9 JÜHI '»9^ CoA R E P O R T AERO No. 209
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THE COLLEGE OF AERONAUTICS
CRANFIELD
OPTIMUM DESIGN OF FIBRE REINFORCED
CORRUGATED COMPRESSION PANELS
by
CoA R e p o r t A e r o No. 209 J a n u a r y , 1969.
THE COLLEGE OF AERONAUTICS CRANFIELD
Optimum Design of F i b r e Reinforced C o r r u g a t e d C o m p r e s s i o n P a n e l s
by
D. M. R i c h a r d s
SUMMARY
Monofilament composite m a t e r i a l s offer the possibility of significant s t r u c t u r e weight s a v i n g s , e s p e c i a l l y for applications where stiffness is important.
The optimum design of wide c o r r u g a t e d c o m p r e s s i o n panels i s c o n s i d e r e d , fabricated from l a m i n a t e d monofilament sheet.
Local and flexural buckling modes a r e accounted for, and the a n a l y s i s is extended to include an over riding m a x i m u m s t r e s s limitation.
E x p r e s s i o n s a r e obtained for optimum c o r r u g a t i o n and lamination g e o m e t r y , and the r e s u l t s a r e s u m m a r i s e d in t e r m s of equivalent s u r f a c e weight.
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CONTENTS Page S u m m a r y 1. Introduction 1 2. Buckling s t r e s s 1 3. Optimisation 2 4. Maximum s t r e s s limitation 4 5. Conclusions 4 6. R e f e r e n c e s 5 F i g u r e s
1. Introduction ' The practical application of monofilament composites to major aircraft structural
components is now being actively considered. U-J 1?] HI L^J [S]
These new materials offer improved tensile strength-weight ratios compared with currently used isotropic metals, but the most significant advantages are likely to be gained when stiffness is an important factor, l^.l
This paper considers the design of wide compression panels (figure 1) which are simply supported at regular intervals by transverse ribs. The applied compressive loading is uniform.
Attention is concentrated on panels formed from a single trapezoidally corrugated sheet. This configuration has an inherently high buckling efficiency, LU while presenting few serious jointing problems.
Recent work L°Jon the buckling of simply supported fibre reinforced flat plates in compression, uses netting analysis to show that maximum resistance to buckling is given by the orientation of fibres - 45° to the direction of loading. However, this arrangement lacks axial strength,
Consequently, the plate lay-up shown in figure 2 has been adopted. A central core of axial fibres is braced by two outer layers of - 45° material, located at the plate surfaces where the effect on plate bending stiffness is greatest. •'••
The axial core effectively c a r r i e s the applied load, and provides resistance to buckling in the flexural mode. The bracing layers prevent buckling in the local mode and the neglect of the axial fibres in this mode allows the interaction between
adjacent corrugation flats to be accounted for. 2. Buckling s t r e s s
Local buckling of the corrugation in terms of core s t r e s s , ignoring the contribution of axial fibres, is given by:
, . K „ ' KoV,E(j) ,j) where
2
_ A ^ •?•! K i ^ ö t - i l d + Q o irYTr-ilTT- o n r \ r N r t - r « + Q i - l ^ - ' ^ o ~ — for - 4 5 " fibre plate, simply supported
6
K a [9J
TT- ) is a function of 7- only, for which i s o t r o p i c r e s u l t s a r e used
o
E = e l a s t i c modulus of fibre -.., :••< -, •• . •;' V-= fibre volume fraction in b r a c i n g l a y e r s
t = total plate thickness
t j = core thickness
a = corrugation flat width b = corrugation flank width.
F l e x u r a l instability s t r e s s , a s s u m i n g simply supported conditions at supptirt pitch L, and accounting only for the contribution of axial f i b r e s , is given in t e r m s of c o r e s t r e s s by: - , . . .
f , - ^ . s i n ^ e ( 3 t ^ 1 ) . ^ ^ ^ ( b . ^ 2 - : ^ ^ ; _ : : . . ( 2 ) .
