1973 Back Pressure Distribution

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Int. J. mech. ScL Pergamon Press. 1973. VoL 15, pp. 993-1010. Printed n G r e a t BrRain

P R E S S U R E D I S T R I B U T I O N A N D D E F O R M A T I O N S

O F M A C H I N E D C O M P O N E N T S I N C O N T A C T

N . BACK, M . B U R D E K I N a n d A . C O W L E Y

Mechanical Engineering Department , Centro Tecnologico, Universid ade Federal de Sant a

Catarina, and Mechanical Engineering Departmen t, Univ ersit y of Manchester Institu te of

Science and Technology, Manchester, England

(Received 1 February 1973, and in revised form 9 July 1973)

Summary--The paper presents a general method for calculating the pressure distribution

at contacting machined surfaces and the resulting deformations of components subjected

to ext ernal loads. It is well known t hat when machined surfaces are compressed one

against the other t hen due to the deformation of the asperities the approach of the surfaces is

related to t he interface pressure by a non-linear function. Another effect that must be taken

into accoun t is that the actual area of contact is not known previously and is, of course,

different from the apparent area. For the met hod presented, the contact or connexion

between the components is simulated with finite elements, springs or plates defined as a

funct ion of the surface roughness and surface deformations. The syst em so establis hed is

solved in an iterative way using the finite element method and this enables the pressure

distribu tion at the cont act and the resulting deformations of the whole component to be

determined. Finally, examples of some problems solved by different methods are given

and the results compared with the experimental data.

A ~

a.

C

d(t:,D

E , E . , E ~F , F .

F rk

Lm

pa

P.,P~t

x, y

Iz

O .

N O T A T I O N

area of influence of a pair of nodes

cross-section of the springs

constant relating the surface deformation and interface pressure

displacements

modulus of elasticity

forces

friction force

modulus o f the elastic foundation

length

constant of the power law relationship between contact deformation and

interface pressure

apparent interface pressure

pressure at the surface in contact

thickness of the finite plates

displacements

co-ordinates

coefficient of fricti on

contact deformation

rotation

I . I N T R O D U C T I O N

M ACHI NE t o o l s a r e n o t g e n e r a l l y m a n u f a c t u r e d a s o n e c o n t i n u o u s c a s t i n g o r

f a b r i c a t io n a n d t h e r e a s o n s f o r t h i s a r e t h e d i f f ic u l ty i n m a n u f a c t u r i n g a n d

t r a n s p o r t a ti o n , a n d a l s o f o r f u n c t io n a l p u r p o s e s s u c h a s g u i d e w a y s . S u c h

c o n s t r u c t i o n s , t h e r e f o r e , n e c e s s i t a te c o u n e x i o n s b e t w e e n t h e b a s ic e l e m e n t s o f

6e 993



994 N. BACK, M. BURDEKIN and A. COWLEY

the machine and these can be classified as bolted or fixed joints and sliding

joints in the case of guideways. In both fixed and sliding connexions, forces

are transmitted across the interfaces and the overall static and dynamic

characteristics of the machine tool can be considerably influenced by compliance

at the individual connexions.Much of the work which has been carried out on establishing the

characteristics of machine tool joints has concentrated on the normal com-

pliance of contacting surfaces and its dependence on surface topography,

hardness, interface pressure and materials. Rele vant work is presented in

refs. (1)-(7) and (13).

Connolly7 and Levina M have used simple structural models in an att emp t

to show the influence of the interface compliance on the overall stiffness of a

structure. However, in their analyses the components in contact are assumed

to be rigid which is invalid for most practical cases encountered. When thecomponents are not rigid, the resulting model becomes more complicated since

the pressure distribution between the surfaces will be quite different from the

assumed and for certain areas of the surface contact can be completely lost.

Another aspect that will increase the complexity of the solution, as it is shown

in the references mentioned, is that the surface compliance is nonlinear and

given by the following equation

= c(100P) m (1)

In equation (1), 2 is the approach of the surfaces in tma, c and m surface

characteristics dependent upon the pair of materials and surface finish. ]t has

been shown ~ th at for cast iron c = 0-3-2.0 and m = 0.3-0.7 for all conventional

mach ined surfaces. P is the interface pressure in kgf /mme. A more detailed

analysis of the relationship (1) and t he parameters c and m can be found in

refs. (1) and (2).

