twisted timeshenko beams

7/28/2019 Twisted Timeshenko Beams

1/11

1 9 0 R . S . G U PT A A N D S . S . RA Ow h e r e x ' x ' a n d y ' y ' a r e t h e a x e s i n c li n e d a t a n a n g l e O, t h e a n g l e o f tw i s t, a t a n y p o i n t i n t h ee l e m e n t , t o t h e o r i g i n a l a x e s x x a n d y y a s s h o w n i n F i g u r e l ( c ) . T h e v a l u e o f / = , , , = 0 a n d t h ev a l u e s o f l~ , ~ , a n d I , ,y , c a n b e c o m p u t e d a s

b ( ~ ) h ' ( ~ ) II x , ~ , ( z ) - - l ' - - - - - ~ - 1 2 Y [ a ' z 4 + a 2 I z 3 + a 3 1 2 z 2 + a 4 1 3 z + a s ] ' (7 )w h e r e

w h e r e

a , = ( b 2 - b , ) ( h 2 - h , ) 3 , a z = b , ( h 2 - h , ) 3 + 3 ( b 2 - b , ) ( h 2 - h , ) 2 h , ,a 3 = 3 { b , h ~ ( h 2 - h , ) 2 + ( 6 2 - b , ) ( h 2 - h ~ ) h ~ } ,a , = 3 b , h~(h2 - h , ) + (b 2 - - 6 , ) h ~ , a s = b x h ~ ,

~ , . , . ( z ) = h ( z ) . b 3 ( z ) = ~ [ a , z" + a 2 t : + a ~ " z 2 + d , t 3 + a , t ' ] ,12 1214

( 8 )( 9 )

t h e e l e m e n t s t if fn e s s m a t r i x c a n b e e x p r e s s e d a s[ A K ] [0 ] [ D K ] [0 ]

[ o ] [ C K ] [ 0 ] [ 01 1 ,[ K ] = [ D K ] [0 ] [ B K ] [0 ] /1 6 x 1 6

[o 1 [o 1 [o 1 [ C K l ]

a n di l a 2 w b \ [ a " vb

w h e r e [ A K ] , [ B K ] , [ C K ] a n d [ D K ] a r e s y m m e t r i c m a t r i c e s o f o r d e r 4 a n d [ 0] i s a n u l l m a t r i xo f o r d e r 4 . T h e e l em e n t s o f m a t r i c es [ A K ] , [ B K ] , [ C K ] a n d [ D K ] a r e f o r m u l a t e d i n A p p e n d i xA .

0 5 )

( 1 6 )

d , = ( h2 - h , ) ( 6 2 - b , ) 3 , d 2 = h , ( b 2 - b , ) 3 + 3 ( h 2 - h , ) ( 6 2 - b , ) 2 b , ,d 3 = 3 {h , b , ( 6 2 - 6 , ) 2 + ( h 2 - h , ) ( b 2 - 6 , ) 6 ~} ,d 4 = 3 h, b ~ ( b 2 - b , ) + ( h 2 - h , ) b ~ , d s - - h , b ] . (113)

B y s u b s t i t u t i n g t h e e x p r e s s i o n s f o r w b , w : , v b , v s, A , I x x , I xy a n d I yy f r o m e q u a t i o n s ( 2 ) , ( 4 )a n d ( 6) in e q u a t i o n ( 3) , t h e s t r a in e n e r g y U c a n b e e x p r e s s e d a s

v = 8 9 ( l l )w h e r e u i s t h e v e c t o r o f n o d a l d i s p l a c e m e n t s u , , u2 . . . . . u , 6 , a n d [ K ] is t h e e l e m e n t a l s t if f n e ssm a t r i x o f o r d e r 1 6. I n t e r m s o f th e i n t e g ra l s d e f i n e d a s

f [ a 2 w ~ \ 'E I , , , ,~ - - ~z~J d z = [u~u2u3u4l r [AK] [u~u2u3u4] , ( 1 2 )z / 0 2 . \ 2

i . l a w , \ 2~ A G \ / - ~ - z d z = [ u , u 6 u 7 u s ] [ C K ] [ u , u 6 u 7 u s ] , ( 1 4 )0


2/11

TAPERED AND TWISTED TIMOSHENKO BEAMS 1912. 3. ELEMENT MASS MATRIX

T h e k i n e t i c e n e r g y o f t h e e l e m e n t T , i n c l u d in g t h e e ff e ct s o f s h e a r d e f o r m a t i o n a n d r o t a r yin e r t i a , i s g iv e n b y

o ; L w +p /a 'w~\/azv~'~ p[= [a~w ,\q+ g i , , , t a - - - ~ t ) t - ~ . ~ ] + . ~ g l O - - ~ ) l d z . ( 1 7 )