( ^ . 1 )
w h e r e ' ••". ,.•• •-•-..••' ••>"••
6 = c o r r u g a t i o n angle . '
V(j= axial c o r e fibre volume fraction.
The r e l a t i o n between applied load p e r unit width u , and c o r e s t r e s s intensity i s :
I
. u, 1 (^ + cos 0 ) i -^ T-- T -^ (3) . .. (.2. + 1) . b 3. OptimisationThe r e q u i r e m e n t that both modes of instability o c c u r at the s a m e load l e v e l , so that fi " fo " ^' allows the elimination of the two v a r i a b l e s b and t, giving finally for the optimum c o r e s t r e s s ÏQ,
fo = I ^ • 0 ^ .VcVd ( - s i n ' e [ 3 ^ + l ] [ ^ + c o s e 3 ^ ^ . r 2^2
(4)
Since the s t r u c t u r e is a c o m p o s i t e , a t r u e reflection of s t r u c t u r a l efficiency is weight p e r unit projected surface a r e a , including both fibres and m a t r i x . This is given by: Wo = ^ -!(V^-V^) ( l - p ) + ^ | V c + ( l - V c ) p | i (5)
~r- {(Vd-Vc)(l-P)+^ rVc + d-VcJpjl
w h e r e P j = fibre m a t e r i a l density p .= m a t r i x density m P =_^m = r e l a t i v e density of rfiatrix. , . ''fThe effect of the c o r r u g a t i o n g e o m e t r i c p a r a m e t e r s 6, ^ / b on the surface weight is e x p r e s s e d completely by v a r i a t i o n of the ( b r a c k e t e d ) t e r m in equation (4). F i g u r e 3 shows how s t r u c t u r e weight depends on t h e s e t e r m s , and indicates optimum values to b e :
( ^ ) = 0, 85
t> opt , . ^ , e . = 58.°3
• 3
-Substitution of these v a l u e s into (4), (3) and (2) gives optimum s t r e s s and s i z e s ; fo = 0.909 to = 1.011 ( 1 - t ) ^ 4
{v.v.ii^'}^ [ ^ f
I v ^ V n t (1-t ) J L bo = 1. 44 4 ;Vdt (1-t ) V c ( l - t ^ ) 1 ^ tt)L E V d ^ - t ^ r wL 3 Ï E (6) (7) (8)Substitution of (6) into (5) shows that the effect of the c o r e t h i c k n e s s p a r a m e t e r t on s u r f a c e weight is r e p r e s e n t e d by the e x p r e s s i o n W K 1 + r t w h e r e
rt(i-t^i^
Vd - Vc Vc + _ e _ 1-p (9) (10)Differentiating (9) gives the condition for optimum c o r e t h i c k n e s s : 3 . 3^+ 1
t o " + ? r t o - Ï = 0
for which a positive r e a l root is given by:
"•O 2 1 + (l+r'^) 3 \ 2 1 - ( l + r ^ ) 2
-f]
(11)
(12) F o r round f i b r e s , volume fraction is r e s t r i c t e d to the r a n g e :
0 <V„ H < — — ( - 0.91) ' c , d
2/3"
F i g u r e 4 shows the optimum c o r e t h i c k n e s s to be c o m p a r a t i v e l y insensitive to differences in volume fraction between composite l a y e r s .
Note that when Vc = V^, r = 0 from (10) and (12) gives t o = ^ i ^ ' ( - 0-63)
Substitution of (6) into (5) gives optimum surface weight in the form
Wo = W w h e r e W = 1. 10 (1-P) (VcVd) " (to - to
{
T- \ ( V d - V c ) t o + V^^T^}
(13) ( 1 4 '- 4 '
Using (12), W is plotted a s figure 5, i l l u s t r a t i n g the advantage of a high volurhe fraction in both composite e l e m e n t s ,
4, Maximum s t r e s s limitation
R e s t r i c t i n g attention to c o m p o s i t e s of uniform volume fraction, so that Vc = Vd = V g i v e s , from (6)
^ ^ fm (15) f„ = 1. 1971 ^ ^ "
' o — • L
n
w h e r e f^^^ i s the m a x i m u m allowable c o m p r e s s i v e s t r e s s in the c o r e m a t e r i a l , which will itself be a function of volume fraction.