In order to include the nonlinear compliance characteristics of the surfaces,

elasticity of the components and the fact th at the surfaces are stiff only in

compression, the simplest way is an ite rative techn ique using the finite element

method. In this case the co ntact is simulated by some connecting elements

th at represent the surface behaviour.The finite element method will not be discussed here since it is adequately

covered in refs. (8-10) and (12). A discussion regarding the precision, prepara-

tion of the dat a and the computer programs of the finite element method used

in this work are given in refs. (]), (8) and (9).

Three meth ods used for the purpose of calculating the pressure distribution

and deformation in joints, are described herein.

2 . H Y D R O S T A T I C M E T H O D

T h i s t e r m i n o l o g y h a s b e e n c h o s e n s i n c e t h e r e p r e s e n t a t i o n o f t h e s m ' f k c e i n t e r f a c e i s

a n a l o g o u s t o a h y d r o s t a t i c b e a r i n g s y s t e m .

C o n s i d e r t h e r e s u l t i n g p r e s s u r e d i s t r i b u t i o n a t t h e a p p a r e n t a r e a of c o n t a c t a n d thed e f o r m a t i o n s o f t h e c o m p o n e n t s s h o w n i n F i g . l ( a ) d u e t o t h e a p p l i e d f o r c e F . F o r s i m p l i c i t y

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

c a n b e c o n s i d e r e d b u t t h i s i s n o t a l i m i t a t i o n .



Pressure distributi on and deformations of machined components in contact 995

The first step in the solution is the division of both components into finite elements.

As is shown in Fig. l(b), both components can be divided independentl y but along the

apparent length of contact it is recommended that the nodes coincide forming pairs.

The distan ce between the consecutive pairs of nodes along the contact can be different,bu t as is shown later the best way is to choose equal distances.

o)

b )

.I I

FI A. . . . . } . . . . . . . . . . " "

lY 'V x, u d)

•~ ~ / P l ( i + n . J + n)

F ~A

I I I l l : I J . L _ L

7 - - r - - ' , - - - l - , - - I - 7 - 7 - I ] ; t- - - - - i - - - -- - i - - - - - - i- - ~ - i - - ~ - - i - - I - i + I -

L - - ' - - _ I _ _ + ~ _ 4 _ _ , , -. - ' - , - 4 - - L

e )

..... : g ~ i - i - t

I I 1 I I , , _ . , .. L J _ ~

. - - , - - - , , - - - , - , - , - - , , - , .

I _ _ _ _ ~ _ _ ~ , . . . t J , - -

k / 5 th i t e r o t i on

B L \ / / ~ ,h

"~ ,

g )

c )

FIe. 1. (a) Jo int model; (b) finite element division of the join t

comp onen ts; (c) displacements at th e surface in contact of the join t

comp onen ts; (d) deflexions at the conta ctin g surface used to generatea new pressure distribution; {e) calculated pressure distribution at the

surface in co ntac t; (f) concent rated loads at the nodes; (g) sequence of

the pressure distri but ion in the successive iterations.

If the system is symmetric then the displacements u alongA B

are zero and thecomponents are analysed independe ntly with the points A and B completely fixed as

shown in Fig. l(b).

For the first iterat ion a pressure distri but ion mus t be assumed, for example, a unifo rm

pressure, and from this pressure distrib ution the equ ivalent concentrated node forcesF(i+,~ are calculated and applied as it is shown in Fig. l(b). At this poi nt two ind epen dent

problems are solved using the finite element method. Firstl y, the deflexions v of the nodes

along the surfaces in conta ct are calculated as shown in Fig. l(c). Duo to sym met ry only

the deformations at one surface, at the right-hand side of the component, will be

represented.

The nex t step is to calculate the sums of the displacements of each pair of nodes or

d(~+n,~+~ = v(~+n)+ vu+~. (2)

Whe n th e mi ni mu m sum is ob tained , say for example d(~s~, th en the differences given

in the following equa tion are calcul ated

J D ( i + , , t + n ) = d [ i + n , j + n ) - d ( i . ~ ) (3)

and when plotted result in the curve in Fig. l(d).