B y d e f i n i n g

a n d

! 2

J - ~ - \ ~ - / d z = [ a , a ~ a 3 a , ]" [ a M ] [ a , a , a 3 a , ],0l 2

(3 T \ 0 - - f f ~ / d z = [ a , u z a , ~ , ] r [ B M ] [ a , u z u 3 t i , ] ,0

!f T \ a - ~ - ~ / d z = [a 9 a ,o a , , a , = ] " [ C M ] [ a g . , o . , , a , . ] .o

(18)

(19)

(20 )

I t is t o b e n o t e d h e r e t h a t a l l t h e f o r c e d b o u n d a r y c o n d i t i o n s c o u l d b e s a ti sf ie d b y t h e p r e s e n tm o d e l . A m o n g t h e n a tu r a l b o u n d a r y c o n d i t i o n s , i f t h e c o n d i t i o n o f z e r o b e n d in g m o m e n t i st o b e e n f o r c e d a t a f r e e e n d , t h e e l e m e n t d u e t o T h o m a s a n d A b b a s [2 0] is e x p e c t e d t o b eb e t t e r t h a n t h e p r e s e n t o n e .

f r e e e n d : O wd O z =O a n d Ov,/Oz=O; (24)c l a m p e d e n d : w , = 0, wb = 0, v, = 0, vb = 0, Owb/Oz 0 a n d OvdOz= 0 ; (25 )

h in ge d en d : w, = 0 , wb = 0 , v , = 0 an d vb = 0 . (26 )

c o n d i t i o n s :

1

J o g \ O z Ot] ~Oz Ot] d z = [ a , a z z~ 39 , ] r [ D M ] [ 9 9 9 , o u , , ~ , = 1 , ( 2 1 )w h e r e ~ d e n o t e s t h e t im e d e r i v a t iv e o f th e n o d a l d i s p l a c e m e n t tq , i = 1 , 2 . . . . . 1 6, t h e k in e t i ce n e r g y o f t h e e l e m e n t c a n b e e x p r e ss e d a s

T = 8 9 f i, ( 2 2 )w h e r e [ M ] i s t h e m a s s m a t r i x g i v e n b y

[ [ A M ] + [ a M ] [ A M ] [ D M ] [0 1 ]/ /= [ [ A M ] [ A M ] [ A M ] [0 1 1[M] ( 2 3 )/ [DM] tAM ] tAM] + [ C M ] t A M ] / '~6x16/ /L [0] [0] [ A M ] [AMlJ

a n d [ A M ] , [ B M ] , [CM ] a n d [ D M ] a r e s y m m e t r i c m a t r i ce s o f o r d e r 4 w h o s e e l em e n t s a r ed e f i ne d in A p p e n d i x A .2.4. BOUNDARYCONDmONS

T h e f o l l o w i n g b o u n d a r y c o n d i t i o n s a r e t o b e a p p l i ed d e p e n d i n g o n t h e t y p e o f e n d


3/11

192 R .S. GLIPTAAND S. S. RAO3 . N U M E R I C A L R E S U L T S

T h e e l e m e n t s ti ff ne s s a n d m a s s m a t r ic e s d e v e l o p e d a r e u s e d f o r t h e d y n a m i c a n a l y s i s o fc a n t i l e v e r b e a m s . B y u s i n g th e s t a n d a r d p r o c e d u r e s o f s t r u c t u r a l a n a l y s i s , t h e e i g e n v a l u ep r o b l e m c a n b e s t a t e d as

( iX] - - o ,2 [M] ) U = 0 , ( 27 )w h e r e [ K ] a n d [ M ] d e n o t e t h e s t i ff n e ss a n d m a s s m a t r i c e s o f th e s t r u c t u r e , r e s p e c t i v e l y , Ui n d ic a t e s n o d a l d i s p l a c e m e n t v e c t o r o f t h e s t ru c t u r e , a n d w i s t h e n a t u r a l f r e q u e n c y o fv i b r a t i o n .