(15) m a y be r e - w r i t t e n in t e r m s of the loading index
J i ^ n . 6 9 8 ' m = ,u_. , g.
When — ^ ( — )o, f ~ fjn ^'^'^ the c o r e t h i c k n e s s r a t i o t m u s t be chosen to account for L L
t h i s . Introducing t h i s condition into (6) gives the c o r e t h i c k n e s s subject to s t r e n g t h linaitation
[(TrW(f)o}
^-p^)/'T^o]
which i s plotted in figure 6.
E x p r e s s e d in the s a m e form as equation (13), the surface weight i s given b y :
W*= 0.835 '" "/^-"^^^ J l ^ L l T i l Ü Z o i I (18)
V 2 C o r r u g a t i o n d i m e n s i o n s a r e given byto = 0.768 P JZ^ ^^Vo \ I I - ! ^ I (19)
b = 1 . 6 5 7 ] 1 1 ' 1 - ^ ' I (2°)
l(£)/(f:)o
Equations (18), (19) and (20) a r e shown plotted in figures 7 and 8, 5. Conclusions
Explicit e x p r e s s i o n s have been derived for optimum g e o m e t r y , dimensions and weight of f i b r e - r e i n f o r c e d c o r r u g a t e d c o m p r e s s i o n panels, failing in local and flexural buckling m o d e s . The a n a l y s i s has been extended to include the effect of an o v e r - r i d i n g limitation on m a x i m u m s t r e s s , when the important p a r a m e t e r s a r e found to be fibre volume fraction and loading index,
R e f e r e n c e s
1. Noyes, J. V. 'Design of an a i r c r a f t utilising f i b r e g l a s s reinforced p r i m a r y s t r u c t u r e ' . J o u r n a l of A i r c r a f t , Vol. 3, No. 5, S e p t e m b e r - O c t o b e r , 1966.
Atkin, H. P. 'Boron filament reinforced p l a s t i c c o m p o s i t e s for a i r c r a f t s t r u c t u r e s ' . M e t a l s Engineering Q u a r t e r l y A. S. M.
F e b r u a r y , 1967.
R o g e r s , C. W. 'Advanced composite m a t e r i a l applications to a i r c r a f t s t r u c t u r e s ' . J o u r n a l of A i r c r a f t , Vol. 5, No. 3, May-June, 1968.
Swan, B. E, 'The application of reinforced p l a s t i c to a i r c r a f t s t r u c t u r e s ' , College of A e r o n a u t i c s T h e s i s . September, 1968. S a y e r s , K. H. ' T e s t and a n a l y s i s of g r a p h i t e - f i b e r , e p o x y - r e s i n composite and a i r f r a m e s t r u c t u r a l e l e m e n t s ' . A. I. A. A. P a p e r No. 68-Hanky, D. P . 1036, October, 1968. Dow, N. F . , and Rosen, B. W.
'Evaluation of filament reinforced c o m p o s i t e s for a e r o s p a c e a p p l i c a t i o n s ' . NASA CR-207, 1965.
7, E m e r o , D. H. , and
Spunt, L.
'Optimisation of m u l t i - r i b and m u l t i - w e b wing box s t r u c t u r e s under s h e a r and moment l o a d s ' . A. I. A. A. 6th S t r u c t u r e s and M a t e r i a l s Conference, P a l m Springs, California, April, 1965.
8. Rothwell, A. 'Optimum fibre o r i e n t a t i o n s for the buckling of thin p l a t e s of composite m a t e r i a l s ' . (to be published).
9. Engineering Sciences Data (Aeronautical Series) S t r u c t u r e s OL 01.19.
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