On “A grid framework analogy for laterally loaded plates”

Int. J. Mech. Sci. Pergamon Press Ltd. 1965. Vol. 7, PP. 153-154. Printed in Great Britain

LETTERS TO THE E D I T O R On "A grid framework analogy for laterally loaded plates" (Received 27 J u n e 1964)

THE w r i t e r was v e r y i n t e r e s t e d to r e a d L i g h t f o o t ' s p a p e r , 1 since w o r k i n g u n d e r his supervision t h e w r i t e r d e v e l o p e d a g e n e r a l p r o g r a m m e for grillages a n d utilized it, following L i g h t f o o t ' s suggestion, for t h e s o l u t i o n of p l a t e p r o b l e m s . T h e following e x t r a c t f r o m t h e w r i t e r ' s t h e s i s 2 m i g h t t h r o w a n e w l i g h t o n L i g h t f o o t ' s p a p e r a n d give a b e t t e r idea of t h e a c c u r a c y a c h i e v e d in solving p l a t e s b y a n e q u i v a l e n t grillage. " . . . W h e n slabs are m a d e of a m a t e r i a l of zero P o i s s o n ' s r a t i o w h i c h is n e a r l y t r u e for c o n c r e t e slabs, t h e y c a n b e a n a l y s e d as o p e n grids w i t h o u t v i o l a t i n g c o m p a t i b i l i t y e q u a tions, a n d c a n t h u s b e s o l v e d b y t h e g r i d - f r a m e w o r k s p r o g r a m m e . " F o r c o m p a r i s o n w i t h t h e o r e t i c a l v a l u e s a u n i f o r m l y l o a d e d s q u a r e slab h a s b e e n chosen. E a c h side of t h e s q u a r e was d i v i d e d i n t o a n u m b e r of e q u a l p a r t s a n d a n e q u i v a l e n t g r i d w o r k m e m b e r was t a k e n a t t h e c e n t r e of e a c h s t r i p as e x p l a i n e d in Ref. 12 [Ref. 2 of L i g h t f o o t ' s p a p e r ~] w h i c h is i n c l u d e d in A p p e n d i x 2. T h e r e s u l t s s u m m a r i s e d in T a b l e 4.1 show t h e p e r c e n t a g e o v e r e s t i m a t e of c e n t r a l deflections o v e r t h e t h e o r e t i c a l v a l u e s w h e n t h e sides of t h e s q u a r e s are d i v i d e d i n t o 3, 5 a n d 7 s t r i p s e a c h for cases of rigidly-fixed a n d s i m p l y - s u p p o r t e d b o u n d a r i e s . T h e a c c u r a c y is seen t o increase w i t h t h e n u m b e r of s t r i p s i n t o w h i c h e a c h side is d i v i d e d , j u s t as in t h e finite-difference a p p r o a c h . TABLE 4.1

3 strips 5 strips 7 strips

Rigidly fixed

Simply supported

5.59o 2.9 °,o 1.7 ~)

10"3 ~o 3.9?/o 3-4%

'%Vhen P o i s s o n ' s r a t i o does n o t e q u a l zero, a p p r o x i m a t e s o l u t i o n s for p l a t e s c a n still b e o b t a i n e d . T h e flexural stiffness G J of a n e q u i v a l e n t m e m b e r of t h e g r i d w o r k is t a k e n as e q u a l to t h e flexural stiffness E l l ( 1 - v 2) w h e r e v is t h e P o i s s o n ' s r a t i o of t h e m a t e r i a l of t h e p l a t e . W i t h t h e s e values, t h e c o m p u t e r s o l u t i o n gives a n a p p r o x i m a t e deflection profile for t h e p l a t e considered, t h e a c c u r a c y d e p e n d i n g o n t h e t y p e s of b o u n d a r y c o n d i t i o n s a t t h e edges: free b o u n d a r i e s give t h e w o r s t a g r e e m e n t a n d fixed b o u n d a r i e s t h e b e s t agreem e n t , w i t h t h e t h e o r e t i c a l solution. " T h e deflection profile m a y t h e n b e t r e a t e d as in t h e finite difference a p p r o a c h , m o m e n t s a n d s h e a r forces b e i n g o b t a i n e d b y s u b s t i t u t i o n in t h e w e l l - k n o w n finite difference approximation formulae. Alternatively, moments may be obtained from the computer v a l u e s in t h e 'x' a n d 'y' d i r e c t i o n s f r o m t h e r e l a t i o n s : M~=

