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Flexure of long flat curved plates with built-in curved edges
MECHANICS RESEARCH COMMUNICATIONS 0093-6413/80/060359-06502.00/0
F
~
Vol.
7(6),359-364,1980. Printed in the USA. Copyright (c) Pergamon Press Ltd
OF LONG FIAT CURVgD P l A T e S PITH BUILT-IN CURVED EDGES
B. G. I~£KASH 8 t r u e t u r s l . k n s l y s l s Group M o t h • n i s e i Systems D i v i s i o n I n d i s n SlXLCe R o s e • r o b O r g s n i e s t i o n , Ben•slots560 058, I I D I 1
Sstollito
Centre
(Received 28 March 1980; accepted for print 21 September 1980)
INTRODUCTION S t r e s s snslysi8 i n s e c t o r p l a t e s u s i n g s n g u l s r o i g e n f u n c t i o n s of t h e b i h ~ r m o n t c o q u & t i o n h s s boon t h e s u b j e c t of d i s c u s s i o n o f mA,y ~ u t h o r e end the l i t e r s t u r e is replete with references. Bthsrnonie el•onf u n c t i o n s i n t h e oJagulsr e o - o r d i n s t e e • n e l s e be u s e d t o d e s l w i t h t h e s t r e s s s n 6 1 y s t s i n c u r v e d beams a n d c u r v e d p l s t o s . However, i t i s t n p o r t a n t t o n o t s %hiLt t h e i f t h e l e n g t h o f t h e c u r v e d b e • n / p i e r s s s n e s s u r s d • l o n g t h e m i d d l e l i n e i s v e r y much g r e e t e r t h e n i t s w i d t h , t h e n t h e • n g u l ~ r s i g e n f u n e t i o n s do mot mske much s e n s e . In such e s s e s one h s s t o d e v e l o p t h e e i g e n f u n c t t o n s o f t h e b i h s r n o n t c o q u & t i o n i n the rsdt~l co_ordinsto. R s d i g l e i g e n f u n o t i o n s f o r problems • • • e e l • b e d w i t h c u r v e d be•ms and p l s t e s h s e been r e p o r t e d i n [ 1 - 4 ] . I n [ 5 ] i t i s i n d i c • t e d how o o n s t r u © t i n g s b i o r t h o g o n s l i t y r e l • t i o n , s o l u t i o n s f o r t h e b e n d i n g of s e a t - I n f i n i t e r e e t s n g u l s r p l & t s e el•roped • l o n g t h e l o n e e d g e s e s n be o b t a i n e d . An & t t e m p t i 8 msde i n t h e p r e s e n t p s p e r t o o b t s t n 8 o l u t l o n s f o r t h e f l e x u r e of l o n g f l s t c u r v e d pl&tog w i t h b u i l t - i n c u r v e d odgee, on l i n e s 8 t m i l s r t o t h e one s u g e e s t s d i n t h e • b o y s l~pero Some numericL1 r e s u l t s f o r the f l o x t t r o u n d e r edge l o ~ d i n g &re p r e s e n t e d . JL~IJ&L%818
C o n s i d e r t h e f l e x u r e of • l o n g f i s t r e g i o n a ~ r ~_ b ,
0 ~ O .
curved plste
occupying the
~o s h e l l &esume t h & t t h e
pl~tO &s m o • s u r o d a l o n g t h e middle l i n e r n ( b ~ l ) / 2 ¢Onl~red t o t h e w i d t h o f t h e p l • t o , governing dtfferenti•l
i.e.
length of the
iS l s r e o when
(b-s)/(b+s)~
equstion for the flexure
.
The
of t h e c u r v e d p l s t e
i s g i v e n by ~&w where ~
ffi 0 ,
•~
r~
i s the hsrmonio e p e r s t o r
b , O~ 0
.
.
.
.
(1)
in polar co-stalin•tee.
When t h e c u r v e d e d g e s of t h e p l ~ t o &re el•roped, t h e b o u n d • r y c o n d i t i o n s 359
360
B.G.
PRAKASH
o r e g i v e n by w - v
=
0
,
on
r
=
•
•nil
r
=
b
(2)
Me s h a l l seek t h e o x p s n s i o n of t h e d o f l e o t i o n f u n o t i o n i n t h e form
v =~
exp (ske) rPk(r)
sk
. . . .