9 9 6 N . B A C K , M . B U R D E K I N a n d A . C O W LE Y

T o g e n e r a t e t h e p r e s s u r e d i s t r i b u t i o n f o r t h e s e c o n d i t e r a t i o n e q u a t i o n (1 ) a n d F i g . l ( d )

m u s t b e a n a l y s e d t o g e t h e r i n t h e f o l lo w i n g w a y . F r o m F i g . l ( d ) i t c a n b e s ee n , w i t h o u t

d i f fi c u lt y , w h e t h e r o r n o t t h e c o m p o n e n t s a r e f l e x ib l e o r r i g id a n d m o r e o r l es s if t h e

c o n t a c t w i l l r e m a i n o v e r a l l t h e n o m i n a l a r e a o f c o n t a c t . H e r e i t i s s u p p o s e d t h a t t h e

c o n t a c t w i l l r e m a i n o v e r th e n o m i n a l a r ea . T h e n a l in e a r e x t ra p o l a t i o n i s m a d e a s s h o w n

i n F i g . l ( d ) . F o r p r a c t ic a l p u r p o s e s t h i s is a l w a y s a g o o d a p p r o x i m a t i o n b e c a u s e t h ep r e s s u r e a t t h i s p a r t o f t h e s u r f a c e w i l l b e s m a l l a n d c o n s e q u e n t l y t h e s u r f a c e w i ll r e m a i n

m o r e o r le ss s t r a i g h t . A t t h e p o i n t w h e r e t h e m i n i m u m s u m o f d i s p l a c em e n t s o c c u rs i t is

a s s u m e d t h a t t h e s u r f a c e c o m p r e s s i o n i s A (i,j~, a n a r b i t r a r y v a l u e t h a t i n g e n e r a l m u s t b e

m a n y t i m e s l a r g e r t h a n t h e f in a l v a l u e e x p e c t e d . F r o m t h i s v a l u e , a s t r a i g h t l i n e i s

d r a w n s u c h a s i s s h o w n i n F i g . l ( d ) a n d a t t h e i n t e r s e c t i o n w i t h t h e c u r v e t h e p r e s s u r e

f a ll s t o z e r o. T h e d i s t a n c e s b e t w e e n t h e s e l i n e s a r e t h e c o n t a c t d e f o r m a t i o n s a n d c a n b e

c a l c u la t e d b y t h e f o l l o w i n g e q u a t i o n

A (i+ , , j+, ) = A<i,j) - D ~ i + n , j + , ~ l . (4 )

A c c o r d i n g to e q u a t i o n (1) t h e p r e s s u r e a t t h e p a i r s o f n o d e s c a n b e c a l c u l a t e d a s

f o l l o w s

• 1 ( . ~ ( i + n , j + n , ] l h nP ( i+ , , J + , ) = ]-65 \ ~ ! " (5)

P l o t t i n g t h e v a l u e s f r o m e q u a t i o n (5 ) t h i s g i v e s t h e d i a g r a m s h o w n i n F i g . l (e ) . t V h e n

t h i s p r e s s u r e d i s t r i b u t i o n i s i n t e g r a t e d o v e r t h e a r e a i n c o n t a c t t h e f o r c e s h o u l d b e l a r g e r

t h a n t h e e x t e r n a l a p p l i e d f or ce , o t h e r w i s e t h e i n it i a l a s s u m p t i o n w a s w r o n g . T o e v a l u a t e

t h e f o r c e s t h a t s h o u l d b e a p p l i e d a t t h e n o d e s f o r t h e s e c o n d i t e r a t i o n , t h e i n t e g r a l o f t h e

p r e s su r e d i s t r i b u t i o n m u s t b e c o r r e c te d w i t h o u t a l t e r i n g t h e a c t u a l d i s t r i b u t i o n s o t h a t

t h e r e s u l t is i n e q u i l i b r i u m w i t h t h e e x t e r n a l f o r ce . T h i s c a n b e d o n e w i t h t h e f o l l o w i n g

e q u a t i o n

" - q P ( i m A '= 7 - - . + . , j + . ) p . + . , , + . ) = F , (6)n~OP ( i , J )

w h e r e q i s t h e n u m b e r o f n o d e p a i r s i n c o n t a c t a n d A ( ~+ ., j +n ) i s t h e a r e a o f i n f lu e n c e o f

e a c h p a i r .

