Optimum Michell frameworks for three parallel forces

r t

NOV. 1963

CoA. Report Aero. No. 167

T H E C O L L E G E O F A E R O N A U T I C S

C R A N F I E L D

OPTIMUM MICHELL FRAMEWORKS FOR THREE PARALLEL FORCES

by

THE C O L L E G E OF A E R O N A U T I C S

CRANFIELD

Optimum Michell Frameworks for Three Parallel Forces b y

-H.S.Y. Chan, B.A.

SUMMARY

The design of Michell optimum structures to carry three coplanar forces has been the most popular topic in this field. However, the mathematical technique involved so far has been rather limited in as much as only strictly symmetrical cases have been considered. It is the intention of the present paper to apply the general theory of optimum design to a rather complicated problem of this type, and study in full detail the outcoming mathematical concepts, both numerical and theoretical.

V', CONTENTS 1. 2 . 3 . 4 . 5 . 6. 7, Summary Introduction Geometrical layout

Calculation of the virtual displacements Volume of the Michell structures

The limiting case for /d large Acknowledgment References Table 1 Appendix A Appendix B Figures Page 1 1 2 6 8 10 10 11 12 14

1. Introduction

The fundamental problem of structural design is the determination of

structures of minimum weight which safely equilibrate a given system of external forces. In the study of two-dimensional optimum Michell structures, it is of some advantage to make use of the analogy with the theory of plane plastic flow, which states that the members of a Michell frame lie along lines which have the sazne form, as the slip lines in the plastic case. Several cases of practical interest were studied in great detail by A.S. L. Chan^^'using this method. The present work is concerned with the optimum design of a framework under a further case of three force loading, which is derived by an extension of the slip line field for one of the classical designs of Michell.

2. Geometrical layout

Consider the loading problem of Figure 1. The points of application of the forces lie on a straight line, withOP< po*. The forces are all perpendicular to the line OO' and in equilibrium. The problem is to construct a Michell structure which equilibrates these forces.

It is already known that when OP = P O ' , the optimum structure is as shown in Figure 2. The corresponding slip line field is presented in Figure 3, which shows that the slip lines in the region A C P A ' C ' consist of circular a r c s and radii, with AC' perpendicular to A ' C , whereas the slip lines in the squares OAPA' , O ' C P C ' are naerely orthogonal straight segments.

It is proposed to extend this slip line field outside its original region. The procedure is explained below, and is illustrated in Figures 4 and 5, because of the symmetry, it is sufficient to consider only the region above O O ' .

(2.1) Beginning with Figure 1, determine first a point E on P O ' , so that OP = P E . Then draw from O, P , E, the slip line fields OAP, APC and CPE, identical with those shown in Figure 3.

(2.2) Since the point O is a point of application of force, it is possible that it is a singular point similar to the point P . This suggests the introduction of a region OAB similar to APC.

(2.3) Two orthogonal a r c s AB and AC, are now given, and so the slip line field can be extended to the whole region BACD in the manner of Figure 16 of Reference 2.

(2.4) The straight segment CE is perpendicular to CD, so the slip line field can now be extended to the whole region DCEF. Here, one set of the slip lines are straight segments which envelope an 'evolute'; the other set of slip lines are then 'involutes', (see Reference 2, p. 5).

Up to now, the extensions of the slip line fields on either side of OO' are separate from one another. Similar fields under the line OO' are also shown in Figure 5. At this stage, they must, if possible, be brought together to complete the final layout.

2

-s o a -s l i p line field can be obtained from t h e m , each c u r v e of which will i n t e r -s e c t O O ' a t 4 5 ° .

The layout of the r e q u i r e d s t r u c t u r e m a y now be obtained from the s l i p line field of F i g u r e 5. T h e n e c e s s a r y s t r u c t u r e for the p r o p o s e d p r o b l e m i s given in F i g u r e 4 .