A s t u d y o f t h e c o n v e r g e n c e p r o p e r t i e s o f t h e e l e m e n t i s m a d e b y t a k i n g t h e s p e c ia l c a s e o f au n i f o r m b e a m w i t h a le n g t h o f 0 .2 5 4 0 m , b r e a d t h o f 0 .0 7 62 m , d e p t h o f 0 .0 7 0 4 m , E = 2 .0 7 10 ~ I N /m z , G = 3E/8, m a s s d e n s i t y o f 8 00 k g / m 3 , / a = 2 / 3 a n d 0 = 0 ~ F o r t h i s b e a m , t h ef i r s t , s e c o n d , t h i r d a n d f o u r t h n a t u r a l f r e q u e n c i e s o b t a i n e d b y t h e p r e s e n t m e t h o d ( w i t h 4e l e m e n t s ) h a v e b e e n f o u n d t o h a v e 0 " 0 0 y , 0 . 07 ~ o, 0 . 3 0 ~ a n d 0 . 6 0 ~ e r r o r s , r e s p e c t i v e l y .T h e f i rs t f o u r n a t u r a l f r e q u e n c i e s o b t a i n e d b y u s i n g 8 e l e m e n t s a r e 8 4 5. 8 , 39 8 9 .5 , 8 8 3 6. 8 a n d1 38 27 .1 H z w h i l e t h e e x a c t v a l u e s a r e 8 4 5 .8 , 3 9 8 8 .9 , 8 8 3 4 .2 a n d 1 38 18 .1 H z , r e s p e c t i v e l y[2 0]. T h e c o n v e r g e n c e o f th e n a t u r a l f r e q u e n c i e s o f a p r e t w i s t e d d o u b l y t a p e r e d c a n t i l e v e rb e a m h a s a l s o b e e n s t u d i e d a n d t h e r e s u lt s a r e s h o w n i n T a b l e 1 . I n t h is c a s e t h e n a t u r a lf r e q u e n c i e s g i v e n b y t h e m e t h o d o f re f e r e n c e [ 21 ] h a v e b e e n f o u n d t o b e s l ig h t l y h i g h e r t h a nt h o s e p r e d i c t e d b y t h e p r e s e n t m e t h o d . I t c a n a l s o b e s ee n t h a t r e a s o n a b l y a c c u r a t e r e s u l tsc a n b e o b t a i n e d e v e n b y u s i n g f o u r f in it e e l e m e n t s .

T A ~ L z 1Natural frequencies of a tapered and twisted Tirnoshenko beam ( H z )

N u m b e r o f e le m e n ts F i r s t m o d e S e co n d m o d e T h i r d m o d e F o u r t h m o d e1 304-8 1187.0 2259.3 4519-22 298.7 1146.8 1685-3 4046.53 298-1 1139.2 1652.2 3647.44 297.9 1137.9 1647.3 3593.55 297.8 1137.5 1646.0 3585.66 297.8 1137.4 1645.3 3578.87 297.8 1137.3 1645.1 3578.58 297-8 1137-3 1645-0 3578.3A c c o r d i n g t o t h e m e t h o d o fre ~r en ce [21] 299.1 1142-8 1653"3 3595.7

D ata: length o f beam = 0.1524 m, breadth at ro ot = 0-0254 m, depth tap er ratio = 2.29, breadth taperratio = 2"56, tw ist = 45 ~ shear coefficient = 0.833, mass density = 800 kg]m 3, E = 2.07 x 10 l N]n l2,G = (3/8)E.T a b l e 2 s h o w s t h e f r e q u e n c y r a t io s o f a n u n t w i s te d t a p e r e d b e a m f o r v a r i o u s c o m b i n a t i o n s

o f d e p t h a n d b r e a d t h t a p e r r a ti o s . S ix fi ni te e l e m e n t s a r e u s e d t o m o d e l t h e b e a m . I t i s o b s e r v e dt h a t f o r c o n s t a n t d e p t h t a p e r r a t i o t h e f r e q u e n c y r a t io o f al l t h e f o u r m o d e s i n c re a s e s w i thb r e a d t h t a p e r r a ti o w h i le f o r c o n s t a n t b r e a d t h t a p e r r a t io t h e f r e q u e n c y r a t i o d e c r e a s e s f o rt h e f i r s t m o d e a n d i n c r e a s e s f o r t h e s e c o n d , t h i r d a n d f o u r t h m o d e s w i t h a n i n c r e a s e i n t h ed e p t h t a p e r r a t i os . T h e s h e a r d e f o r m a t i o n e f f ec ts r e d u c e t h e f r e q u e n c y o f m o d a l v i b r a t i o n .T h e p r e s e n t r es u l ts c a n b e s e en t o c o m p a r e w e l l w i t h t h o s e r e p o r t e d b y M a b i e a n d R o g e r s [ 6]w h i c h a r e a l s o i n d i c a te d i n T a b l e 2 . F i g u r e 2 s h o w s c o m p a r i s o n o f th e r e s u lt s g i v e n b y t h ef in i te e l e m e n t m e t h o d w i t h t h o s e r e p o r t e d b y R o s a r d [9 ] f o r a t w i s t e d b e a m o f 0 . 0 25 4 m x0 . 00 6 3 5 m c r o s s - s e c t i o n a n d 0 . 2 79 4 m l e n g t h . I t c a n b e s e e n t h a t t h e t w o s e t s o f re s u l ts a r eq u i t e c o m p a r a b l e .


