THE USE OF COMPLEX VARIABLE THEORY

I N SEVERAL AREAS OF PLANE ELASTICITY

Thomas C l i f f o r d C h a r l t o n

B.Sc., S imon F r a s e r U n i v e r s i t y , 1 9 6 8

A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

i n t h e D e p a r t m e n t

o f

Ma thema t i c s

Thomas C l i f f o r d C h a r l t o n 1 9 7 2

SIMON FRASER UNIVERSITY

September, 1 9 7 2

APPROVAL

Name: Thomas C l i f f o r d Charlton

Deg ree : Master o f Science

T i t l e of Thesis: The Use o f Complex Variable Theory

i n Several Areas o f Piane E i a s t i c i t y

Examining Committee: LL

'chairman: N. R . Rei l l y

Dr. ~harma Senior Supervisor

R. W, Lardner

.D? Shadman

M. Singh

ABSTRACT

I n t h i s paper t h e use o f complex v a r i a b l e theory i n s o l v i n g problems

i n p lane e l a s t i c i t y and t h e r m o e l a s t i c i t y i s discussed. Two approaches

t o s o l v i n g the p lane e l a s t i c i t y problem a re presented. One i s t h a t

developed by N.E. M u s k h e l i s h v i l i us ing i n t e g r a l equations and t h e

other u t i l i z i n g or thogonal polynomials by Stippes-Shadman. A method

o f s o l u t i o n f o r t h e p lane thermoelast ic problem, again us ing complex

v a r i a b l e theory and developed by B.E. Gabywood i s then s e t f o r t h .

The paper concludes w i t h t h e a p p l i c a t i o n o f complex v a r i a b l e theory

t o t h e s o l u t i o n o f t h e problem o f t o r s i o n o f polygonal bars. Th is

i s done through t h e use o f or thogonal polynomials and i l l u s t r a t e s how

C r e a d i l y t h i s method i s adaptable t o numerical computations

ACKNOWLEDGEMENTS

I would l i k e t o express my thanks t o D r . D. Sharma and D r . D. Shadman,

Mathematics Department, Simon Fraser U n i v e r s i t y f o r t h e i r ass is tance

and encouragement du r ing the p repa ra t i on o f t h i s t hes i s . A lso t o

Miss M. Grat ton, Computing Centre, Simon Fraser U n i v e r s i t y , f o r her

ass is tance d u r i n g the computer ana lys i s phase o f t h e thes i s .

Thanks a r e a l s o due t o t h e Simon Fraser U n i v e r s i t y P res iden t ' s

Research Grants Committee f o r t h e i r f i n a n c i a l support du r ing t h i s f'

per iod . I

F i n a l l y I wish t o express my app rec ia t i on t o Mrs. S. Deschamps f o r

t y p i n g t h e e n t i r e work.

TABLE OF CONTENTS

Page . ....................................................... APPROVAL ii

....................................................... ABSTRACT iii

............................................... ACKNOWLEDGEMENTS i v

INTRODUCTION ................................................... 1

........ ............ ...An.& * - t * rn r ln l C I A C T T ~ T T V 2 A . COMPLEX V A K ~ A B L ~ ~ IN I W U ~ ~ ~ ~ c n a r v n n ~ L L ~ U ~ ~ V L ~

1 . M u s k h e l i s h v i l i ' s Method ................................ 2

2 . Orthogonal Polynomials ................................. 11

....................................... 3 . Ther rnoe las t ic i ty 20 f'

B . TORSION OF AN HEXAGONAL BEAM (2D) .......................... 28

........................................ 1 . Basic Equations 28

2 . Method o f S o l u t i o n ..................................... 31

................ 3 . Determinat ion o f Orthogonal Polynomials 37

4 . Determinat ion o f H ( r r O ) ................................ 43

5 . Determinat ion o f Stresses .............................. 48

6 . Conslusion ............................................. 49

C . RESULTS .................................................... 50

.............................................. 1 . TABLE I 50

2 . TABLE I 1 .............................................. 51

3 . TABLE I11 .............................................. 54

4 . TABLE I V .............................................. 57

5 . TABLE V .............................................. 58

D . REFERENCES ................................................. 59

INTRODUCTION

6. Kolossof f [I 1 I as e a r l y as 1909 I proposed t h a t complex ' v a r i a b l e

theory be used i n t h e s o l u t i o n o f problems i n p lane e l a s t i c i t y . I t

wasn't u n t i L n e a r l y f o r t y years Later t h a t thi; p roposa l was brought

t o a successfuL conc lus ion by t h e work o f N. I . M u s k h e l i s h v i l i E21.