So f a r , no explanation h a s been given a s to the method of c o n s t r u c t i n g the layout a n a l y t i c a l l y . A s would be expected for such a complicated c a s e , n u m e r i c a l solution of the a n a l y t i c a l equations will b e c o m e unavoidable. H o w e v e r , with the powerful g r a p h i c a l method developed in the t h e o r y of p l a s t i c i t y - s e e for i n s t a n c e R e f e r e n c e 3 , C h a p t e r 6 - one can obtain s u c h a layout with sufficient d e g r e e of a c c u r a c y . T h e s l i p line field in F i g u r e 5 w a s c o n s t r u c t e d by t h i s g r a p h i c a l method with an i n c r e m e n t of 10 between adjacent s l i p l i n e s .

3 . Calculation of the v i r t u a l d i s p l a c e m e n t s

F o r the s a k e of c o n v e n i e n c e , s o m e b a s i c f o r m u l a e a r e s t a t e d and notations explained, using F i g u r e 6.

T h e layout l i n e s for the s t r u c t u r e can be taken s e p a r a t e l y in each r e g i o n a s c o - o r d i n a t e c u r v e s of a c u r v i l i n e a r c o - o r d i n a t e s y s t e m ( a , P ) .

Denote ;$ = " a + P ( D which i s the angle between the positive a - d i r e c t i o n and the x - a x i s . The r a d i i

of c u r v a t u r e of the a, p c u r v e s a r e denoted b y A and B r e s p e c t i v e l y . T h e y a r e r e l a t e d by

T h e v i r t u a l d i s p l a c e m e n t s along the a , P c u r v e s a r e denoted by u and v. T h e y satisfy the following e x p r e s s i o n s

{

ÖV a>(a , p ) = = -Ae Aoj 2e(a • t + P ) + du ÖP - ^ è P a>(o.o) V = -BCD (3) Be

w h e r e (u( a , p ) = 2e(a + P ) + a>(o.o) (4) i s the r o t a t i o n at any point ( a , p ) within the r e g i o n , and ± e denote the p r i n c i p a l

s t r a i n s . In what follows, the ( a , P ) c o - o r d i n a t e s y s t e m s a r e so chosen that the d i r e c t s t r a i n in the a - d i r e c t i o n i s a l w a y s - e , and that in the p - d i r e c t i o n i s a l w a y s +e, ( s e e F i g u r e 7).

( 3 . 1 ) S t a r t from r e g i o n OAB, a s s u m i n g , the o r i g i n O to be fixed. Since the ( a , P ) c o - o r d i n a t e s coincide with p o l a r c o - o r d i n a t e s , the v i r t u a l d i s p l a c e m e n t s satisfy the following r e l a t i o n s .

*

è u öV u -r— = -e , — r — + — = e , da a d p a and u(o.o) = o = v ( o , o ) . T h e solution i s ""^ "' V.'' T " . . , ^ ^

èu

a

èp

V a = O (5) v( a , p ) = 2 e a P + ka ^®' w h e r e k i s a c o n s t a n t .

( 3 . 2 ) C o n s i d e r next the r e g i o n O A P , and t a k e the point A a s o r i g i n . The ( a , P ) c o - o r d i n a t e s a r e s i m p l y C a r t e s i a n c o - o r d i n a t e s , and so

ÖU è v è u è v _. -_. - — = -e , T— = e , -T-- + ^— = O (7)

èa èp èp èa

D e r i v i n g the b o u n d a r y condition on OA from (6), one obtains the following r e s u l t . 'u( a , '-v( a ,

ru(P)

tv(P)

p ) = - e a + e r - k p p ) = e p + k a - k r = u ( o , r ) = r ( e - k) = v ( o . r ) = r ( e - k)

As r ' (9)

the constant k i s b e s t chosen s o that the point P i s at r e s t , s i n c e t h i s simiplifies the r e m a i n i n g c a l c u l a t i o n s c o n s i d e r a b l y ,

( 3 . 3 ) In r e g i o n s A P C and C P E , the layout i s the s a m e a s in ( 3 , 1 ) and ( 3 , 2 ) .

T h e r e i s no need, t h e r e f o r e , to go into the s a m e d e t a i l . After s i m p l e calculation, the following r e s u l t s a r e obtained.