131
E q u s t i o n e (1) sod (2) c~n now be w r i t t e n i n t e r m s of t h e f u n o t i o n Pk snA i t 8 d o t t y • t i r e s
,*;
so 8
+
+
Pk-l'~-o
with
on r . .
÷
,,.
•hA
r n b
whore • prime d e n o t e s t h e o p e r s t o r r ~
o
..
(,)
..
(~)
.
The s o l u t i o n of e q u s t i o n (4) oubJoot t o t h e oond£tion8 (5) • r e g i v e n by Fk - s i n ( I k ln~) . h ( ~ )
-
oo8 kT[ .h(ln~) .in(m k ~ )
(6)
..
where the #k*8 &re the roots of the tr~nnoondent•l equstiom sin(ok
The f i r s t
ln-5 - % ~o, k ~
,h(]~b)
-
o
..
(7)
t e n r o o t s of o q u s t i o n (T) l y i n ~ i n the p o s i t i v e q u m h s n t hove
boon o s l c u l s t o d on s Aigit&l oomputer u s i n ~ t h e s m y u l ~ o t i e f o r n u l s f o l l o w e d by New, onto i t o r s t i o •
The r o o t s of the oqu&tion f o r t h e
8 s p o r t r e t i e -b - 3 o r e l i s t e d out i n T~blo 1. Let f i end f i k r O S l m o t i v e l y Aenote ~vo f o u r component r o o t e r f u n o t i o n 4 . The oomponents of f i b e i n g v~ ~ e ' ~ / D •hA Ve/D (D b e i n g t h e f l o x u ~ • l risidity
of t h e m J t o r i ~ l o f t h e p l o t s ) t v h o r o • s f i k i s comprise4 of
Vk *~Ok ' MOk end Vek.
ii " ~"k k
Vo s h e l l s o d
"*P ('k e) ~*k
Prom t h e r e l a t i o n
t h e OXl~noion8
. .
""
(3) I t i J osoy t o i d e n t i f y the ooupononts o f f l k i n
term8 of t h e f u n o t i o n Pk end i t s b r i v j t i v o o flk ~
Wk "
(s)
rFk
.
.
sO f o l l o w s : .
.
(9)
CURVED PLATES WITH BUILT-IN
f2k = (~ ek =
CURVED EDGES
361
. . . .
Sk l'k
(~o)
1
',k " ".,. " - ; t ~ ~; + (1+ ~) ,; + c',+.~) ,~ t snd
..
¢11)
sk
vhoro
~ d o n o t e s the F o i s s o n ' s r s t i o
o f tho m s t o r i s l of the p l s t e .
Consido~ tho i n t o ~ s l
(Vk Ye;j + ~e:j Mek) dr
88
b
"
-
+c,+¢
+ , ~ [ ~ ,~ + c,+ ,) ,~ + (,+.~) ,~ ]l ~
-.
('~)
Upon t n t o g z a t t n g t h s r i g h t head s i l o o£ tho ,bovo r e l a t i o n by l ~ r t s sn4 u s / a ~ t h e o o n d i t i o n s ( 5 ) , vo obt&/n t h e r o l ~ t i o n
(14)
..
Hovoyoz, t h o r i g h t b u d
sib
:oprosonts aT/nborgts [6]snslogous
o r t h o j o n s l £ t y r o l s t i o n f o r ourvod p l s t o s ,
so t h s t vo hsvo t h e i n n e r
orthopnslityrolst£on
./(v
kve~+
E ej%k ) a:.o
for
j~k (15)
ee
=
Nk/2 f o , j .
k
l n t e r e h s n d i n 8 j ~na k end a d d i n g t h e r e s u l ~ i n g r e l a t i o n t o ( 1 5 ) , we 8or
S,
('k Ve,1 + (e:l Mek ÷ V:l vek + ~ek MeJ) ~ = o
~or ,1 ~ k
= Nk~Or :j = k vhAoh i s t h e r o q u i r e d b i o r b h o g o n ~ l i t y rtlfbtiom.