T h e u n k n o w n i n e q u a t i o n (6 ) i s p . , j ~ a n d w h e n c a l c u l a t e d i t w i l l d e f in e t h e p r e s s u r e

d i s t r i b u t i o n o v e r t h e s u r f a ce b y t h e f o l l o w i n g e q u a t i o n

P~'+"'~+"~ (7)P " + ~ '~ + ' ~ = P " ' J ) P ~ m

t h a t i s a l so p l o t t e d i n F i g . l ( e ) .

T h e s e q u e n c e o f p r e p a r a t i o n o f t h e d a t a f o r a n e w i t e r a t io n c a n b e s i m p l i f i e d w h e n

e q u a t i o n s ( 2 )- (5 ) a r e s u b s t i t u t e d i n e q u a t i o n s (6 ) a n d (7 ) a n d w h e n t h e l e n g t h o f t h e f i n i te

e l e m e n t s a l o n g t h e s u r fa c e i n c o n t a c t a r e t h e s a m e .

F r o m t h e p r e s s u re d i s t r i b u t i o n o f F i g . l ( e ), t h e e q u i v a l e n t c o n c e n t r a t e d l o a d s inF i g . l ( f ) a r e d e t e r m i n e d a n d a g a i n t h e f i n it e e l e m e n t m e t h o d i s a p p l i e d a n d t h e d e f o r m a -

t i o n s o f t h e s e c o n d i t e r a t i o n a r e o b t a i n e d .

T o p r o c e e d w i t h t h e i t e r a t i o n s o n e w a y i s t o r e d u c e t h e v a l u e 2 (i,~); i t c a n b e f a s t e r

a t t h e b e g i n n i n g a n d s l o w e r w h e n t h e c o n v e r g e n c e is r e a c h ed . S i m i la r ly , t h e r e d u c t io n i n

~ , , j ) c a n b e c o n s i d e r e d a s r e d u c i n g t h e l e n g t h o f c o n t a c t . T h e l a s t f o r m i s r e c o m m e n d e d .

A t t h e p a i r o f n o d e s , w h e r e t h e s u m o f t h e d i s p l a c e m e n t s i s a m i n i m u m , t h e n f r o m

o n e i t e r a t i o n t o a n o t h e r , p ~,.,~ d e c r e a s e s a n d P c~ ,J) i n c r e a s e s , a n d , t h e r e f o r e , f o r c o m p l e t e

c o n v e r g e n c e th e s e t w o v a l u e s m u s t b e e q u a l . T h e f o l l o w i n g i d e n t i t y i s t h e n o b t a i n e d .

P(~m = ] -0"6 - - " (8)

T h e s e q u e n c e o f t h e p r e s s u r e d i s t r i b u t i o n o b t a i n e d f r o m t h e s u c c es s iv e i te r a t io n s c a nb e r e p r e s e n t e d a s s h o w n i n F i g . l ( g ) .

F o r t h e l a s t i t e r a t io n t h e d e f o r m a t i o n s o f e a c h c o m p o n e n t a r e d e t e r m i n e d a n d t h e

s u r f a c e c o m p r e s s i o n i s c a l c u l a t e d . F r o m t h e s e r e s u l t s i t i s p o s s i b l e t o a s s e m b l e t h e w h o l e

d e f o r m e d s y s t e m w i t h t h e r e s p e c t i v e p r e s s u re d i s t r ib u t i o n a t t h e c o n t a c t s u r f a ce .



Pressure distribu tion and deformations of machined components in contact 997

If, for example, the relative displacement between points A and B on Fig. l(a) is

required it is obtained from

Deformation = d(i,~}÷Av,~). (9)

When non-syrametric systems are to be solved the so-called hydrostatic method

described above is not recom mended and to overcome these difficulties the followingmethod s will be used.

3. PLATE METH OD

To describe this metho d let us consider the deformations and the pressure distrib utionbetween the components shown in Fig. 2(a). Here again it is assumed that the twocomponents are of the same unit wid th an d t hat it is a plane state of stress.

