186 unknowns i s a c h i e v e d w i t h l e s s than 3 percent l o s s i n accuracy o f the corresponding d e f l e c t i o n s .
F o r the n i n e - p l y laminates,
9 terms o f the
GLM a r e s u f f i c i e n t t o r e t a i n an a c c u r a t e p r e d i c t i o n o f d e f l e c t i o n s w i t h less
than
1 percent l o s s i n accuracy,
and the
number o f
unknowns i s
reduced by more than 60 p e r c e n t i n comparison w i t h a complete layerwise model. To f u r t h e r v e r i f y the implementation o f the g l o b a l - l o c a l model, the s t r e s s e s o f the f o u r - p l y l a m i n a t e a r e o b t a i n e d by u s i n g 13 terms o f the global functions.
I t i s found t h a t t h e y a r e i d e n t i c a l t o those u s i n g 4
l a y e r w i s e c u b i c shape f u n c t i o n s , see T a b l e 9 . 5 . T h i s i s expected because i d e n t i c a l number o f unknowns i s u s e d i n b o t h c a s e s which i m p l i e s that no c o n s t r a i n t i s e f f e c t i v e l y a p p l i e d t o the l a y e r w i s e model. F o r l a m i n a t e d p l a t e s w i t h s=100,
we a n t i c i p a t e t h a t the f i r s t two
l i n e a r terms o f t h e g l o b a l t h r o u g h - t h i c k n e s s f u n c t i o n s a r e s u f f i c i e n t to g i v e a c c u r a t e d e f l e c t i o n s as w e l l a s i n - p l a n e s t r e s s e s . As demonstrated i n Table 9.6,
the present r e s u l t s ,
i n c l u d i n g d e f l e c t i o n s and s t r e s s e s ,
a r e w i t h i n 2 p e r c e n t d i f f e r e n c e from the e x a c t s o l u t i o n s . (*)
Centre d e f l e c t i o n w GLM
%e r r o r
GLM
%e r r o r
%r e d u c t i o n
5
11,.432
f o u r - p l y laminate .395
2.1
62
9
11,.475
.413
1.7
31
13
11.,764
.491
LM
11..764
,491
exact
11..767
,491
5
11.547
n i n e - p l y laminate .924
82
9
12.188
,047
68
12
12.216
,054
57
LM
12.288
,079
exact
12.288
,079
T a b l e 9.4.Convergence o f c e n t r e d e f l e c t i o n s o f the g l o b a l - l o c a l model. (*) p = number o f terms t a k e n from e x p r e s s i o n ( 9 . 3 ) . GLM=globall o c a l model , LM=layerwise model • % error = percentage difference between the GLM and the corresponding LM(4x4 mesh with one cubic element per material layer.) % reduction = percentage reduction in the number of unknowns of the GLM with respect to the corresponding LM.