.e
(16)
362
B.G.
PRAKASH
DETERMINATION OF PARTICIPATION CONSTANTS
Using .the r e l a . t i o n s (8) and ( 1 6 ) , i.t i s easy .to o b t a i n the f o l l o w i n g reid.riCh on
e = o : b
~k Nk =
&f
(Wb Vek +
E e b Hek + Meb E e k + Veb Vk) dr
..
(17)
where .the s u b s c r i p t b deno%es .the bounasry func.tion. The f o l l o w i n g .three p~/rz of p h y s i c a l l y m e a n i n g f u l combination of boundary f u n c . t i o n s i n d e f l e c . t i o n , s l o p e , moment and shear f o r c e can be p r e s c r i b e d on e = 0 : Problem (1) wb and Meb p r e s c r i b e d Problem (2) wb and ~ e b p r e s c r i b e d Problem (3) Mob and Vob p r e s c r i b e d Using (8) end (17) i.t i s f a i r l y
easy to show .tha.t .the l ~ r t i c i p s . t i o n
c o n s t a n t s s k i n the e i g e n f u n c t i o n exl~nzionz a r e determined i n c l o n e d form f o r problem (1) and a r e g i v e n by b
2
ak = Nk J (vb vek + Meb ~ ek) dr
.
.
.
(18)
.
For problems (2) and (3) t h e deternLtnation of .the constantts r e d u c e s t o s o l v i n g .the f o l l o w i n g l i n e a r a l g e b r a i c sys.tem of e q u a t i o n s :
b
b
2 s,J"(Vb V e k + C eb Mek) dr + ~k 2 3~. = Nk " a;;I /
(Mej ~" ek + VejVk)~r ..
(19)
and b
2 j (.eb
Sk = ~ k
Ok + Veb Wk )&r + Nk'2 ~ j
&J f / (wj VGk+4~ej Mek) dr ..
(20)
,,HU~RICAL R~SULT8 As s n u m e r i c a l exsmplet we hsve c o n s i d e r e d .the f l e x u r e of c u r v e d p l ~ under edge l o a d i n g . The consto, n te a k i n (18) were evslusted when sere d e f l e c t i o n and u n i t moment a r e p r e s c r i b e d on .the r a d i a l edse e = Oo The sooure~y wi.th whioh d e f l e o t i o n and momen.t s r e reproduoed on e m 0
C U R V E D P L A T E S WIT}{ B U I L T - I N
CURVED EDGES
363
Boots of t he e q u s t i o n ( 7 ) AlYmlTbOtiO form i s given by
2J~÷1)~
•k k
Re•l
+
i 1,1 [-2-1~.1_-)-~--s~ (-l-.n--b~!]
2
I ( P•zt
xn b/• Inmsin~ry l~zt
k
J /
Resl l~rt
in b
Insginszy Psrt
1
0.28P29 E 01
0.79763 g O0
6
0.17158 g 02
O. 81030 Z O0
2
0.57181 E 01
0.80684 g OO
7
0.20017 E 02
0.81042 K O0
3
0.85785 g 01
0.80895 E OO
8
0.22877 B 02
0.81050 Z O0
4
0.11438 E 02
O. 80973 g O0
9
O. 25736 g 02
0.810~5 I
5
O. 14298 g 02
0,81010 E O0
10
0.28596 g 02
0.81059 Z O0
T* LX 2 Numozicsl Convergonoe obtained using Ton E i g e n ~ l u e s l~esortbed
£ u n o t i o n v - O, Me -- 1, 1 ~ r ~ 3 ,
Z
w
Me
1.2
0 . 1 9 1 0 E-O4
0 . 9 5 8 9 R 00
1.4
-0.2317 g-05
0 . 1 1 ) 7 g 01
1.6
0.8788 g-05
0 , 9 1 6 6 g O0
1• 8
O. 4693 g - 0 5
O, 1041 g 01
2.0
- 0 . 1 1 4 7 F--05
0.9631 g O0
2.2
O. 6225 R-05
O. 1070 g 01
2.4
O. 2288 g - 0 5
O. 8672 2 O0
2.6
-0.7195 g-04
0 , 1 1 6 7 Z 01
2.8
0 . 8 6 2 8 g-O4
0 . 9 6 4 3 g O0
0 n O
00
364
B.G.