Ft F3

2 .) F2

F~ I F3

[-~-+$-i-~-I--i-4I-i:-i-i-/:~i-T?iI ' - I ' , , I l l l ~ . . . .

i ~ ~ , | , , i i ~ ~ i , i . ~ _ ,

l---r--~i~ *~*~-i---, ---,r--II - - - * - - - / - ~ * ~ - ~ ' - -~ - ~ * :~ - -} . . . . . * - - 4I , ~ I I ~ I

/ I I I ~ I i I f , , , , , t a

b )

~: I ' E? , E .

e)

d)

. . . . . . . • . . . . . . .

. . . . . . . . . . . . •

i : I [ i , , , , ,

I 2 3 4 5 6 ? 8 9 I0

N o o f i t e r o t i o r ~

Fie. 2. (a) Joi nt model; (b) finite element division of the joint andrepresentation of the d umm y plate at the surface in contact; (c) uniformcontac t def ormation s for the first ite rati on; (d) displacements of theconta cting nodes norma l to th e surface from the first itera tion ; (e) dis-placements of the contacting nodes normal to the surface for the tr ialiteration; (f) calculated pressure distribution; (g) sequence of contactdeformations at a general pair of nodes without improving the

convergence.

The finite element division mus t follow the same rules described in the first meth od.

However, in this case both components are connected by a dummy plate with thebehaviour equivalent to the surface in contact. When this plate is inserted the dimensionsof the sy stem are not modified. The thickness of the p late is only considered in thecalculations in order to define the rigidity and no b endin g action is implied by the ter mplate. The thickness and the elastic modulus of this plate must be chosen so tha t the



9 9 8 N . B A C K , I V [ . B U R D E K I N a n d A . C O W L E Y

b o h a v i o u r o f t h e c o n n e x i o n r e m a i n s t h e s a m e a s t h e s u r f a c e i n c o n t a c t . T h e p r o p e r t i e s o f

t h e p l a t o a r e s u c h t h a t i t m u s t h a v e h i g h r i g i d i t y a n d b e n o n l i n e a r i n c o m p r e s s i o n ,

h a v e z e r o r i g i d i t y i n t e n s i o n a n d n o s h e a r r i g i d i t y i n t h e d i r e c t i o n t a n g e n t i a l t o t h e

s u r f a c e s o t h a t t h e f r i c t i o n f o r c e c a n b e t a k e n i n t o a c c o u n t . T h e m e t h o d d e s c r i b e d b e l o w

w i l l n o t i n c l u d e a l l t h e s e c o m p l e x i t i e s b u t t h e m a i n p r o p e r t i e s c a n b e t a k e n i n t o

c o n s i d e r a t i o n .

F o r t h e f i r s t i t e r a t i o n t h e t h i c k n e s s o f t h e p l a t e i s c h o s e n w h i c h c a n a l w a y s r e m a i n

t h e s a m e . T h e m o d u l u s o f e l a s t i c i t y i s c h o s e n a s a f u n c t i o n o f t h e s u r f a c e f i n i s h , p a i r o f

m a t e r i a l s a n d p r e s s u r e d i s t r i b u t i o n . A t t h e b e g i n n i n g s o m e a s s u m p t i o n s f o r t h e p r e s s u r e

d i s t r i b u t i o n m u s t b e m a d e . T h i s c a n b e c o n s t a n t o r a n y v a r i a b l e f o r m c a l c u l a t e d s o t h a t

i t e q u i l i b r a t e s t h e e x t e r n a l l o a d w h e n t h e c o m p o n e n t s a r e a s s u m e d t o b e r i g i d .

F o r a n i s o t r o p i c m a t e r i a l t h e f o l l o w i n g r e l a t i o n s h i p i s v a l i d :

h E

P = t × l O a " ( 10 )

W h e n e q u a t i o n s (1) a n d ( 10 ) a r e c o m p a r e d t h o f o l l o w i n g i d e n t i t y c a n b e e s t a b l i s h e d .