The purpose o f t h i s paper i s t o show n o t o n l y hou MuskheLishvSLi

a p p l i e d complex v a r i a b l e theory t o p lane e l a s t i c i t y problems, b u t

a l s o t o i l l u s t r a t e one a l t e r n a t e method t o M u s k h e l i s h v i l i ' s , and

how complex v a r i a b l e theo ry was a p p l i e d t o t h e r m o e l a s t i c i t y by

B.E. Gatewood E33.

I n a d d i t i o n t h e t o r s i o n problem o f an hexagonal beam i n two,

dimensional e l a s t i c i t y i s so lved through the use o f complex p o t e n t i a l s

and or thogona l polynomials.

A . COMPLEX VARIABLES IN TWO DIMENSIONAL ELASTICITY

1. Muskhel i shv i t i 's Method

We w i l l f i r s t d iscuss t h e approach used by M u s k h e l i s h v i l i C51 *

i n t h e s o l u t i o n o f t h e p lane e l a s t o s t a t i c ~ p r o b l e m i n l i n e a r

e l a s t i c i t y .

I n t h e above case t h e r e l e v a n t d i f f e r e n t i a l equat ions and

boundary c o n d i t i o n s take t h e form

and

where Ta(S) a r e known f u n c t i o n s o f t h e a r c parameter S and

v i s t he rec tangu la r c a r t e s i a n component o f t h e e x t e r i o r B u n i t normal t o C , t h e boundary o f some p lane r e g i o n D.

* In p lane s t r e s s t h e displacements and s t resses a r e average ones and A a l s o changes i t s va lue.

* * Repeated i n d i c i e s mean summation w i t h Greek L e t t e r s rang ing over 182 and t h e Roman over 1,2,3.

G.B. A i r y CB1 noted t h a t t he re e x i s t s a f u n c t i o n U(xI,x2) such

t h a t if

then equat ions (11, (21, and (3) reduce t o

E. Goursat C71 f i r s t ob ta ined a rep resen ta t i on f o r t h e biharrnoni c

f u n c t i o n U i n terms of two a n a l y t i c f unc t i ons o f a complex va r iab le .

I n p a r t i c u l a r we w i l l represent U i n t h e form [ I 0 1

- 2~ = z Q W + z 6(z) + ~ ( z ) + i ( z ) (6)

where Q(z> and $(z) a r e a n a l y t i c funct ions. Equat ion (6) i m p l i e s

t h a t

where we have replaced ~ ' ( z ) by $(z). Usi

t h e s t resses become

; ( z)

ng t h i s representa t

(7)

i o n

, The d i sp lacemen ts now t a k e t h e f o r m

When complex p o t e n t i a l s a r e used t h e boundary c o n d i t i o n (3) f o r

t h e f i r s t boundary v a l u e proSLem becomes,

whi Le f o r t h e d i sp l acemen t boundary v a l u e p rob lem we have

where Tat and u a r e t h e c a r t e s i a n components o f t h e s u r f a c e %.. .

- 2

t r a c t i o n s and d i s p l a c e n e n t s , r e s p e c t i v e l y , on t h e boundary C .

We w i h L f i r s t use t h e method proposed by Muskhe l i shv in i and

s o l v e t h e problem f o r a s imp ly connected domain.

Le t t h e r e g i o n D be mapped conformal ly on to t h e u n i t c i r c l e

(~14 by .the a n a l y t i c f u n c t i o n I I

If D i s f i n i t e w(< ) can be w r i t t e n i n the form o f a power

s e r i e s

whi le f o r D i n f i n i t e

S u b s t i t u t i n g (13) i n (7) and (10) we g e t

and i n t o t h e boundary c o n d i t i o n s (11) and (12 ) t o g e t

$l(c) = $J(w(<) )

and F ( 0 ) . G(0) represents t h e t rans formed va lue o f t h e r i g h t hand

s ides o f equat ions (11 ) and (12 ) r e s p e c t i v e l y .