M=+vM~

M~= Mu+vM~

w h e r e IV/~ a n d 3Iy are t h e c o m p u t e r m o m e n t s in t h e x a n d y directions, .TV/~a n d M~ t h e a c t u a l m o m e n t s a c t i n g in t h e slab a n d v t h e P o i s s o n ' s ratio. " T h e a d v a n t a g e of t h i s m e t h o d o v e r t h e finite difference a p p r o a c h is t h e ease of d e a l i n g with the boundary conditions .... " 153

154

L e t t e r s to t h e E d i t o r

I t is q u i t e easy to d e m o n s t r a t e t h a t a n e x a c t analog}' b e t u c e n p l a t e s a n d a ~grillage c o m p o s e d of t w o parallel sets of i n t e r a c t i n g m e m b e r s is n o t possible by c o n s i d e r i n g a I)ur(~ m o m e n t a p p l i e d a t opposite edges of a r e c t a n g u l a r slab a n d , ' q u i v M e n t grillagt ., see Figs. l(a) a n d (b).

(a)

(b)

Fit;. 1. (a) M o m e n t applied to slab. (b) M o m e n t applied to ~rillagc. A n t i - e l a s t i c c u r v a t u r e is d e v e l o p e d in a slab, Fig. l(a), for a n y t'(~isson's r a t i o ml(.(t,,ad to zero, w h e r e a s n o m e c h a n i s m exists for p r o d u c i n g t h i s effect in ~ grillage, Fig. l(b).

Departme~t of Cicil E~gi~teeri~g Leeds University REFERENCES 1. E. LIGHTFOOT, l~t. J. Mech. Sci. 6, 201 (1964). 2. F. Sh~rKO, A~(dysis oj" Gr/llages a~d Related Structttres. M.Sc. thosis, Leeds I r n i v t , r s i t y (19(~0).

Reply to comment on "A grid framework analogy for laterally loaded plates"

(Receipted 21 July 1964) I AM glad o f this o p p o r t u n i t y to r e p l y to S a w k o ' s l e t t e r a n d to deal w i t h one (,' txxo p o i n t s o m i t t e d f r o m t h e originM p a p e r .

Accuracy of the grid frameworks analogy S a w k o ' s q u o t a t i o n frorn his M.Sc. thesis shows t h a t t h e grid f r a m e w o r k s a n a l o g y (G.F.A.) gives t h e c e n t r a l deflexion of a u n i f o r m l y l o a d e d s q u a r e slab of u n i f o r m section a n d m a t e r i a l w i t h i n c r e a s i n g a c c u r a c y , as m o r e a n d m o r e e q u a l s t r i p s are t a k e n in d i r e c t i o n parallel to t h e sides. B o t h t h e s i m p l y - s u p p o r t e d a n d t h e e n c a s t r 6 cases are considered a n d t h e results suggest a s p e e d y c o n v e r g e n c e to T i m o s h e n k o ' s a n a l y t i c a l vMues ( t h o u g h an e x t r a p o l a t i o n would suggest a lower v a l u e t h a n t h e 3.4 p e r c e n t q u o t e d in T a b l e 4.1). Values f r o m t h e finite differences m e t h o d w o u l d b e i n t e r e s t i n g for c o m p a r i s o n ; only ,me r e s u l t is a v a i l a b l e to t h e writer, f r o m S a l v a d o r i a n d B a r o n ' s Numerical 3Iethod,s i,, Engineering, p. 202, w h i c h gives t h e c e n t r a l deflexion in t h e c n c a s t r 6 case for a n S × S m e s h as 13 p e r c e n t a b o v e t h e correct, vMue (to b e c o m p a r e d w i t h 1.7 p e r cent b y Sawko for t h e 7 x 7 case, w h i c h is a l m o s t e q u i v a l e n t ) . I t would a p p e a r t h a t P o i s s o n ' s r a t i o m e r e l y affects t h e p l a t e r i g i d i t y v a l u e in t h e s e two e x a m p l e s , so t h e stone p e r c e n t a g e s w o u l d b e o b t a i n e d w h a t e v e r t h e v value. I t would be i n t e r e s t i n g to k n o w w h y t h e r e s u l t s for t h e e n c a s t r 6 p l a t e are m o r e a c c u r a t ~ t h a n t h o s e for t h e s i m p l y - s u p p o r t e d p l a t e , w h e n t h e s a m e grid f r m n e w o r k is used.

Rectangular grid and plate with end mome~ts L e t us consider a r e c t a n g u l a r slab of u n i f o r m section a n d m a t e r i a l w h i c h is simplys u p p o r t e d a t its ends, like t h e grillage s h o w n b y Sawko in Fig. l(b). L e t m o m e n t s : l / p e r