4/11

Ig

III1

,+--

'0

,~

"000IE~"do0Io

~c~

,o~o~

~j~3

.~0~

0

.

~j

000


5/11

1 9 4 R. S. GUPTA AND S. S. RAO9

8 -

7

6.o

g4 ~

3

2

I

1 I

Third mode

Second mode

First mode

I II0 ZOAngle of twis t (degrees)

30

Figure 2. Comparison of results for an uniform twisted beam. -- -- , Values by Rosard method; - -values by present method.

F i g u r e s 3 a n d 4 s h o w t h e v a r i a ti o n o f m o d a l f r e q u e n c ie s w i t h b r e a d t h t a p e r r a t io f o r b e a m sh a v i n g 0 ~ 3 0 ~ 6 0 ~ a n d 9 0 ~ t w i s t w i t h c o n s t a n t d e p t h t a p e r r a t i o w h i l e F i g u r e s 5 a n d 6 s h o ws i m i l a r v a r i a t i o n s f o r b e a m s w i t h c o n s t a n t b r e a d t h t a p e r r a t i o a n d v a r y i n g d e p t h t a p e rr a t i o . H e r e t h e l e n g t h o f t h e b e a m i s t a k e n a s 0 . 2 5 4 m a n d t h e r o o t c r o s s - s e c t i o n a s 0 .0 7 6 x0 . 0 3 8 t i m e s t h e l e n g t h o f t h e b e a m . A g a i n t h e e ff ec t s o f b r e a d t h a n d d e p t h t a p e r s a re s e e n t o

I i I

4r - Second mode

O~o6O*

F First mod e/ ] / - 9 o ~' / / / // / / s .

/ l l O / / / / l l ~ - . ~ . ~f i l l / 1~ .~ , - ~ - - = - -

I I I2 3 4

Breadth toper rQliOF ig u re 3 . E f f ec t s o f s h e a r d e fo r m a t i o n a n d b r e a d t h t a p e r r a t i o o n t h e f i rs t a n d s e c o nd n a t u r a l f requenc ieso f a t w i s te d b e a m . e , M e t h o d o f r e fe r en c e [ 2 2 ]; , w i t h o u t s h e a r d e f o r m a t i o n ; . . . . , T i m o s h e n k o b e a m ;d e p t h t a p e r r a t i o = = 3.


6/11

T A P E R E D A N D T W I S T E D T I M O S H E N K O B E A M S1 4 I I I

13 \ \\ \ ~. \12 \ \ \

/ f / f - Fourth mode11 - \ \ \ ~ ' ' , -0 I0 ~ . ~

9 - ~"t~

8 / f / f -Th i rd mode~ +

6 - 3 0 "

4 I i I I2 3 4 5Breadth toper ratio, l

195

Figure 4. Effects of shear deformation and breadth taper rat io on the third and fourth natural frequenciesof a twisted beam. , Without shear deformation; . . . . . , Timoshenko beam; depth taper ratio cr = 3.

be pronounced at higher modes of vibration. Here also the effect of shear deformation isseen to reduce the moda l fr equencies at higher rates in higher modes of vibrat ion in all thecases. The results found by the meth od of Carnegie and T hom as [22] for the first two nat ura lfrequencies are also indicated in Figures 3 and 5. It can be seen that the present resultscompa re excellently with those of Carnegie and Tho mas.

. 9

u~

I I I

~ 9 0 ~Firsl mode -~ I~, 60 ~

I I I2 3 4Depth taper ratio,a

Figure 5. Effects of shear deformation and depth taper rat io on the first and second natural frequencies of atwisted beam. , Without shear deformation; . . . . , Timoshenko beam; breadth taper ratio fl = 3. o,Method of reference [22].


7/11

196

a twisted beam.