On AB AC CE u = - e r , V = e r ( 2 p + 1) u = - e r ( 2 a + 1) , v = e r u = - e a - e r (1 + « ) , v = . as(A) = e e a (1+Jt ) + e r (10) (11) (12) w h e r e u, v and a , P in the above equations a r e consistant with the notations for the v i r t u a l d i s p l a c e m e n t s and c o - o r d i n a t e s used in r e g i o n s BACD and D C E F , with points A and C r e s p e c t i v e l y taken a s o r i g i n s , ( s e e F i g u r e 7).

At t h i s s t a g e , the c l a s s i c a l solution of the s t r u c t u r e in F i g u r e 2 i s a l r e a d y d e t e r m i n e d . F r o m (12), the v i r t u a l d i s p l a c e m e n t s of the point E (which c o r r e s p o n d s to point O ' in F i g u r e 2) a r e

u = - e r ( 2 + K ) , V = e r ( 2 + K ) (13) T h e r e s u l t a n t v e r t i c a l d i s p l a c e m e n t i s

4

-and s o the volume V of the s t r u c t u r e is (2 + It) d .

2f

a . f . f

(14)

w h e r e f i s the allowable s t r e s s in the m e m b e r s of the s t r u c t u r e .

( 3 . 4 ) F o r r e g i o n BACD, the c o m p l e t e a n a l y s i s had been c a r r i e d out a l r e a d y in R e f e r e n c e 2, and the r e s u l t s a r e a s follows.

<o

I

{

2 e ( a + P ) + a>(A) = 2e( a + p ) + e A{ a , p ) B( a . p )

_{Io(2,/5T) +^[f- Ii(2VÏ7 )}

(15) (16) u( a , p ) = - e r (1 + 2 a ) 1 ^ ( 2 7 ^ ) + 2 i/ET^ IJL(2 ^ ~ p ~ " ) v( a , p ) = e r (1 + 2p ) y 2 i/öTp) + 2V5rp I i ( 2 i / a p ) (17)

T h e b o u n d a r y conditions on CD can be obtained by s e t t i n g a = T ' ^ " ^' ( 3 . 5 ) In r e g i o n D C E F , the ( a , p ) c o - o r d i n a t e s a r e shown in F i g u r e 7. By H e n c k y ' s t h e o r e m ' ^ ' , the v a r i a b l e P is identical with that in r e g i o n BACD on c o r r e s p o n d i n g a - l i n e s . Since one s e t of s l i p l i n e s a r e s t r a i g h t s e g m e n t s , the calculation of the layout i s quite s t r a i g h t f o r w a r d , taking into account the b o u n d a r y conditions on CD, ( R e f e r e n c e 2, § 4 , c a s e 2). The r e s u l t s a r e A( a , p ) = 1 B( a , P ) = a + B(o, P ) = a + r o) = 2ep + a)(C) = e ( 2 p + l + n ) (18) (19) u( a , p ) = - e a + u(o, p ) = - ea - e r ( I + K ) lo(i^2lp)+ •2«p I i ( ^ 2 « p ) 1 v( a , p ) = am + v<o. p ) = (2p + 1 + K ) e r + e r (1 + 23) y / 2 ^ ) + y 2 « p Ii(V'2rtD (20) ( 3 . 6 ) F i n a l l y , in r e g i o n F E F * , the ( a , p ) c o - o r d i n a t e s a r e shown a l s o in F i g u r e 7. By H e n c k y ' s t h e o r e m , the v a r i a b l e p i s s t i l l i d e n t i c a l with that of the p r e v i o u s c a s e for c o r r e s p o n d i n g a - l i n e s . F r o m (18), the r a d i u s of c u r v a t u r e B of t h i s new c o - o r d i n a t e s y s t e m s a t i s f i e s

B(o, P ) = r 1 + y / 2 n p ) + / ^ I i (1/2 «p ) (21)

Because of symmetry, A( a , P ) = B( p , a ) , (22) and so A ( a . o ) = B(o, a ) = r 1 + 1^(72 « a ) +Jf^ It(VSrii^) 1 (23)