A t t h i s s tage t h e r e a r e two procedures t h a t can be fol lowed. The

f i r s t i s t o assume t h e func t i ons P I ( < ) and $1 ( 5 ) can be represented

i n t h e fo rm

i f D i s f i n i t e , w h i l e if D i s i n f i n i t e t h e f o l l o w i n g r e p r e s e n t a t i o n

must be used.

t r a c t i o n s and 8', B, and C* a r e r e l a t e d t o t h e s t r e s s d i s t r i b u t i o n

a t i n f i n i t y . The s u b s t i t u t i o n o f these rep resen ta t i ons i n t h e

boundary c o n d i t i o n s leads t o a system o f equat ions f o r t h e

c o e f f i c i e n t s an and bn.

In genera l t h i s approach i s cumbersome whi l e t h e f o l l o w i n g procedure

r e s u l t i n g f rom t h e convers ion of t he boundary condi t i o n s i n t o

f u n c t i o n a l equat ions, Leads t o cons iderab ly more e f f e c t i v e methods

of s o l u t i o n .

Let 6 = 1 i n t h e case of t h e s t r e s s boundary va lue problem and

6 = K f o r t h e displacement boundary va lue problem and s im i l a r l y

H(8) = F(0) o r H(0) = G(0) r e s p e c t i v e l y .

' d' and a f t e r M u l t i p l y t h e boundary c o n d i t i o n (18 o r 19) by ~u i n t e g r a t i n g ove r t he boundary o f u n i t d i s c we g e t

't- o

I

t hen (25) i n bo th cases becomes

On d i f f e r e n t i a t i n g w i t h respect t o o and L e t t i n g o-tt on C (27)

y i e l d s a Freedholm i n t e g r a l equat ion

The ex i s tence o f t h e s o l u t i o n Q o (t) now foLLows d i r e c t l y f rom

Freedholm theory . Once Q o ( t ) and thus Ql ( t ) i s determined then

(01 i s g i v e n by

The case o f m u l t i p l e &- connected domains i s more complicated. One ..

approach i s t h a t used by S.G. M i k h l i n 18, 91 i n which he mod i f ies

M u ~ k h e l i s h v i L i ' s i n t e g r a l equat ions so t h a t they w i l l be v a l i d f o r

b o t h m u l t i p l y and s impty connected regions. He used t h e concept

0, t h e complex Green's f u n c t i o n f o r the reg ion which i s v a l i d f o r

b o t h m u l t i p l y and s imp ly connected domains. Some examples o f i t s

use can be found i n C l l , 121.

2. orthogona t Po lynorni - a t s

One d i f f i c u l t y t h a t a r i s e s i n Muskhel ishv i li ' s method i s f i n d i n g

the requ i red conformal mapping. A way .of a v o i d i n g t h i s d i f f i c u l t y

i s through t h e use o f o r thogona l polynomials. The idea of us ing

or thogonal poLynomiaLs was f i r s t advanced by S. Bergman Cl31.

The f o l l o w i n g method was l a t e r a p p l i e d t o the s o l u t i o n s of p lane

problems f o r i s o t r o p i c e l a s t i c bodies C5, 141.

Let t h e r e g i o n D be s imp ly connected and a compact subset o f t h e

complex plane. I t f o l l o w s t h a t t he p o t e n t i a l s Q ( z > and + ( z > a r e

holomorphic i n D and i f i n a d d i t i o n one assumes they a r e o f t h e

c lass L2 t hen they belong t o a H i l b e r t space H. I f the s e t o f

f u n c t i o n s P (2) form a bas i s i n H then we can w r i t e Cn 1

where

':sing Green's formula we get

Assuming Q'(z) E H we w r i t e

Subs t i tu t ing t h i s i n (32) the fo l lowing r e s u l t i s obtained:

For m u l t i p l y connected bounded domains the approac h i s t h e same

b u t more complicated. Suppose D i s a (p+l) - f o l d connected

domain bounded by a smooth r e c t i f i a b l e Jordan curves yo,y1,y2, ...yp

where yo conta ins t h e p o i n t a t i n f i n i t y ' . A lso suppoze z1 , z 2 1 ~ 3 1 . . . a r e f i x e d p o i n t s i n t h e f i n i t e components of t h e compliment o f D.

