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P l a te o n V i s co e la s ti c o u n d a t i o n u n d e r a M o v i n g L o a dy SATYANARAYAN M A N D A L 1

Summary - In this note the deflection of a simply supported rectangular elastic plate on visco-elastic foundation has been obtained du e to a moving load.Introduction

T h i s n o t e c o n c e r n s w i t h t h e d e f l e c ti o n o f a n e l a s t i c p l a t e o n v i s c o - e l a s ti c f o u n d a t i o n .T w o d i f f e r e n t t y p e s o f f o u n d a t i o n s h a v e b e e n c o n s i d e r e d . I n t h e f i r st c a s e [1 ] 2 t h ed i f fe r e n ti a l o p e r a t o r B h a s b e e n t a k e n a s li n e a r o f th e f o r m B -a s +a2 6/6t w h e r ea j a n d a2 a r e m a t e r i a l c o n s t a n t s . T h e s e c o n d c a s e d ea l s w i t h v i s c o -e l a st i c f o u n d a t i o no f t h e P a s t e r n a k t y p e . I n a d d i t i o n t o t h e W i n k l e r h y p o t h e s i s , w h e r e a t e a c h p o i n t t h ep r e s s u r e p i s d i r ec t l y p r o p o r t i o n a l t o t h e d e f l e ct io n w , P a s t e r n a k a s s u m e s t h e e x i s te n c eo f s h e a r i n t e r a c ti o n s b e t w e e n t h e s p r i n g e l e m e n t s in t h e f o u n d a t i o n . T h i s is d o n e b yc o n n e c t i n g t h e e n d s o f th e s p r i n g s t o a p l a t e c o n s i s t i n g o f i n c o m p r e s s i b l e v e r t i c a le l e m e n ts , w h i c h d e f o r m s o n l y b y tr a n s v e r s e s h ea r a n d a s e c o n d f o u n d a t i o n c o n s t a n t ,t h e S h e a r m o d u l u s G i s c o n s i d er e d . S i n ce t h e m o v e m e n t o f t h e f o u n d a t i o n i s v e r ys l ow , t h e i n e r t i a t e r m s a r e n e g l e c t e d .

T h e d i f f e r e n t ia l e q u a t i o n o f a l a t e r a l l y l o a d e d t h i n e l a s ti c p l a t e r e s ti n g o n a v i s c o -e l a s t i c P a s t e r n a k - f o u n d a t i o n i s g i v e n b y [ 2 ] ,

V 2D V * w - ~ t w + q ~ + c w = qw h e r e / ~ i s a v i s c o s i t y c o e ff ic i e nt , r / v i s c o - c o m p r e s s i b i l i t y c o e f fi c ie n t . I n b o t h t h e c a s e s,a n o r m a l l o a d h a s b e e n a p p l i e d a n d i t m o v e s w i t h a c o n s t a n t v e l o c i t y a l o n g t h e l in ep a r a l l e l t o o n e e d g e o f t h e p l a te .

T h e p r o p e r t y o f t h e D i r a c d e l ta f u n c t i o n p l a y a n i m p o r t a n t r o l e i n s o l v in g th ep r o b l e m . I t h a s b e e n f o u n d v e r y h e l p f u l t o u s e L a p l a c e t r a n s f o r m .

EquationsT h e d i f f e r e n ti a l e q u a t i o n f o r t h e d e f l e c t i o n w o f a n e l a s ti c p l a t e o n a v i s c o e l a s ti c

i) Hatuganj M .N.K . High School, P.O. Hatuganj, Dt. 24-Parganas, W est Bengal.~) Num bers in brackets refer to R eferences, page 59.