From (19), CD = 2e(a + p ) + a)(E) = 2e( a + P) + e (1 +« ) (24) Using equation (2) and the boundary conditions (21), (23), A, and B can be

obtained from the following formula (Reference 2, equation (27): a

A ( a . p ) = A ( o , o ) I o ( 2 y ^ ) + ƒ lji2^(a - 6 ) p ) ^^Li_iO>dS +

o P

+ r y 2 v ' a (p - T)) )B(o, Ti)dTi (25)

Details of this integration are presented in Appendix A, and the result gives, from (A.4), (A.10),

A ( a . p ) = r | ( 2 + - ^ ) y 2 t ' a p ) + J l " I i ( 2 Va p ) + ^ ^ ^ ] i ( / 2 P ( 2 a + «)) " , .k . \ l + k / ^ v(l+k)/2 k=o ,k / \l+k / „o \(l+k)/2 k=o » - ,.k / .2+k . „ J l + k ) / 2 •^

I ^ • ( i ) • ( ^ ) • i:..<^^^TÏJTT), }

k=o

The formulea for the virtual displacements are obtained by adding equation (3) in pairs - | r + I T - = - Ae - Be (2a + 2p + 1 + „ ) r èa èp

I

(27)

- ^ + - ^ = Be + Ae (2a + 2p + 1 + It) èa dp Letting a = cx+ fi , r = a - p , (27) becomes 2 - | ^ = - Ae - Be (2a + 1 + n ) oa èv , , <28) 2 - — = Be + Ae (20 + 1 + «) oa

6

-T h e o r e t i c a l l y s p e a k i n g , once the v a l u e s of u and v on the b o u n d a r y a r e known, t h i s s e t of equations can be i n t e g r a t e d along the line T = constant to obtain u ( a , P ) and v ( a , P ). Owing to the complexity of equation (26), it d o e s not s e e m p o s s i b l e to obtain an a n a l y t i c a l e x p r e s s i o n for u, v. However, for the solution of the p r e s e n t p r o b l e m , it i s of i n t e r e s t to calculate only t h o s e v a l u e s of u, V on the line T = o.

L e t t i n g a = — = p in (28) and noticing (22), the following i n t e g r a l e x p r e s s i o n i s d e r i v e d ,

2[x

u(fi , M) = u(E) - I r (20 + 2 + It) A ( ^ | , I ' j d a

o 1 = - ( 2 + ] t ) e r - e F ^i (4 (i t + 2 + «) A ( ^t, Mt) dt, (29) w h e r e

A(^t, ^t) = r j(^2 + ^ X ( 2 ^lt) + I ^ ( 2 ^ x t ) + / ' | ^ ~ ^ I^ (V2 n t(2 p t + « )

+ k=o ^ / n k r „ , -, / „ x l + k / , , x ( l + k ) / 2 . (30) F i n a l l y , by s y m m e t r y , v ( ^ , t i ) = - u ( n , ^ ) (31) Detailed c o n s i d e r a t i o n of the n u m e r i c a l c a l c u l a t i o n s i s given in Appendix B .

T h e r e s u l t s a r e shown in T a b l e 1. 4 , Volume of the Michell s t r u c t u r e s

Before c a l c u l a t i n g the volume of the s t r u c t u r e s , it i s n e c e s s a r y to r e l a t e the ( a , p ) c o - o r d i n a t e s to d i s t a n c e s along O P O ' .

R e f e r r i n g to the c o - o r d i n a t e s y s t e m in r e g i o n F E F ' , the d i r e c t i o n c o s i n e s of the tangent l i n e s t o the c o - o r d i n a t e c u r v e s a r e given by, ( R e f e r e n c e 2, equation (22)).