PRAKASH
u s i n g t h e e i g e n f u n o t i o n s e r i e s &re gLTen i n Tsble 2. v , s o s r r i e d out t a k i n g t h e P o i s e o n ' s r e t i e r&%io b&
= 3.0 .
The OOmlXat~tion
~ = 003 ~nd t h e ~speot
Ton eigenT&lues were u s e s t o s t u d y t h e n u m e r i c , 1
c o n v e r g e n c e of t h e e i g e n ~ u n o t i o n e x p a n s i o n s w i t h 8 u m B t i o n h e i n 8 oxt4nded over t h o s e e i g e n v s l u e s whose r e e l l~g~ i s n o g s t i v e .
I% v , s
f o u n d th~% Mo d e e s y s r s p i 4 1 y w i t h the t n o r e s s e of O slon~ t h e middle line (b~)/2
of the c u r v e d p l ~ t e which e x p l s i n s t h e 8~int-¥enLn%
l~rinoiple. it~CZS
1.
8 ~ r ~ t P.V.B.A.8. p R~maohsndz~ R4~o, B.S. sad G o l ~ l s o h s r y u l u , S . ,
Trot. J. B ~ g . 2.
SOL**
13 , 1975, p 149.
Pralmsh, B.G. snd lt6umohandzs B6o, B. 8 . , J . M6th. e~nd Phy. Sci.~
10, 1976, p 333. 3.
R~mJoh~ndzs R~o, B. S. snd Ps'Lk~sh, B. G., J . Sv,s'uot. Mech., 6,
1978, p 133. 4.
S~Z~L, P.V.B.JL.S., ih~emohandra R6o, B.S. ~nd 9OlXb]4~h~zyulu, S . ,
SIAM J. J,pjpl. Ms,t h e . , 26, 1974, p 568. ~.
R~m~oh~nd~s Rao, E~S. end 8 ~ i d h s z s , J . K . , J.J~ppl. Mech., 38, 1971,
p 70~.
6.
(~tnbezs, G.A., PI,M J . , 1T, 19~3, p 211.
F
~
Vol.
7(6),359-364,1980. Printed in the USA. Copyright (c) Pergamon Press Ltd
OF LONG FIAT CURVgD P l A T e S PITH BUILT-IN CURVED EDGES
B. G. I~£KASH 8 t r u e t u r s l . k n s l y s l s Group M o t h • n i s e i Systems D i v i s i o n I n d i s n SlXLCe R o s e • r o b O r g s n i e s t i o n , Ben•slots560 058, I I D I 1
Sstollito
Centre
(Received 28 March 1980; accepted for print 21 September 1980)
INTRODUCTION S t r e s s snslysi8 i n s e c t o r p l a t e s u s i n g s n g u l s r o i g e n f u n c t i o n s of t h e b i h ~ r m o n t c o q u & t i o n h s s boon t h e s u b j e c t of d i s c u s s i o n o f mA,y ~ u t h o r e end the l i t e r s t u r e is replete with references. Bthsrnonie el•onf u n c t i o n s i n t h e oJagulsr e o - o r d i n s t e e • n e l s e be u s e d t o d e s l w i t h t h e s t r e s s s n 6 1 y s t s i n c u r v e d beams a n d c u r v e d p l s t o s . However, i t i s t n p o r t a n t t o n o t s %hiLt t h e i f t h e l e n g t h o f t h e c u r v e d b e • n / p i e r s s s n e s s u r s d • l o n g t h e m i d d l e l i n e i s v e r y much g r e e t e r t h e n i t s w i d t h , t h e n t h e • n g u l ~ r s i g e n f u n e t i o n s do mot mske much s e n s e . In such e s s e s one h s s t o d e v e l o p t h e e i g e n f u n c t t o n s o f t h e b i h s r n o n t c o q u & t i o n i n the rsdt~l co_ordinsto. R s d i g l e i g e n f u n o t i o n s f o r problems • • • e e l • b e d w i t h c u r v e d be•ms and p l s t e s h s e been r e p o r t e d i n [ 1 - 4 ] . I n [ 5 ] i t i s i n d i c • t e d how o o n s t r u © t i n g s b i o r t h o g o n s l i t y r e l • t i o n , s o l u t i o n s f o r t h e b e n d i n g of s e a t - I n f i n i t e r e e t s n g u l s r p l & t s e el•roped • l o n g t h e l o n e e d g e s e s n be o b t a i n e d . An & t t e m p t i 8 msde i n t h e p r e s e n t p s p e r t o o b t s t n 8 o l u t l o n s f o r t h e f l e x u r e of l o n g f l s t c u r v e d pl&tog w i t h b u i l t - i n c u r v e d odgee, on l i n e s 8 t m i l s r t o t h e one s u g e e s t s d i n t h e • b o y s l~pero Some numericL1 r e s u l t s f o r the f l o x t t r o u n d e r edge l o ~ d i n g &re p r e s e n t e d . JL~IJ&L%818
C o n s i d e r t h e f l e x u r e of • l o n g f i s t r e g i o n a ~ r ~_ b ,
0 ~ O .