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56 Satyanarayan M andal (Pageoph,f o u n d a t i o n i s g i v e n b y [ t ]

~ 2 ~ 2 ~ zB ~x2x + ~ y 2 j w x , y , t ) = q x , y , t ) - o h ~ w x , y , t ) (1)w h e r e B i s a d i f f e r e n t i a l o p e r a t o r .T h e d i f f e r e n t ia l e q u a t i o n f o r t h e d e f l e c t i o n w o f a n e la s t i c p l a t e o n v i s c o - e l a s t i c

f o u n d a t i o n o f t h e P a s t e r n a k t y p e is g i v e n b y [2 ]O ~ wV z w + t l + c w = q x , y , t ) (2 )D V ~ w # ~ t ~ -

w h e r e q x , y , t ) i s t h e i n t e n s i t y o f t h e l a t e r a l l o a d a n d D i s f l e x u r a l r ig i d i t y o f t h e p l a t e ,/~ v i s c o s i t y c o e f f ic i e n t, r / v i s c o - c o m p r e s s i b i l i t y c o e f f ic i e n t .

M e t h o d o f s o lu t io nT a k i n g t h e x y - p l a n e a s t h e m i d d l e p l a n e o f th e p l a t e [3 ] o f c r o s s -s e c t i o n a x b , l e t

u s s u p p o s e t h a t t h e l o a d i s m o v i n g w i t h a c o n s t a n t v e l o c i t y v a l o n g t h e l in e Y = Y oa n d l e t i t b e r e p r e s e n t e d b y

q x , y , t ) = P ~ x - v t ) 6 y - Y o ) (3 )w h e r e P i s a c o n s t a n t .

W e a s s u m e t h a t t h e d e f l e c ti o n o f t h e p la t e w is o f t h e f o r mw x , y , t ) = ~ ~ A , , n t ) s i n m ~ X s i n n ~z y - (4 )

m l n l a ba n d t h e l o a d q is o f th e f o r m

q x , y , t ) : ~ ~ , P m , ( t ) s i n m _ _7 ~ s i n _ _ nc y (5)m l n l a b

L e t t h e d i f f e r e n t i a l o p e r a t o r B b e t a k e n a s l i n e a r , i . e . ,B - a 1 + a 2 ~ l ~ t

w h e r e a ~ a n d a 2 a r e c o n s t a n t s .S u b s t i t u t i n g t h e s e r i e s ( 4 ) , ( 5 ) a n d ( 6 ) i n t h e e q u a t i o n ( 1 ) w e g e t

( m 2 n2 ~2 6 2a l + a 2 ~ / O t ) a2~ + b2,] ~z4 Am n( t) = P m n t ) - - 0 h ~ t ~ A m n t ) .

T a k i n g t h e L a p l a c e tr a n s f o r m d e f in e d b yoo

f p ) = f e - p t f t ) d tql]

(6 )

(7 )

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Vo l. 93, 1972/I) Plate on Viscoelastic Foun dation und er a M oving Load 57a n d a p p l y i n g it t o t h e e q u a t i o n ( 7) w e h a v e

& , ( p ) = P ~ . ( P )~ 4 ( a l + a 2 P ) ~ + ~ z + ~ h P z

A n y p a r t i c u l a r c o e f f ic i e n t P m . ( t ) o f the ser ies (5) is given b y [3]a bf f m n x. . 0 = ~ q ~ , y , t ) s in - - a

U s i n g t h e r e l a t i o n (3 ) w e g e t f r o m t h e r e l a t i o n (9 )

r e m e m b e r i n g

s in n n y d x d y .b

4 P n n y o . m n v tP , . . ( t ) = - s i n - s a n -a b b aoo

f 6 ( t - - c ) f ( t ) d t = f ( c ) .

