6 , 200 45 I f the r a d i a l e l a s t i c foundation acted only f o r out-ward displacements of the r i n g , equation (24) would govern and the c r i t i c a l a x i a l load would be p c r r = J~2 J K r E l ' = 11,500 pounds per inch of width F a c t o r of s a f e t y = 1 : L » 6 Q 0 » 1-85 6,200 I t i s apparent from the preceeding c a l c u l a t i o n s that n o n - l i n e a r i t y of the e l a s t i c f oundation can be an important item i n the dete r m i n a t i o n of the f a c t o r of s a f e t y a gainst b u c k l i n g of a s t e e l c u l v e r t i n a deep f i l l . A No. 3 gauge corrugated c u l v e r t i s adequate only i f the e l a s t i c f oundation i s completely l i n e a r . I f not, a heavier gauge corrugated p l a t e or an i n -crease value of the modulus of r a d i a l s o i l r e a c t i o n i s r e q u i r e d . EXAMPLE 2 Consider the same problem as i n Example 1, and i n -clude the e f f e c t of a t a n g e n t i a l e l a s t i c f oundation with a modulus of 15 pounds per square i n c h . Example 1 showed that a No. 3 gauge corrugated p l a t e had an adequate f a c t o r of s a f e t y a g a i n s t y i e l d s t r e s s . The determination of the c r i t i c a l uniform l o a d i n g f o r a c u l v e r t supported by a r a d i a l and t a n g e n t i a l e l a s t i c 46 foundation can be made from equation (11). Equation (11) f o r the c o n d i t i o n s of the problem reduces to Per - < n 2 _ 1 ) + K r r 4 + K t r 4 E 1 ( n Z -1) I T n 2 ( n 2 - l ) EI or Pcr = ( n 2 -1) EI + K r r + K t r ( n 2 - l ) n 2 ( n 2 - l ) The c o r r e c t number of modes of b u c k l i n g to give the minimum c r i t i c a l uniform pressures must be made by a t r i a l procedure because none of the graphs w i t h i n the t h e s i s were p l o t t e d using the p a r t i c u l a r con-d i t i o n s of t h i s example. C a r r y i n g out a t r i a l for n we f i n d that the governing value i s n = 8. The c a l c u l a t i o n s f o r n = 8 are as f o l l o w s : Pcr = ( 8 2 - l ) ( 3 0 x 10 6)(.146) + 15(180) + 15(180) ( 1 8 0 ) 3 ( 8 2 - l ) 8 2 ( 8 2 - l ) = 47.3 + 42.8 + .7 = 90.8 pounds per inch of width and the f a c t o r of s a f e t y against b u c k l i n g i s = 90.8 = 2.64 34.4 The above example shows that e f f e c t of the t a n g e n t i a l s p r i n g constant i s very small unless the number of modes of b u c k l i n g i s a l s o s m a l l . Even when n » 2, the maximum c o n t r i b u t i o n the t a n g e n t i a l s p r i n g constant can make i s 1/4 the value of the r a d i a l s p r i n g constant when both constants are the same. 47 BIBLIOGRAPHY 1. Stevens G.W.H., " S t a b i l i t y of a Compressed E l a s t i c Ring and of a F l e x i b l e Heavy S t r u c t u r e Spread by a System of E l a s t i c Rings", Q u a r t e r l y J o u r n a l of  Mechanics, V o l . 5 (1952). 2. Hetenyi M., Beams on E l a s t i c Foundations, U n i v e r s i t y of Michigan, S c i e n t i f i c S e r i e s XVI, The U n i v e r s i t y of Michigan Press. 3. Meyerhof G.G. and F i s h e r C.L., "Composite Design of Underground S t e e l S t r u c t u r e s " , E n g i n e e r i n g J o u r n a l , (Sept. 1963), The J o u r n a l of the E n g i n e e r i n g I n s t i t u t e of Canada. 4. Hahn L., "Flambage des anneaux c i r c u l a i r e s dans un m i l i e u e l a s t i q u e " , P u b l i c a t i o n s , I.A.B.S., V o l . 11, Z u r i c h (1951) . 5. Cheney J.A., "Bending and B u c k l i n g of Thin-Walled Open-Section Rings", Proceedings, A.S.C.E., V o l . 89, No. EM5 (Oct. 1963) . 6. H a l l and Woodhead, Frame A n a l y s i s , John Wiley & Sons, 1961. 48 7. Timoshenko and Gere, Theory of E l a s t i c S t a b i l i t y , E n g i n e e r i n g S o c i e t i e s Monographs, 2nd Ed., McGraw-H i l l , 1961. 8. Terzaghi K., " E v a l u a t i o n of C o e f f i c i e n t s of Subgrade R e a c t i o n " , Geotechnique, V o l . 5 (1955). 9. B a i k i e L.D., "Strength of Curved P l a t e s Bearing Against Dense Sand", Unpublished M.E. T h e s i s , Nova S c o t i a T e c h n i c a l C o l l e g e , H a l i f a x , N.S., (1961). 10. Z. angew. Mathematik und Mechanik V o l . 3, p. 227 (1923) 

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