Then i f XneYn denote the components o f the r e s u l t a n t o f t h e L- I e x t e r n a l fo rces a c t i n g on t h e contour y_ a n one can w r i t e

Where Qo(z) and $Jo(z) a r e holomorphic i n D L l 4 l . Here we

recognize two cases. The f i r s t , when t h e e x t e r n a l f o rces

a c t i n g on each boundary component a re balancedl i n which case

Q(z) = Q0(z) and +(z) = $,(z). So t h a t f rom equat ions (11)

and (12)

f"? In -the second case when t h e e x t e r n a l f o rces o f ye do n o t

vanish equat ions (11) and (12) assume t h e form

and where f:)($) has t h e form

,

Assuming (Pn(z)) i s an or thonormal b a s i s f o r t h e H i l b e r t space

L2 o f holomorphic f u n c t i o n s on Dl we ge t

where t h e sense of d e s c r i p t i o n o f yo i s assumed t o be counter-

c lockwise whereas ya, a # o i s assumed t o be descr ibed i n a

c lockwise d i r e c t i o n . Adding t h e (p+l ) equat ions o f t h e t ype

(38) we o b t a i n

Employing Green's formula t h e above equat ion reduces t o

As b e f o r e assume Qo(z) E LZ and expand i t i n t h e s e r i e s

S u b s t i t u t i n g t h i s equat ion i n equat ion (40) y i e l d s

Equat ions (42) form an i n f i n i t e system o f equat ions f o r t h e

* unknown pn . I f t h i s system o f equat ions can be so l ved f o r

p, t hen t h e s t resses and displacements a r e r e a d i l y determined.

The s u b s t i t u t i o n of pn i n equat ion (41) determines Q i ( z ) and

when t h e boundary values o f Q i ( z ) a r e known, q o ( z ) can be

eva lua ted by an argument s i m i l a r t o t h e one used i n t h e

prev ious case.

For t h e f i r s t boundary va lue problem

* For t h e displacement boundary va lue problem t h i s system o f eqs. a l s o con ta ins a d d i t i o n a l unknowns X and Y.

I

where fl(EL)(B) and f:EL) (B) a r e ob ta ined from t h e components o f

t h e t r a c t i o n s on t h e boundary components ya so t h a t (42) reduces

Only one of t h e cons tants ca may be assigned a r b i t r a r i l y ; t h e

r e s t must be determined f rom t h e c o n d i t i o n s o f t h e problem.

U n t i l now n o t h i n g has been s a i d o f t h e c o n s t r u c t i o n o f t h e

or thogona l po lynomia ls P,(z). I t can be shown t h a t t h e s e t o f

f u n c t i o n s I ,z,z2,z3, .. . form a complete s e t w i t h respect t o a

c lass o f l2 f u n c t i o n s on a f i n i t e s imp ly connected domain whose

compliment i s a c losed domain [ I S ] . S i m i l a r l y

from a complete s e t w i t h respect t o L ~ ( D ) where D i s a m u l t i p l y

connected domain bounded by s imple c losed a n a l y t i c curves cl...c n

and a,, i s a p o i n t i n s i d e t h e h o l e o f D which i s surrounded by cv.

The s e t of or thonormal polynomials i s e a s i l y found now by us ing

t h e GramSchmidt o r t h o g o n a l i z a t i o n process on your complete s e t I

(1 5) :

3. Thermoe las t i c i t y

Another major area o f e l a s t i c i t y t o which the theory o f complex

v a r i a b l e s can be a p p l i e d i s t he rmoe las t i c i t y . We w i l l so l ve t h e

problem o f a c y l i n d r i c a l body, whose length i s l a rge compared t o t h e

g rea tes t l i n e a r dimension o f i t s cross-sect ion, sub jec t t o a temperature

change independent o f t h e length o f t h e c y l i n d e r (z ax i s ) . The cross-

i s a constant o f such va lue t h a t t he r e s u l t a n t normal f o r c e over t h e

cross-sect ion i s zero. Stresses near t h e ends w i l l n o t be considered.

Le t To(x,y) = T(x,y,t,) be t he i n i t i a l temperature and Let T(x,y,t) be

t h e temperature a t some Later t ime t dimin ished by To(x,y). The

Duhamel-Neuman Law connect ing s t ress , s t r a i n s , and displacements i s

as f o lows*:

1 e . . = $ui ,j + u .) 11 j , I

where

* No summation over repeated i n d i c i e s .