13

12

II

I 0._o

9

7

6

5

4

R. S. GUP TA AND S. S. RAO

I I I

FcxJrth mode~ ~ ~)o

T h i r d m o d e --~ o

i i I2 3 4D e p t h t a p e r r a t io

Figure 6. Effects of shear deformation and depth taper ratio on the third and fourth natural frequencies of, Without shear deformation; . . . . . , Timoshenko beam; breadth taper ratio/Y = 3.4. CONCLUSION

The finite element procedure developed for the eigenvalue analysis of doubly tapered andtwisted Timoshenko beams has been found to give reasonably accurate results even with fourfinite elements. The effects of breadth and depth taper ratios, twist angle and shear de forma-tion on the natura l frequencies of vibration of cantilever beams have been investigated.The present results are found to compare very well with those report ed in the literature. Theelement developed is expected to be useful for the dynami c analysis of blades of roto-dyna micmachines.

REFERENCES1. J. S. RAo 1965 Aeronau t i ca l Quar ter l y 16, 139-144. The fundamental flexural vibration ofcantilever beam of rectangular cross-section with uniform taper.2. G. W. HOUSNERand W. O. KEIG}{T[.EY1962 Proceedings o f the Ame rican S ocie ty o f Civi l Enghz-eers 88, 95-123. Vibrations of linearly tapered beam.3. N. O. MYKLESrAD1944 Journa l o f Aerospace Sc i ence 2, 153-162. A new method for calculatingnatural modes of uncoupled bending vibrations of aeroplane wings and other types of beams.4. A. I. MARTIN1956Aeronaut ical Quarterly 7,109-124. Some integrals relating to the vibration ofa cantilever beams and approximations for the effect of taper on overtone frequencies.5. J. S. RAO and W. CARNEGIE 1971 Bul le t in o f Mec hanica l Engineerhtg Education 10, 239-245.Determination of the frequencies of lateral vibration of tapered cantilever beams by the use ofRitz-Galerkin process.6. H.H. MAB1E,and C. B. ROGERS1972Journa l o f t he Acous t i ca l Soc ie t y o f Am er ica 51, 1771-1774.Transverse vibrations of double-tapered cantilever beams.7. H. H. MABm and C. B. ROGERS 1974 Journa l o f t he Acous t ica l Soc ie t y o f Amer ica 55, 986--988.Vibration of doubly tapered cantilever beam with end mass and end support.8. m. MENDELSONand S. GENDLER 1949 N A C A TN-2185. Analytical determination of coupled

bending torsion vibrations o f cantilever beams by means of station functions.9. D. D. ROSARD 1953 Jottrt ta l o f App l ied M echan ics 20, 241-244. Natural frequencies of twistedcantilevers.10. R. C. DI PRIMA and G. H. HANDELMAN 1954 Quar ter l y on App l i ed Mathemat i c s 12, 241-259.Vibration of twisted beams.


8/11

T A P E R E D A N D T W I S T E D T I M O S H E N K O B E AM S 19711. W. CARNEGIE1959 Proceedings o f the Inst i tu te o f i t lech anica l Engineers 173, 343-346 . V ib ra t ionof p r e twi s t ed ca n t i l eve r b l ad ing .12. B . DAW SON 1968 Journal of Mechanical Engineerhtg Science 1 0 , 381-388. Cou pled bend ingv i b r a t i o n s o f p r e t w is t e d c a n t il e v e r b la d i n g t r e a t e d b y R a y l e i g h - R i t z m e t h o d .13. J. S. RAG 1971 Journal o f the Aeronautical Socie ty o f lndia 23 , 62 -64 . F l exura l v ib r a t ion o fp r e twi s t ed beam s o f r ect angu lc . r c ross - sec t ion .14. W . CARNEGIE an d 3". THOMAS 19 72 Journal of Engineerhtg for Industry, Transactions o f the .Am erican Socie ty o f Mechanical Enghwers 94 , 255-266 . The coup led bend ing -bend ing v ib r a t ionof p r e twi s t ed t ape r ed b l ad ing .15. R . M cCALLEY 1963 General Electr ic Company, Schenectady, N ew York, Repo rt No. D IG /SA ,6 3 - 7 3 . R o t a r y i n e rt i a c o r r e c ti o n f o r m a s s m a t r i c e s.16. J . S . ARCHER 1965 Am erican Institute o f Aeronautics an d Astronautics Journal 3, 1910-1918.Con s i s t en t m a t r ix f o rm ula t ions fo r s t r uc tu r a l ana lys i s u s ing fin it e e l emen t t echn iques .17. K . K . KAPUR 1966 Journa l o f the A cous t ica l Soc ie ty o f A mer ica 40, 1058-1063. Vib rat ion s of aT i m o s h e n k o b e a m , u s i n g f in i te e l e m e n t a p p r o a c h .18. W. C A R N E G I E , J . THOMAS and E. DOCUMAKI 196 9 Aeronautical Quarterly 2 0 , 321-332. Ani m p r o v e d m e t h o d o f m a t r i x d i s p l a c em e n t a n a l y s is i n v i b r a ti o n p r o b l e m s .19. R. NICKEL an d G . SECOR 1972 bzternational Journal o f Nume rical M ethods in Enghwering 5 ,