/ 1 ex 1 èy

C O S V = -^ —•• — •— M- •

A èa B öp (32)

, . l e y -1 ex

sin p = -T Tr~ = 'TT -^— '^ A èa B èp

ƒ

( a , P )

' (A cos j< da - B sin/^ d p )

( a , p ) (33) (A sin /5 da+ B cos ^ d p )

which t a k e s a r a t h e r s i m p l e form when i n t e g r a t e d along the line E O ' . F o r any point O ' ( a , p ) on E O ' , the following r e l a t i o n s a r e s a t i s f i e d .

a - (I - P

•] cos ^ = 1 , sin ^ = 0 (34)

A( n , n ) = B( ^, ^ )

Substituting (34) into (33) gives

x = ƒ A( a , a ) d a = ƒ B(P , P)dp = y (35) o o

T h e i n t e g r a l

x = ƒ A ( a , a ) d a = j A( ^ t , jit) dt (36) o o

can be c a l c u l a t e d in e x a c t l y the s a m e way a s that of equation (29). R e f e r r i n g to F i g u r e 7, the final r e l a t i o n i s

d = E O ' / x a + y 2 = •2x (37)

Notice that the angle \x m u s t be l e s s than 135 , o t h e r w i s e the s l i p line fields at point O will o v e r l a p . T h i s shows that the length, t , will attain an upper l i m i t when ^ = 135 . A c u r v e i s plotted in F i g u r e 8, showing t h i s r e l a t i o n s h i p , and the c o r r e s p o n d i n g n u m e r i c a l r e s u l t s a r e shown in Table 1,

Another but l e s s a c c u r a t e way of obtaining the r e l a t i o n b e t w e e n * /d and \i i s to m e a s u r e the g r a p h i c a l layout of F i g u r e 5, for s e v e r a l v a l u e s of [i. With the help of i n t e r p o l a t i o n , the r e l a t i o n at any position of E O ' can be found.

T h e volume of the s t r u c t u r e s can now be d e t e r m i n e d . A s s u m i n g O, P , O , a r e the points of action of the f o r c e s , t h e n , r e f e r r i n g to F i g u r e 1 once m o r e ,

T h e r e s u l t a n t d i s p l a c e m e n t of the point O ' i s , from ( 2 9 ) ^ ( 3 1 ) . along the line of action of the force F g and i t s m a g n i t u d e i s equal to - / 2 u ( n , M ) . Since

Tl

VLt:c-,

the points O and F a r e at r e s t , the volum.e of the r e q u i r e d s t r u c t u r e i s

vru(

JA . H ) ^ ' ef -/'2d d + *

u(n

) ^ ' ef (39) T h e n u m e r i c a l r e s u l t s a r e shown a l s o in T a b l e 1, 9 indicating the c o m p l e t e s e t of s o l u t i o n s . A c u r v e i s plotted in F i g u r e

An a p p r o x i m a t e e s t i m a t e of the volume r e q u i r e d m a y be obt ained f r o m F i g u r e 10. T h i s h a s been c o n s t r u c t e d using the g r a p h i c a l method, ( R e f e r e n c e 2, § 6), and from t h i s , the l o a d s c a r r i e d by each m e m b e r can be calculated by

s t a t i c a l a n a l y s i s s t a r t i n g from point O ' . The length of each m e m b e r can be obtained by n a e a s u r e m e n t . T h i s gives an a p p r o x i m a t e solution for the r e q u i r e d s t r u c t u r e . An e x a m p l e of such a calculation i s r e c o r d e d by the n u m b e r s attached on e a c h m e m b e r of F i g u r e 10. Some v a l u e s a r e tabulated below for a d i r e c t c o m p a r i s o n with the t h e o r e t i c a l r e s u l t s . * / d 1 1.5 2 2 . 5 3 3 , 5 4 4 . 5 5 The l i m i t i n g Vf / Fd <^^ 2 . 5 7 0 8 3.1366 3.5896 3.9662 4 . 2 8 8 2 4 . 5 7 0 2 4 . 8 2 1 0 5,0462 5.2495 c a s e for (39)) lid l a r g e .

i < «

-3.269 3.63 4 . 0 8 2 4 . 3 9 8 4 . 6 8 4 . 9 3 8 5,182 5,402 •— ( g r a p h i c a l solution) 5.