Under the cond i t i ons o f the body yxZ = O. y = o, and eZ = Y z

constant; a l s o i f the body fo rces a re zero the e q u i l i b r i u m

equat ion becomes

The bounaa r y condi t i on ; s

where Xi a r e the app l i ed sur face t r a c t i o n s and n t h e u n i t outward

normal. A f u r t h e r c o n d i t i o n

which f o l l o w s from the assump t i o n t h a t the r e s u l t a n t f o r c e i n t he

z d i r e c t i o n i s zero, must be s a t i s f i e d . F i n a l l y t h e c o m p a t i b i l i t y

equat ion takes the form

- 22 -

Solve i n (44) f o r o- ( i=j=3) and r e s u b s t i t u t i n g i n equat ion (44)

(i, j=1,2) then (44) i s d i v i d e d i n t o two p a r t s

and

Now i f ( 4 5 ) . ( 4 6 ) . (48) and (49) are solved then equat ions (47 )

and (50) can a l s o be solved.

It i s w e l l known t h a t t he A i r y s t ress f u n c t i o n F s a t i s f i e s ( 4 5 ) .

S u b s t i t u t i n g F i n (49) and t h e r e s u l t i n (48) we get

The boundary c o n d i t i o n s i n terms o f F become

Note t h a t oZ and eZ can be determined f rom equat ions (47) and

(50) and can be w r i t t e n as

I t i s convenient t o make a f u r t h e r s i m p l i f i c a t i o n o f t h e d i f f e r e n t i a l

equat ion (51).

Let

F = U-V

where V i s any s o l u t i o n o f t h e d i f f e r e n t i a l equa t i on

V ~ V = kT (56)

t hen (51) i s e q u i v a l e n t t o a system o f t w 6 d i f f e r e n t i a l equat ions

Ema where k = .

The boundary c o n d i t i o n s (52) now become

Now t h e problem has been reduced t o s o l v i n g (57) s u b j e c t t o (58).

I t i s n e c e s s a r y t o f i r s t f i n d a p a r t i c u l a r s o l u t i o n o f t h e

equa t i o n

s ince then t h e boundary c o n d i t i o n s (58) f o r U(xty) a r e determined.

Therefore assume V(x,y) i s known, Then t h e f u n c t i o n U(x,y) can

be obta ined by s o l v i n g v 4 u = 0 s u b j e c t t o t h e boundary c o n d i t i o n

(52) which can be w r i t t e n i n t h e form

or if these d e r i v a t i v e s a r e s i n g l e valued, i n t h e form

where s i s t h e l e n g t h on t h e boundary c o f t h e c ross -sec t i on D

of t h e bodyt w i t h c a simple c losed curve enc los ing t h e r e g i o n D.

Th i s p a r t o f t h e problem i s now e x a c t l y t h a t problem discussed

e a r l i e r which has been so lved by M u s k h e l i s h v i l i u s i n g complex

p o t e n t i a l s .

Thus i t remains o n l y t o determine V(xty> i n terms o f i t s f i r s t

d e r i v a t i v e s , s ince i t i s o n l y those f u n c t i o n s which appear i n

the boundary condi t i ons ( 5 9 ) , f rom

A p a r t i c u l a r i n t e g r a l o f (60) i s

where ( x , y ) i s a p o i n t i n DI (xl,yl) i s a v a r i a b l e p o i n t i n D,

p2 = (x-x1l2 + (y-y1)2, and D i s t h e area o f t h e c ross-sec t ion

under cons idera t ion . I f z = w(s) maps D conformal ly on t h e

unit d i s c o f t h e then i n terms o f t h e v a r i a b l e s o f t h e

s-p Lane (61 becomes

where s = S + i~ = rei8, t = Il + i = Rei8, S' t h e unit d i s c - -

and p2 = (s - t (s - t ) . D i f f e r e n t i a t e (62) t o o b t a i n

Since bo th V(x,y) and U(X,Y) a r e now known the displacements and I

s t r e s s can be found w i t h L i t t l e t r o u b l e .

I n t h e preced ing d i scuss ion i t should be no ted t h a t knowledge of

t h e temperature f u n c t i o n T has been assumed.