243-253 . Co nvergen ce o f cons i s ten t ly de r ived Timo she nko beam f ini te e l emen t s .20. J. THOMASand B. A . H . ABBAS 1975 Journal o f So un da nd Vibration 4 1 , 2 9 1 - 2 9 9 . F i n i te e l e m e n tm o d e l f o r d y n a m i c a n al ys is o f T i m o s h e n k o b e a m .21. 3 . S. RAG 1972 Journal o f Engineerhtg for bMustry , Transactions o f the American Socie ty o fMe chanical E ngineers 94 , 343-346 . F l exu ra l v ib r a t ion o f p r e twi s t ed t ape r ed can t i l eve r b l ades.22. W . CARNrGm an d J. THOMAS1972Journa lo fEng ineer ing for lndus try , Transac tions o f the Amer i -can Socie ty o f Mechan ical Engineers 94 , 367-378 . The e f f ec t s o f shea r de fo rmat ion and ro t a ryine r t i a on the l a t e r a l f requenc ies o f can t i l eve r beams in bend ing .

A P P E N D I X A : E X P R E S S IO N S F O R [ AK] , [ BK] . . . . , [ D M ]T h e f o l lo w i n g n o t a t i o n is u se d f o r c o n v e n i e n c e :

1w , = J z ' - ' d z , i = 1 ,2 . . . . . n , ( A I )0

L l = I t - l , i = 1,2 .. .. . n, (A2)[ z ]Vl = f z ' - ' co s 2 (02 - 0 , ) + 01 dz , i = 1 , 2 . . . . . n , ( 1 3 )0 , [ z ]St = f z l-~ sin 2 (02 - 01) + 01 d z , i = 1 , 2 . . . . . n , ( 1 4 )0

w h e r e O~ a n d 0 2 d e n o t e t h e v a l u e s o f p r e t w i s t a t n o d e s I a n d 2 , re s p e c t i v e ly , o f th e e l e m e n t .As wb, w~, Vb a n d v~ a r e a l l th e s a m e i n n a t u r e e x c e p t f o r t h e i r p o s i t i o n s i n t h e s t if fn e s s a n d

m a s s m a t r i c e s , o n e c a n u s e )~ t o d e n o t e a n y o n e o f t h e q u a n t i t i e s w b, w s, v b o r v s a n d i n as im i l a r m an n er t he se t (111,112,113, a4 ) ca n be us ed to r ep res en t a ny on e o f t he se t s (u~, u2, u3, u4),(us, u6, u7, us) , (ug, U , o , u l , , u12) o r (u13, u14, u ,s , U16 . T h u s

1] 13 _113 -3 or .2 12Z) q_~3(3122_223)_~__~4(23 -i f (z ) = ~ - ~ ( 2 z 2 - 3 1 z 2 + ) ~ ( ~ - , . , ~ + 1 1 2" IZ2), (AS)d ~ 111 ~ ( 3 Z 2 - - 4 1 z " --~z = - f ~ (6 z 2 - 6 1 z) - + 1 2 ) + ~ ( 6 1 z - 6 z 2 ) - ~ ( 3 z 2 - 2 1 z ) , ( A 6 )

d 2 if, l] 1 . a3 t]2 a 4d z 2 = I-S ( 1 2 z 6 1 ) - . ~ ( 6 z - 4 1 ) + i f ( 6 1 - 1 2 z ) - 7 ~ ( 6 z - 2 1 ) . ( t 7 )