curved plste
occupying the
~o s h e l l &esume t h & t t h e
pl~tO &s m o • s u r o d a l o n g t h e middle l i n e r n ( b ~ l ) / 2 ¢Onl~red t o t h e w i d t h o f t h e p l • t o , governing dtfferenti•l
i.e.
length of the
iS l s r e o when
(b-s)/(b+s)~
equstion for the flexure
.
The
of t h e c u r v e d p l s t e
i s g i v e n by ~&w where ~
ffi 0 ,
•~
r~
i s the hsrmonio e p e r s t o r
b , O~ 0
.
.
.
.
(1)
in polar co-stalin•tee.
When t h e c u r v e d e d g e s of t h e p l ~ t o &re el•roped, t h e b o u n d • r y c o n d i t i o n s 359
360
B.G.
PRAKASH
o r e g i v e n by w - v
=
0
,
on
r
=
•
•nil
r
=
b
(2)
Me s h a l l seek t h e o x p s n s i o n of t h e d o f l e o t i o n f u n o t i o n i n t h e form
v =~
exp (ske) rPk(r)
sk
. . . .
131
E q u s t i o n e (1) sod (2) c~n now be w r i t t e n i n t e r m s of t h e f u n o t i o n Pk snA i t 8 d o t t y • t i r e s
,*;
so 8
+
+
Pk-l'~-o
with
on r . .
÷
,,.
•hA
r n b
whore • prime d e n o t e s t h e o p e r s t o r r ~
o
..
(,)
..
(~)
.
The s o l u t i o n of e q u s t i o n (4) oubJoot t o t h e oond£tion8 (5) • r e g i v e n by Fk - s i n ( I k ln~) . h ( ~ )
-
oo8 kT[ .h(ln~) .in(m k ~ )
(6)
..
where the #k*8 &re the roots of the tr~nnoondent•l equstiom sin(ok
The f i r s t
ln-5 - % ~o, k ~
,h(]~b)
-
o
..
(7)
t e n r o o t s of o q u s t i o n (T) l y i n ~ i n the p o s i t i v e q u m h s n t hove
boon o s l c u l s t o d on s Aigit&l oomputer u s i n ~ t h e s m y u l ~ o t i e f o r n u l s f o l l o w e d by New, onto i t o r s t i o •
The r o o t s of the oqu&tion f o r t h e
8 s p o r t r e t i e -b - 3 o r e l i s t e d out i n T~blo 1. Let f i end f i k r O S l m o t i v e l y Aenote ~vo f o u r component r o o t e r f u n o t i o n 4 . The oomponents of f i b e i n g v~ ~ e ' ~ / D •hA Ve/D (D b e i n g t h e f l o x u ~ • l risidity
of t h e m J t o r i ~ l o f t h e p l o t s ) t v h o r o • s f i k i s comprise4 of
Vk *~Ok ' MOk end Vek.
ii " ~"k k
Vo s h e l l s o d
"*P ('k e) ~*k
Prom t h e r e l a t i o n
t h e OXl~noion8
. .