The most impor tan t r e s t r i c t i o n on the theory t h z t has been

developed above i s t h a t t h e mapping f u n c t i o n be r a t i o n a l . T h i s

r e s t r i c t i o n was i n t roduced t o p rov ide a system of equat ions which

cou ld be so tved e x p l i c i t l y f o r t h e des i red func t i ons . T h i s

r e s t r i c t i o n can be dropped w i t h t h e r e s u l t t h a t t h e mapping

f u n c t i o n can be w r i t t e n i n t h e form

which maps any simply-connected reg ion b u f f i c i e n t l y regu la r )

o n t o t h e unit d i s c t o any d e s i r e d degree of accuracy. Hence

fo r a l l p r a c t i c a l purposes any the rmoe las t i c problem f o r s imply

connected reg ions can be solved.

- 28 -

B. TORSION OF AN HEXAGONAL BEAM (20)

A L I t h e techniques i L l u s t r a t e d i n t h e preceding s e c t i o n have, i n

t h e i r approach, one aspect i n common. They a l l make use of complex

p o t e n t i a l s .

In a l l cases i t i s necessary t o assume some form f o r these p o t e n t i a l s .

The most obvious one i s a power se r ies . I t i s a l s o p o s s i b l e t o

express these p o t e n t i a l s i n another form. That i s i n terms o f a s e t

o f o r thogona l polynomials which form a bas i s i n t h e r e g i o n under

cons idera t ion .

As an i l l u s t r a t i o n o f how or thogona l polynomials can be used i n two

dimensional e l a s t i c i t y , t he t o r s i o n problem f o r an hexagonal beam

w i l l be solved. The approach used here i s due t o S. Bergman L161.

1. Basic Equations f o r t h e Tors ion o f a Bar

Consider a s e t of rec tangu lar c a r t i s i a n coord ina tes x , y, z

w i t h t h e z a x i s be ing perpend icu la r t o t h e cross-sect ion o f

t h e bar .

According t o S a i n t Venant t h e components o f t h e displacement

vec to r u, v, w a?e g i v e n by the expressions

U =-ayz

v = axz

where a i s t h e angle o f t w i s t per u n i t length and G(x,y) i s

harmonic over t h e cross-section, D, o f t h e b a r . Using Hooke's

where LI i s t h e modules o f r i g i d i t y .

Now l e t H(x,y) be t h e harmonic conjugate o f G(x,y). Then t h e

non-zero components o f the s t ress tensor can be w r i t t e n as

The boundary c o n d i t i o n on t h e L a t e r a l su r face o f t h e b a r i s

g i v e n by

where - v i s a . u n i t o u t e r normal t o t h e boundary C o f t h e cross-

dG s e c t i o n D and - denotes t h e normal d e r i v a t i v e o f G(x,y) w i t h

dv

respect t o - v. The q u a n t i t i e s cos(x,v) and s in(x lv) a r e t h e

cosines o f t h e ang le between t h e x a x i s and - v, andl t h e y a x i s

and - v r e s p e c t i v e l y . I n terms o f H(x,y) t h e above boundary

c o n d i t i o n assumes t h e f o l l o w i n g form

Therefore, t h e t o r s i o n problem reduces t o t h e de te rm ina t i on of

t h e f u n c t i o n H(x,y) f rom

We n o t e t h a t G(x,y) can be un ique ly determined, a p a r t f rom an

a r b i t r a r y cons tant from H(x,y) by t h e use o f t h e Cauchy-Riemann

equat ions.

2. Method o f Solut ion

Let the bar have an hexagonal cross-section and l e t C be i t s

boundary (see F ig . 1 ) .

Since G(x,y) i s harmonic i n D the funct ion F(z)

def ines an a n a l y t i c funct ion i n D.

D. Then F'(z) can be expanded i n the s e r i e s

F'(z) = an Pn(z) n-o

It can be shown t h a t t h e above s e r i e s converges a b s o l u t e l y on

D and un i fo rm ly on any c losed subregion o f D CIS].

I n t e g r a t i n g equat ion (64 )

F = 1 T 1 F z dxdy + c n=o

where

and

Now w r i t e Tn(z) i n the form

w i t h fn(x.y) and gn(xty) being r e a l funct ions.