9/11

19 8 R. S. GUPTA AND S. S. RAOB y l e t t i n g P ~ , t . k ( i = 1 . . . . 4 ; j = i , . . . , 4 ; k = t . . . . . 7 ) d e n o t e t h e c o e f f ic i e n t o f z k - X l 7 -k f o rt h e a~zT~ t e r m i n t h e e x p r e s s i o n o f ~ 2 , Q~.~ ,k( l = 1 . . . . 4 ; j = i . . . . . 4 ; k = 1 . . . . 5 ) t h ec o e f f i c i e n t o f z k - ~ l 5 - k f o r th e f i ~ t e r m i n th e e x p r e s si o n o f ( d ~ ] d z ) z, R ~ . j , k ( i = 1 . . . . 4 ;

j = i , . . . . 4 ; k = 1 . . . . 3 ) t h e c o e f f i c ie n t o f z k - ~ l 3-k f o r t h e a ~ a j t e r m i n th e e x p r e s s i o n o f( d 2 ~ ' /d z 2 ) 2 , H i . j ( i = I . . . . . 4 ; j = i . . . . . 4 ) t h e i n d e x c o e f f i c i e n t o f l to a c c o u n t f o r t h e d i f f e r -e n c e i n i n d e x o f I d u e t o m u l t i p l i c a t i o n o f r o t a t i o n a l d e g r e e s o f f r e e d o m ~1 a n d a 2 a n d t h ed i s p l a c e m e n t d e g r e e s o f f r e e d o m a 3 a n d t74, t h e v a l u e s o f P ~ . j . k , Q ~ . j .k , R ~ . j , k a n d H ~ ,~ c a n b eo b t a i n e d a s sh o w n in T a b l e s A I a n d A 2 .

T A BL E A 1V a h t e s o f H i . j , R i . j . k , Q l . j. ~

R ~ . ~ ., f o r k = Q t . j . , f o r k =)k Ar "~ r 9i j H~ . j 1 2 3 1 2 3 4 5

1 1 0 1 4 4" 0 - 1 4 4 " 0 3 6 "0 3 6 . 0 - 7 2 - 0 3 6" 0 6 - 0 0 . 01 2 0 - 1 4 4 " 0 1 44 "0 - 3 6 . 0 - 3 6 " 0 7 2" 0 - 3 6 " 0 0 - 0 0 ' 01 3 1 - 7 2 " 0 8 4" 0 - 2 4 . 0 - 1 8 . 0 4 2" 0 - 3 0 " 0 6 ' 0 0 ' 01 4 1 - 7 2 " 0 6 0" 0 - 1 2 " 0 - 1 8 " 0 3 0 ' 0 - 1 2 " 0 0 "0 0 . 02 2 0 144-0 -1 44 "0 36"0 36-0 -7 2" 0 36"0 0 -0 0 -02 3 1 7 2 -0 - 8 4 " 0 2 4 . 0 1 8 :0 - 4 2 " 0 3 0" 0 - 6 - 0 0 ' 02 4 1 7 2 -0 - 6 0 ' 0 1 2" 0 1 8 " 0 - 3 0 ' 0 1 2" 0 0 - 0 0 . 03 3 2 3 6 - 0 - 4 8 " 0 1 6" 0 9 "0 - 2 4 " 0 2 2 . 0 - 8 - 0 1 "03 4 2 3 6" 0 - 3 6 - 0 8 . 0 9 "0 - 1 8 " 0 1 1 .0 - 2 " 0 0 "04 4 2 3 6 -0 - 2 4 " 0 4 - 0 9 "0 - I 2 - 0 4 - 0 0 - 0 0 "0

T A B LE A 2V a h l e s o f P t , j.

P I . j . ~ f o r k =ri ] I 2 3 4 5 6 7

I 1 4 . 0 - 1 2 . 0 9 . 0 4 . 0 - 6 . 0 0 . 0 1 .01 2 - 4 . 0 1 2 .0 - 9 . 0 - 2 , 0 3 .0 0 . 0 0 -01 3 - 2 . 0 7 . 0 - 8 . 0 2 . 0 2 . 0 - 1 - 0 0 -01 4 - 2 . 0 5 . 0 - 3 , 0 - 1 . 0 1 .0 0 . 0 0 -02 2 4 . 0 - 1 2 . 0 9 . 0 0 - 0 0 . 0 0 . 0 0 . 02 3 2 .0 - 7 . 0 8 .0 - 3 . 0 0 . 0 0 . 0 0 . 02 4 2 - 0 - 5 . 0 3 . 0 0 . 0 0 . 0 0 . 0 0 - 03 3 1 -0 - 4 . 0 6 - 0 - 4 - 0 1 -0 0 - 0 0 - 03 4 1 .0 - 3 . 0 3 . 0 - 1 . 0 0 - 0 0 . 0 0 . 04 4 1 . 0 - 2 . 0 1 -0 0 . 0 0 . 0 0 . 0 0 . 0