""
(3) I t i J osoy t o i d e n t i f y the ooupononts o f f l k i n
term8 of t h e f u n o t i o n Pk end i t s b r i v j t i v o o flk ~
Wk "
(s)
rFk
.
.
sO f o l l o w s : .
.
(9)
CURVED PLATES WITH BUILT-IN
f2k = (~ ek =
CURVED EDGES
361
. . . .
Sk l'k
(~o)
1
',k " ".,. " - ; t ~ ~; + (1+ ~) ,; + c',+.~) ,~ t snd
..
¢11)
sk
vhoro
~ d o n o t e s the F o i s s o n ' s r s t i o
o f tho m s t o r i s l of the p l s t e .
Consido~ tho i n t o ~ s l
(Vk Ye;j + ~e:j Mek) dr
88
b
"
-
+c,+¢
+ , ~ [ ~ ,~ + c,+ ,) ,~ + (,+.~) ,~ ]l ~
-.
('~)
Upon t n t o g z a t t n g t h s r i g h t head s i l o o£ tho ,bovo r e l a t i o n by l ~ r t s sn4 u s / a ~ t h e o o n d i t i o n s ( 5 ) , vo obt&/n t h e r o l ~ t i o n
(14)
..
Hovoyoz, t h o r i g h t b u d
sib
:oprosonts aT/nborgts [6]snslogous
o r t h o j o n s l £ t y r o l s t i o n f o r ourvod p l s t o s ,
so t h s t vo hsvo t h e i n n e r
orthopnslityrolst£on
./(v
kve~+
E ej%k ) a:.o
for
j~k (15)
ee
=
Nk/2 f o , j .
k
l n t e r e h s n d i n 8 j ~na k end a d d i n g t h e r e s u l ~ i n g r e l a t i o n t o ( 1 5 ) , we 8or
S,
('k Ve,1 + (e:l Mek ÷ V:l vek + ~ek MeJ) ~ = o
~or ,1 ~ k
= Nk~Or :j = k vhAoh i s t h e r o q u i r e d b i o r b h o g o n ~ l i t y rtlfbtiom.
.e
(16)
362
B.G.
PRAKASH
DETERMINATION OF PARTICIPATION CONSTANTS
Using .the r e l a . t i o n s (8) and ( 1 6 ) , i.t i s easy .to o b t a i n the f o l l o w i n g reid.riCh on
e = o : b
~k Nk =
&f
(Wb Vek +
E e b Hek + Meb E e k + Veb Vk) dr
..
(17)
where .the s u b s c r i p t b deno%es .the bounasry func.tion. The f o l l o w i n g .three p~/rz of p h y s i c a l l y m e a n i n g f u l combination of boundary f u n c . t i o n s i n d e f l e c . t i o n , s l o p e , moment and shear f o r c e can be p r e s c r i b e d on e = 0 : Problem (1) wb and Meb p r e s c r i b e d Problem (2) wb and ~ e b p r e s c r i b e d Problem (3) Mob and Vob p r e s c r i b e d Using (8) end (17) i.t i s f a i r l y
easy to show .tha.t .the l ~ r t i c i p s . t i o n
c o n s t a n t s s k i n the e i g e n f u n c t i o n exl~nzionz a r e determined i n c l o n e d form f o r problem (1) and a r e g i v e n by b
2
ak = Nk J (vb vek + Meb ~ ek) dr
.
.
.
(18)
.
For problems (2) and (3) t h e deternLtnation of .the constantts r e d u c e s t o s o l v i n g .the f o l l o w i n g l i n e a r a l g e b r a i c sys.tem of e q u a t i o n s :
b
b
2 s,J"(Vb V e k + C eb Mek) dr + ~k 2 3~. = Nk " a;;I /
(Mej ~" ek + VejVk)~r ..
(19)
and b
2 j (.eb
Sk = ~ k
Ok + Veb Wk )&r + Nk'2 ~ j
&J f / (wj VGk+4~ej Mek) dr ..