Taking the r e a l and imaginary par ts o f equat ion (65) ? ' ^ --+

where we have used t h e Cauchy-Riemann equat ions and t h e f a c t t h a t

i f B(z) i s an a n a l y t i c f u n c t i o n o f z and

then

Convert ing t h e a r e a l i n t e g r a l s i n equat ions (66) and (67) t o

l i n e i n t e g r a l s by the use o f Green's theorem we ge t

w i t h

where H(s), fn(s), gn(s) denote t h e boundary values o f H(xry)

fn(x,y), g (x,y) respec t i ve l y on C . n

Thus t h e problem has been reduced t o the computation o f H(x,y)

and G(x,y) from equations (68) and (69) which depend on ly on

t h e boundary values o f H(x,y) and on the orthogonal polynomials

P,(z). Obviously i n any p a r t i c u l a r c a l c u l a t i o n the i n f i n i t e

s e r i e s i n (68) and (69) must be t runcated a f t e r a f i n i t e

number o f terms, depending upon the degree o f accuracy required.

Therefore, t h e s o l u t i o n requ i res on ly the complet ion o f t h e

f o l l o w i n g steps:

(1) Determinat ion o f Pn(z) n=0,1,2, .. . and thus

the f u n c t i o n s f n ( x t y ) and gn(xty)

( 2 ) Eva lua t i on o f

(3) Computation o f t h e sums i n equations (68) and

a H a~ (69) and o f t h e p a r t i a l d e r i v a t i v e s and - a Y .

3. Determinat ion o f t h e Orthogonal Polynomials P n (z)

I t i s w e l l known t h a t t h e s e t o f f unc t i ons l lz~z2.z31.. . form a

complete system

process we form

where Dn(z) and

i n D [151. Using the Gram-Schmidt o r t h o g o n a l i t a t i o n

an or thonormal system (Pn(z)) g i ven b y [ I S I .

E a r e determinants. n

The elements, Lnm m,n=0,1,2 ,..., o f these determinants a r e

g i v e n by a r e a l i n t e g r a l s

n -m Lnm = J z z dxdy n,m=0,1,121 ...

D

L e t t i n g z = reie equat ion (73) becomes

L nm =I 1 r nimil cos (n-m) e d r de

Let Re denote ' t h e r e a l p a r t o f ' and I m denote " t h e imaginary

p a r t o f " # then by making use o f t h e symmetry o f t h e cross-

s e c t i o n we can e a s i l y show t h a t

, I n-m

8 1 n-m

and

Therefore

IT

Z

Lnrn n+m+2 l r nfm+2(o) c o n - e de

From t h e geometry o f t h e hexagon i t i s c l e a r t h a t

S u b s t i t u t i n g equat ion (77) i n equat ion (76) we f i n a l l y g e t

- - 41 sec 'nm n+rn+2

n+m*2(6) cash-m)e de

'Z

+ J sec n+m+2

n+m+2(~-2) cos (n-m) e do

These i n t e g r a l s i n equat ion (78) a r e evaluated us ing the Romberg

Method o f I n t e g r a t i o n C171. The values o f Lnm f o r n,m=0,1,2,...r19

a r e g i v e n i n Table 11. ..

Next we expand ;he r i g h t hand s ide o f equat ion (70) i n powers o f

z, t a k i n g o n l y t he f i r s t twenty terms of Pn(z). However, some o f

t h e c o e f f i c i e n t s of zn i n t h i s expansion vanish and we g e t

9

whece n=Ot l t 2 t 3 t ... , I9 and

More e x p L i c i t 1 y t h e polynomials Pn(z) t ake t h e form

The numer ica l values o f t he constants an ... snt n=0r1,2t...t19 are

g i ven i n Table 111.

R e c a l l

S u b s t i t u t i n g equat ion (79) i n equat ion ( 8 1 I t i n t e g r a t i n g and

r e ~ l a c i n c j z by reie we g e t t h e f o l l o w i n g expressions f o r

fn (x ty ) and gn(xty)

Determinat ion o f H(rr 8 )

In o rde r t o determine H(r.0) t h e constants An and I3 must be n .

determined. R e c a l l t h a t these constants a r e g i v e n as l i n e

i n t e g r a l s ,

and tha t t h e boundary values o f H(x,y) on c a r e g i v e n by

1n p o l a r coord ina tes t h i s becomes

I t f o l l o w s then t h a t

Again because o f t h e symmetry o f t h e problem under c o n s i d e r a t i o n

The above i n t e g r a l r e p r e s e n t a t i o n s can be approx imated by t h e

summation

where

I -

The numer i ca l va lues o f An may be found i n Table IV.