It i s shown in T a b l e 1 that the solutions e x i s t only for t h o s e c a s e s satisfying

' * d 149.1764 (40)

P O '

If t h i s r e l a t i o n does not hold, O P b e c o m e s v e r y s m a l l in c o m p a r i s o n with and F i » F a It i s then n a t u r a l to a p p r o x i m a t e the loadings at O and P by a force F p e r unit length continuously d i s t r i b u t e d on the c i r c u m f e r e n c e of a c i r c l e Q, with O P a s d i a m e t e r and c e n t e r Q lying m i d w a y between O and P , ( s e e F i g u r e 11), such that the r e s u l t a n t i s a v e r t i c a l force

at Q.

i è and a m o m e n t F_ . ( » • + * )

Such a loading p r o b l e m i s known a s M i c h e l l ' s C a n t i l e v e r ( R e f e r e n c e 1, ex, 1), the s l i p line field of which c o n s i s t s of e q u i a n g u l a r s p i r a l s . If p o l a r c o - o r d i n a t e s ( p , e ) a r e introduced at c e n t e r Q, the s p i r a l s have the f o r m , ( R e f e r e n c e 4, ( 3 . 1 5 ) ) ,

a - c u r v e s P - c u r v e s , e - 2 P + j p = ke 4 It , k > o i s a constant , - e + 2 a - -r p = ke 4 (41)

.which shows that the c o - o r d i n a t e c u r v e s

p = c o n s t , and 6 = const.

a r e l i n e s of p r i n c i p a l s t r a i n , ( r e p r e s e n t e d by dotted lines in F i g u r e 12), If u, v a r e v i r t u a l d i s p l a c e m e n t s along p and 6 d i r e c t i o n s , then

(42)

( e i s the allowable s t r a i n ) , which gives

u = C,sin0 - C a cos 0 /-„» V = C a s i n © + (^ cos 0 + C3 p + 2eplnp

w h e r e C,^, C g , C 3 , a r e c o n s t a n t s of i n t e g r a t i o n . A s s u m i n g the point Q to be fixed, then Cx= P g = 0- T h e t h i r d constant can be chosen to b r i n g the point O ' to r e s t , that i s C3 = -2e In il + | ) (44) èp è u èv pèe èp Ö V ^ u pèe p - ^ = 2e . P (44), 2

T h e v i r t u a l d i s p l a c e m e n t s at the b o u n d a r y of the c i r c l e Q a r e , by (43) and

{.

J . 2el n ( l + 2 j )

2 d

(45)

T h e volume of the s t r u c t u r e i s then given by the work by F . The r e s u l t i s 2*

efV = r F*. I , 2e In (1 + 2 | ) de

o

a«

= 2eln(l + 2 - ) , J F • I de

= 2 e l n ( l + 2 | ) . ^ • < ! + * > <'*^^

F i n a l l y , by using (38), the above e x p r e s s i o n can be put into a form s i m i l a r to (39),

_ d _ \ 1- / . . o * 1 F d + d '

10

-The numerical values of (47) are plotted as a dotted line in Figure 9, which indicates the similarity of the two solutions. If the original loading problem of § 2 has been formulated in the manner used here, then equation (47) would give the required volume of structure. The layout of the Michell Cantilever can be extended to infinity, and this gives a more complete solution. The present solution, however, covers what may be expected to be a practical range of geometrical layouts.

A further remark may be made. If the point P is situated above OO' and the angle OPO' is greater than a right angle, a similar approach can be adopted. Michell's original structure for the case OP = P O ' and the corresponding extension of the slip line field are shown in Figures 13 and 14.

Acknowledgment

The author wishes to acknowledge the guidance given by Professor W.S, Hemp who first stimulated the author's interest in this subject and has provided invaluable suggestions for the completion of this paper.

References 1. Michell, A.G.M, 2. Chan, A . S . L . 3. Hill, R. 4. Hemp, W.S. 5. C . l . T . 6. Bailey, W.N, 7. Watson, G.N,

The limit of economy of material in frame structures. Phil. Mag. Series VI, Vol, 8, 1904, pp. 589-597

The design of Michell optimum structures, College of Aeronautics Report 142, 1960. Mathematical Theory of Plasticity. Oxford, Clarendon P r e s s , 1950, Theory of structural design.

College of Aeronautics Report 115, 1958. Tables of integral transforms, V.2. Bateman Manuscript Project.

McGraw Hill, 1954.

On integrals involving Bessel functions, Quart, J . Math., Oxford Series, Vol. 9, 1938. pp. 141-147.