(20)
,,HU~RICAL R~SULT8 As s n u m e r i c a l exsmplet we hsve c o n s i d e r e d .the f l e x u r e of c u r v e d p l ~ under edge l o a d i n g . The consto, n te a k i n (18) were evslusted when sere d e f l e c t i o n and u n i t moment a r e p r e s c r i b e d on .the r a d i a l edse e = Oo The sooure~y wi.th whioh d e f l e o t i o n and momen.t s r e reproduoed on e m 0
C U R V E D P L A T E S WIT}{ B U I L T - I N
CURVED EDGES
363
Boots of t he e q u s t i o n ( 7 ) AlYmlTbOtiO form i s given by
2J~÷1)~
•k k
Re•l
+
i 1,1 [-2-1~.1_-)-~--s~ (-l-.n--b~!]
2
I ( P•zt
xn b/• Inmsin~ry l~zt
k
J /
Resl l~rt
in b
Insginszy Psrt
1
0.28P29 E 01
0.79763 g O0
6
0.17158 g 02
O. 81030 Z O0
2
0.57181 E 01
0.80684 g OO
7
0.20017 E 02
0.81042 K O0
3
0.85785 g 01
0.80895 E OO
8
0.22877 B 02
0.81050 Z O0
4
0.11438 E 02
O. 80973 g O0
9
O. 25736 g 02
0.810~5 I
5
O. 14298 g 02
0,81010 E O0
10
0.28596 g 02
0.81059 Z O0
T* LX 2 Numozicsl Convergonoe obtained using Ton E i g e n ~ l u e s l~esortbed
£ u n o t i o n v - O, Me -- 1, 1 ~ r ~ 3 ,
Z
w
Me
1.2
0 . 1 9 1 0 E-O4
0 . 9 5 8 9 R 00
1.4
-0.2317 g-05
0 . 1 1 ) 7 g 01
1.6
0.8788 g-05
0 , 9 1 6 6 g O0
1• 8
O. 4693 g - 0 5
O, 1041 g 01
2.0
- 0 . 1 1 4 7 F--05
0.9631 g O0
2.2
O. 6225 R-05
O. 1070 g 01
2.4
O. 2288 g - 0 5
O. 8672 2 O0
2.6
-0.7195 g-04
0 , 1 1 6 7 Z 01
2.8
0 . 8 6 2 8 g-O4
0 . 9 6 4 3 g O0
0 n O
00
364
B.G.
PRAKASH
u s i n g t h e e i g e n f u n o t i o n s e r i e s &re gLTen i n Tsble 2. v , s o s r r i e d out t a k i n g t h e P o i s e o n ' s r e t i e r&%io b&
= 3.0 .
The OOmlXat~tion
~ = 003 ~nd t h e ~speot
Ton eigenT&lues were u s e s t o s t u d y t h e n u m e r i c , 1
c o n v e r g e n c e of t h e e i g e n ~ u n o t i o n e x p a n s i o n s w i t h 8 u m B t i o n h e i n 8 oxt4nded over t h o s e e i g e n v s l u e s whose r e e l l~g~ i s n o g s t i v e .
I% v , s
f o u n d th~% Mo d e e s y s r s p i 4 1 y w i t h the t n o r e s s e of O slon~ t h e middle line (b~)/2
of the c u r v e d p l ~ t e which e x p l s i n s t h e 8~int-¥enLn%
l~rinoiple. it~CZS
1.
8 ~ r ~ t P.V.B.A.8. p R~maohsndz~ R4~o, B.S. sad G o l ~ l s o h s r y u l u , S . ,
Trot. J. B ~ g . 2.
SOL**
13 , 1975, p 149.
Pralmsh, B.G. snd lt6umohandzs B6o, B. 8 . , J . M6th. e~nd Phy. Sci.~
10, 1976, p 333. 3.
R~mJoh~ndzs R~o, B. S. snd Ps'Lk~sh, B. G., J . Sv,s'uot. Mech., 6,
1978, p 133. 4.
S~Z~L, P.V.B.JL.S., ih~emohandra R6o, B.S. ~nd 9OlXb]4~h~zyulu, S . ,
SIAM J. J,pjpl. Ms,t h e . , 26, 1974, p 568. ~.
R~m~oh~nd~s Rao, E~S. end 8 ~ i d h s z s , J . K . , J.J~ppl. Mech., 38, 1971,
p 70~.
6.
(~tnbezs, G.A., PI,M J . , 1T, 19~3, p 211.