From equat ions (68) and (90)

where H20(rt0) denotes o u r approx imat ion o f H(rnO) and CZ0 t h e

cons tant c 1

. . . .-

On t h e boundary C o f D equa t i on (92) i m p l i e s t h a t

1 where 2 r2te) i s t h e exac t boundary va lue o f HZO(r,B) . Since t h e

a constant. The!iefore we compute t h e d i f f e r e n c e

i n f i n i t e sum i n equa t i on (68) has been rep laced by a f i n i t e sum

we cannot expect t h e r i g h t hand s ide o f equat ion (93) t o remain

a t a number o f p o i n t s on t h e boundary ( in o u r case a t 36 p o i n t s )

and then choose the average o f these values f o r CZO. Hence we

take

where

Using t h i s procedure the f o l l o w i n g "average' va lue f o r CZO i s

obta ined

F i n a l l y we g e t

S ince we have a l r e a d y found t h e va lues o f An and t h e po l ynom ia l s

f (rle)t t h i s e x p r e s s i o n comp le te l y de te rmines t h e harmonic f u n c t i o n n

H ( x ~ y ) . The n u m e r i c a l va lues o f HZO(x,y) on t h e boundary C a r e

g i v e n i n Tab le I. The v a l u e s o f H20(xty) i n D a r e l i s t e d i n

Tab le V. -

5 . Determinat ion o f t h e Stresses

The de te rm ina t i on o f t h e s t resses requ i res t h e e v a l u a t i o n of t h e

d e r i v a t i v e o f H(x1y1.

The above method y i e l d s o n l y f u n c t i o n H20(xty) approximat ing t h e

r e q u i r e d f u n c t i o n . I n s i d e the domain D t h e d e r i v a t i v e H Z O ( x r y )

w i L L i n g e n e r a l approximate the corresponding exact s o l u t i o n q u i t e

s a t i s f a c t o r i l y . Near t h e boundary o r on t h e boundary i t s e l f t h e

approx imat ion ob ta ined f o r t h e d e r i v a t i v e s w i l l , i n many casest

n o t be s a t i s f a c t o r y . I n p a r t i c u l a r a t sharp corners where t h e

rad ius o f c u r v a t u r e o f t h e boundary i s no longer cont inuousl i t

i s necessary t o app ly s p e c i a l methods f o r t he e v a l u a t i o n o f t h e

d e r i v a t i v e s .

6 . Conclus ion

In conc lus ion f t h e t o r s i o n problem o f a ba r o f hexagonal cross-

s e c t i o n has been considered i n o r d e r t o i 1 l u s t r a t e t h e use of

o r thogona l po lynomia ls t o so l ve t h i s problem i n p a r t i c u l a r b u t t

more genera l l y l t h e D i r i c h l e t problem as g i ven by equat ion

(63) .

It should be no ted t h a t no a t tempt was made t o predetermine t h e

degree o f accuracy o f t h e approximations made. Howeverf we n o t e

t h a t due t o t h e n a t u r e o f t he problem the f u n c t i o n H(XIY) must

t ake b o t h i t s maximum and minimum values on t h e boundary- From

t h e geometry o f t h e c ross-sec t ion one eas i Ly sees t h a t

The approximate values o f H(xly) on t h e boundary va ry from

.so00873 t o .6537620 (see Table I ) . The approxi mate vq tues o f

H(x.y) i n s i d e D f a l l w i t h i n t h e exact va lues f o r Hmin and Hmax

(see Table V) .

C . RESULTS

TABLE I

TABLE I1

The values of t h e non-zero constants LAM

AM = 0 , l ,2,3 ,...,I9 where

- .3062297 1 1

Ill I1

1.4369564 11~

.4266647 x 1 o - ~

- .4551075 x

- 52 -

TABLE I1 (Cont 'd . )

- 53 -

TABLE I1 (Cont'd.)

- 54 -

TABLE I11

The values o f t he c o e f f i c i e n t s appearing i n Equat ion (79)

- 55 -

TABLE 111 (Cont'd.)

- 56 -

TABLE I11 (Cont'd.)

- 57 -

TABLE I V

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