Theory of Bessel Functions. Cambridge Univ. P r e s s , 1944, 8. N . P . L . Modern Computing Method.

1» (degrees) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135

T ^ ^ e q , 29)

5.1416 6.9360 9.1725 11.9515 15,3953 19.6527 24.9043 31.3698 39.3156 49.0646 61.0086 75.6220 93.4789 115.2746 141.8499 174.2214 213.6173 261,5221 319.7278 390,3979 476,1433 580.1131 706.1051 858.6982 1043.4130 1266.8977 1537.1629 1863.8560 *(eq. 37) d 1.0000 1.3372 1.7317 2.1934 2,7342 3.3681 4,1113 4,9832 6,0066 7.2082 8.6196 10,2780 12,2274 14.5193 17,2149 20.3862 24.1180 28.5105 33.6820 39.7721 46.9456 55.3969 65.3557 77.0934 90.9303 107.2448 126.4841 149.1764 Vf (eq, 39) Fd 2.5708 2,9677 3.3578 3.7426 4.1228 4.4991 4.8724 5.2430 5.6112 5,9775 6.3421 6.7053 7.0671 7,4278 7,7876 8.1464 8.5046 8.8620 9.2188 9.5751 9.9309 10.2863 10.6412 10,9958 11,3500 11.7040 12.0577 12.4111 Table 1.

12

-Appendix A

I n t e g r a t i o n of equation (25)

F r o m equations (21) and (23), it i s found that

è A ( t .o) è | B(o, tj ) = r

ƒ ! - ix('^27r)+ ^ i . ( / 2 7 r )

1 f L (y2«T) ) 4

Ji^x(/^

and b e c a u s e t h e r e f o r e lim Z.'^ I , ( Z ) z-»o k A ( o , o ) = r ( 2 + | ) 2*^ r (k+1) for any k > o,

Substituting from (A. 1, 2) in (25) : •

( A . l ) (A, 2) (A. 3) (A. 4)

u P

ƒ I^(2V(a- 6)P) ^^^^g^'°^dg + ƒ I^(2ya (p

- T , ) ) B ( O , T ,

)dn

o o

a

= V \ I^(2A a - D P ) / ^ I i ( / r ; i ~ ) + ^ f e ( / 2 l T ' ) di

+ r

ƒ I^j(2ya(p - T,) r 1 + Io<^2«

T " ) + / ^

l i ( ^ 2 « tl)

dn

Next, by s u b s t i t u t i n g | = a s i n ? . t] = P sin % , the two i n t e g r a l s can be combined to give the following

A ( a , p ) - A ( o . o ) I ^ (2 v ^ ) = 2r / ^ I^ (2 V ^ cos C ) [ , J ^ — ^.^ ^

o

(V 2jta sin t ) ^— +

^^hi^_^ ^, , .i„,vr^»i„c,.^f '•'•^i;'"^';

s i n 5 cosf d 5 (A. 5)

T h i s e x p r e s s i o n can be d e r i v e d , for e x a m p l e , by taking limit in S o n i n e ' s f i r s t finite i n t e g r a l , s e e g 1 2 . 1 1 of R e f e r e n c e 7.

The i n t e g r a n d s in (A. 5) have the s t a n d a r d f o r m s , for which the r e s u l t of i n t e g r a t i o n can be found, for i n s t a n c e , in R e f e r e n c e s 5, 6, 7. T h e y a r e

1 P

r ^ IQ (p cos 5 ) sin 5 cos 5 d5 = ~ F lJiZ)ZdZ » - ^ (A.6)

o o

2

F I (p c o s t ) I (q sin f) s i n ^ cos 5 d 5 = , P "^ ^ - (A. 7)

J ° o !. V + q ^

It 2 t ^ k l+2k ƒ I ^ ( p c o s O l x ( q s i n O c o s 5 d 5 -.1 l - ± ^ . - ^ ^ ^ ^ ^ ^ ^ l ^ ^ ^ W 7 ^ ) 2 k l+2k

F I (p c o s U l a ( q 8 i n e ) ^ ? ^ d 5 = 7 7 - ^ - ^ ^ • ^-nTïïTTT ^i ^ J ^ ^ ^ ^ )

J o ^ sin 5 ^ 4 ^ 2\.(k+2) (fy»+q^<l+l^)/2 1+k

k=o

(A. 9)

where p, q are independent of 5 .