A p p l y i n g th e L a p l a c e t r a n s f o r m t o t h e e q u a t i o n ( 10 ) w e g e t4 P m ~ v 1p , . . p ) -

Us ing t he r e l a t i on (11) i n t he r e l a t i on (8 ) we ge t. ( p ) - 4 P m T z v 1a 2 b p h ( p Z + K ) ( p Z _ t _ K 2 p + K a )

4 P m n v I p B- a ~ g ~ ~ A/ --~ p~ ~ c0 1 f]j sin n rc Yo+ p 2 + K z p + K

n ~ y osin - -b

n n y os in - -b

w h e r eK = - - , K I =

= a2 rc4 (rr t 2 n2 ] 2~:~ T i T \ ~ ~ ] ,K - K 1B 2 K K 1 - K 2 - K ~ - K K ~ '

K 1 - K - K 2D 1 = 2 K K 1 - K 2 - K 2 - K K ~

p2 _}_ K 2 P q_ K 1

n 2 2Q l i - 7 + ~K 2A =

K 2C = K 2 + K ~ + K K ~ - 2 K K ,

8 )

(9)

l O )

(11)

(12)

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58 Satyanarayan M andal Pageoph,T a k i n g t h e i n v e rs e L a p l a c e t r a n s f o r m o f t h e r e l a t io n 1 2 ) [4 ] w e g e t

iA m , Q ) 4a2boP ~_t;v [" B[ A c o s , j K t + ~ - - s i n x / K t, / K

+ C e - (g2 /2) t c o s K 3 t + E e - ( ~ 2 / 2 ) t s i n K 3 t ] s i nAw h e r en ~ y o

b

K a - v / 4 K * - K 2 , E _ 2 D a CK 22 2 K a

H e n c e , u s i n g t h e r e l a t i o n 1 3 ) i n t h e s e ri e s 4 ) w e g e too oo

( x , y , t ) m A c o s x / K t + : ~m = l n = l

C e - K E / 2 ) t c o s K 3 t -} - E e - K 2 / 2 ) t s i n K ]n = y o . r n ~ z x n ~ yx s in - - s m - - s i n -b a b

1 3 )

1 4 )

P r o c e e d i n g i n t h e s a m e m a n n e r a n d c o n s i d e r i n g t h e d i f fe r e n ti a l e q u a t i o n 2 ) f o r th ed e f l e c t i o n w ( x , y , t ) o f a n e l a st i c p l a t e o n v i s c o - el a s ti c f o u n d a t i o n o f t h e P a s t e r n a kt y p e a n d t h e l o a d - r e l a t i o n g i v e n b y 3 ) w e o b t a i n t h e d e fl e c ti o n a s

w x , y , t ) = - -

X

w h e r e

co co

4 P ~ z v ~ 1 ; 1 i a~ _ ~ )z b m z n 2= = t l - - , u ~ 2

D 2 -] 9 n ~ YO 9 m 7r XB I e - F t + C 1 c os Q t + ~ - s in Q t .j s m - - s mb a

m T c vQ - am n ~P1 = D ~z4 + C7 + b2 J

n ~ z y- - s i n - -b

F 2 = q - #~ z 2 + , F = F~2'1 F

C1 = i f2 + Q 2, D2 ~- F z + 02

1B 1 - F 2 _1_ Q 2

15)

A c k n o w l e d g e m e n tT h e a u t h o r w i s h e s t o e x p r e s s h is t h a n k s t o D r . S . K . S A R KA R , P h . D . , J a d a v p u r

U n i v e r s i ty , f o r h i s k i n d h e l p a n d g u i d a n c e i n t hi s w o r k .

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Vol. 93, 1972/D Plate on Viscoelastic Foundation under a Moving Load 59

REFERENCES[1] G. E. MASE, Transient Response of Linear Viscoelastic Plates Journal of Applied Mechanics,

Vol. 27, Trans. ASME, Vol. 82 [Series E] 1960), p. 589.[2] Transactions of the ASM E [Series E]. Journal of Applied Mechanics, March 1964. Published bythe American Society of Mechanical Engineers.[3] S. TIMOSHENKOand S. WOINOWSKY-KRIEGER,Theory of Plates and Shells 2nd Edition McGraw-Hill Book Company, Inc., 1959).[4] Cntn~cnmL OperationalMathematies Second Edition McGraw-Hill Book Company, Inc., 1958).Received 30th April 1971)