EVALUATION OF [ B K ]A s t h e p r o c e d u r e f o r th e d e r i v a t i o n o f [ A K ] , [ B K ] . . . . . [ D ~ I ] i s t h e s a m e f o r e a c h , t h e

e x p r e s s i o n f o r [ B K ] i s d e r i v e d h e r e a s a n i l l u s t r a t i o n . O n e h a s

(A8)


10/11

T A P E R E D A N D T W I S T E D T I M O S H E N K O B E A M S 19 9wh e r e ~ = vb a n d

t t 2 = U l o .t 2 3 ) u H9 4 ku12

P u t t i n g t h e v a l u e o f l y r a n d f f i n e q u a t i o n ( A 8 ) g iv e s

i / a 2 r v \ ' E l , % . + ( I , . , . { ( 0 , 0 z + x. . > c o , , , , , 0 , ) ]x g ( 1 2 z - 6 1 ) - ~ ( 6 z - 40 + ~ ( 6 ! - 1 2 z ) - ~7(6z - 20

w i t hd z , ( A9 )

B K x . a = coe f f i c ien t o f a~ ax = coe f f i c ien t o f u9 u9E

1211~+


11/11

2 0 0 R .S . GUP T A AND S . S . RAO3 5

C K , s = I t G ~ ~ [ c , L ( , + j + n , .,, U o - i - j ) O t . j . j ], 1 = 1 , 4 ,9 1 8 ~ 9 . . . ,l = l j = l J = / , . . . , 4 , ( A I 3 )5 3

I=1 1=1I = l . . . . . 4 , J = I , . . . . 4 ,

( A I 4 )3 7

A M , . s = g ~ x ~ [ c , L , . s . m . j i U t u _ , _ s ) P , . , . , ] , I = I . . . . . 4 ,= J=l

5 5B M t s = P ~ ~ [ {d , L c,+ j+ m , ) U o , - , - j ) Q t . s . j } +9 1 2 g P ~ I -1 / = t

+ ( a, - d l ) { L ( t + l . n , . : V o l _ l _ j ) Q t . s . j } ] , I = 1 . . . . 4 ,$ 5

PC M t s =9 12gll----- ~ . ~ [{ a ,L ,+ j+ ,,,.,, U cu _,_ j, Q ,.j.~ } +/ - 1 J = l

+ ( d i - a l ) { L ( i . ~ + n , . ~ ) V ( u - ~ - ~ ) Q 1 . s . j } ] , I = 1 . . . . 4 ,5 5_ P

1=1 1=1

J = I , . . . . 4 , ( A l 5 )

J = l . . . . . 4 , ( A I 6 )

J = I . . . . . 4 , ( A ! 7 )

I = 1 . . . . 4 , J = I , . . . . 4 .

(AI8)A P P E N D I X B : N O M E N C L A T U R E

AbEgGhIx ,,, IT,),, x~,[K ]1L[M]

tUU1)

Wx , yZf r e q u e n c y r a t i oO~f l0Pp

a r e a o f c r o s s - s e c ti o nb r e a d t h o f b e a mY o u n g ' s m o d u l usa c c e l e r a ti o n d u e t o g r a v i t ys h e a r m o d u l u sd e p t h o f b e a mm o m e n t o f i n er ti a o f b e a m c r o ss - se c t io n a b o u t x x , y y a n d x y ax i s , r e sp ec t iv e lye l em en t s t i ff n e ss m a t r ixl e n g t h o f a n e l e m e n tl e n g t h o f t o ta l b e a me l e m e n t m a s s m a t r i xt i m e p a r a m e t e rn o d a l d e g r e e s o f f r e e d o ms t r a i n e n e r g yd i s p l a c e m e n t i n x z p l a n ed i s p l a c e m e n t i n y z p l a n ec o - o r d i n a t e a x e sc o - o r d i n a t e a x is a n d l e n g t h p a r a m e t e rr a t io o f m o d a l f r e q u e n c y t o f re q u e n c y o f f u n d a m e n t a l m o d e o f u n i f o r m b e a mw i t h th e s a m e r o o t c r o s s - s e c t i o n a n d w i t h o u t s h e a r d e f o r m a t i o n e f fe c tsd e p t h t a p e r r a t io , = h d h ab r e a d t h t a p e r r a t i o , = b~[b2a n g l e o f t w i s tm a s s d e n s i t ysh ea r co e f f i c ien tS u b s c r i p t s : b , b e n d i n g ; s , s h e a r .

twisted timeshenko beams

Documents