Comparing these general formulae with equation (A. 5), the final result is

a . p ) - A(o, o) I^ (2V5T ) = r 1 ^ 1 ^ (2 ^ T > + J 2 ^ ^ „ . Ii(/2P(2a+«))

" / i \ k / x l + k . ^ \ ( l + k ) / 2 'L = 0 " / 1 vk / \ l + k / ^ \(1 k=o ,k / ^ v^l+k/ ^ \ ( l + k ) / 2 k=o v(l+k)/2 +

k o

A / n k / \ 2 + k . „ a i + k ) / 2 ^ (A. 10)

14

-Appendix B

N u m e r i c a l a n a l y s i s of equation (29)

Before applying any method of n u m e r i c a l i n t e g r a t i o n , it i s of s o m e i n t e r e s t to e x a m i n e the s e r i e s r=^ A I ^\^ r 0 T / * \ l + k , . ( l + k ) / 2 1+k / „ ^ . ( l + k ) / 2 _ k». • L k+1 2k+4 J • V2 , k=o k=o ( B . l ) n i o r e c a r e f u l l y .

Making u s e of the following r e s u l t , ( s e e R e f e r e n c e 7, § 2.12)

|Jj^(Z)| < l i | i - e ^ l ^ l ^ , f o r k » o , ( B . 2 )

the m o d u l u s of the k - t h t e r m of the s e r i e s is

* vl+k / „ .., v ( l + k ) / 2 (2+ 2) /jJ^"*"VjtL^_ ^ (k+1)'. V 2 / ' V2 ^t+ « J ( l + k ) / 2 , r »l+k ^ ( 2 / i t + w ) \^^^''^ ( H 2 ^ t ( 2 ^ t + 1 . ) e 2 (k+1)

.,ƒ""'*" siHn'*"

T h i s shows that the s e r i e s of ( B . l ) c o n v e r g e s even f a s t e r than the exponential s e r i e s , and i n t e g r a t i o n t e r m by t e r m is t h e r e f o r e p e r m i s s i b l e . F u r t h e r m o r e , s u m m i n g the s e r i e s ( B . l ) from k = N s h o w s , by using ( B . 3), 2 . \<'^*> < (2 + 2 ) e ^ [(k+l)t ]2 k=N k=N

« ?<2'^+'') f^ (K)'^V

< <2 + 2 ) e 2 . KN+l)l(N+2)'^-^]^ k=N • ^ / o . . _ L * \ « 1 + N _ It k - N

(2.i)e^* ^ < i ü i - . v r - 1 ^ 1

- ^ ^ " ^ 2 ' ^ • [(N+l)>.]^' Z.L<N+2)^J _k=N

, J(2.+«) (IM )^^^

( 2 + 2 ) e [(N+1)'. ]2

_{• - 'rm.2f] '"•'"}

provided that (N+2)^ > - p. This result can be used to estimate the number of t e r m s required in nunrierical calculations.

For the numerical integration of equation (29) (and (36)), Simpson's Rule is adopted, which is particularly suitable for automatic computing, (see for

instance Reference 8). The detail of the numerical work can best be explained by the flow diagram of computing below. The result of computing shows that up to seven t e r m s of the s e r i e s ( B . l ) have been used, and the interval of integration

[0,1] has been divided up into 128 equal p a r t s .

set value of

Divide the interval of integration into two equal parts

For each co-ordinate, calculate the integral to the accuracy of four decimal places.

Simpson's Rule of Integration

Test accuracy of the result of integration up to four decimal places

(Satisfy)

( Not ) (Satisfy)

Output Result Half the interval of integration

FIG. I. It e » FIG. 2 T.é/Ji FIG. 3 PRINCIPAL STRAIN - « P P I N C I P A L STRAIN <•«

IDS

MICHELL CANTILEVER

I— <j - 4 —

FIG. II.

X

H

FIG. 12 MICHELL CANTILEVER