100 t x 10 a

o r

IOtA( i /m- i )E - ~ - -

c l /m

I n t h e l a s t r e l a t io n s h i p E i s t h e m o d u l u s o f e l a s t i c i t y i n k g f / m m 2 a n d t t h e t h i c k n e s s

o f t h e d u m m y p l a t o in ra m . T h i s e q u a t i o n c a n b e w r i t t e n i n a g e n e r a l f o r m a s fo l lo w s

10 th~/m- l~En - cl/,,~ (11 )

C o n s i d e r in g t h e p r o b l e m s h o w n i n F ig . 2 ( b) , a n d a s s u m i n g t h a t t h e p r e s s u r e i s u n i f o r m ,

t h e n t h e s u r f a c e c o m p r e s s i o n Am a t t h e n o d e s c a n b e c a l c u l a t e d f r o m e q u a t i o n ( 1) . T h e

m o d u l u s o f e la s t i c i t y E n f o r t h e c o n n e c t i n g f m i t o e l e m e n t s i s c a lc u l a t e d u s i n g e q u a t i o n ( 11 ).

I n t h i s e x a m p l e E n f o r a l l t h e e l e m e n t s i s t h e s a m e a n d r e p r e s e n t e d i n F i g . 2 ( c) .

F o r t h e s y s t e m s o e s t a b li s h e d , w h e n t h e f i n i t e e l e m e n t m e t h o d i s a p p l i e d t h e

d i s p l a c e m e n t s i n t h e y d i r e c t io n o f t h e n o d e s s e p a r a t e d b y t h e p l a t e c a n b e r e p r e s e n t e d

a s s h o w n i n F i g . 2( d ). F r o m t h i s f i g u re i t c a n b e s e e n t h a t a l l f i n i te p l a t e s o v e r t h e s u r f a c e

a r e in c o m p r e s si o n . T h e a m o u n t o f c o m p r e s s i o n c a n b e c a lc u l a t e d b y t h e r e l a ti o n s h i p

An = v(j+,~)-v(i+n). (12 )

W h e n t h e s e v M u e s a r e s u b s t i t u t e d i n e q u a t i o n ( I1 ) t h e n t h e m o d u l u s o f e l a s t i c i t y a t

t h e p a i r o f n o d e s is d e t e r m i n e d . W h e n t h e v a l u e o f An b e c o m e s n e g a t i v e t h e n t h e v a l u e

o f E ~ i s m a d e e q u a l to z e r o a n d m e a n s t h a t t h e e l e m e n t is e l i m i n a t e d . F o r t h e n e x t

i t e r a t i o n , t h e m o d u l u s o f e l a s t i c i t y o f t h e d i f f e r e n t f in i t e p l a t e s w i l l b e c M c u l a t o d a s al i n e a r i n t e r p o l a t i o n o f t h e v a l u e s d e t e r m i n e d a t t h e p a i r o f n o d e s t h a t f o r m t h e b o u n d a r y

o f e a c h p l a t e o r

E ~ + E n + I (13 )E~- 2

M o w a l l t h e d a t a a r e p r e p a r e d f o r t h e s e c o n d i t e r a t i o n . T h i s s e q u e n c e i s f o l l o w e d

u n t i l t h e d i f f e re n c e b e t w e e n t h e v a l u e s o f A , f r o m o n e i t e r a t i o n t o a n o t h e r is le ss t h a n a

p r e d e t e r m i n e d v a l u e . S u p p o s e t h a t a f t e r a c e r t a i n n u m b e r o f i t e r a t io n s t h e d e f o r m a t i o n s

a t S h e s u r f a c e i n c o n t a c t c a n b e r e p r e s e n t e d a s s h o w n i n F i g . 2 (e ). I t i s e v i d e n t t h a t t h e

d e f o r m a t i o n s o f t h e c o m p o n e n t s a r e c a l c u la t e d a t t h e s a m e t i m e .

W h e n t h e v a l u e s f r o m e q u a t i o n ( l l ) a re c a l c u l a t e d f o r t h e l a s t i t e r a t io n t h e n t h e

p r e s s u re d i s t r i b u t i o n a t t h e s u r f a c e in c o n t a c t w i ll b e d e t e r m i n e d b y t h e f o l lo w i n g e q u a t i o n

' ( . /p,~ = ]-0-6

T h i s i s g i v e n i n t h e d i a g r a m s h o w n i n F i g . 2 (f ).



Pressure distribution and deformations of machined components in contact 999

The pressure distri bution can also be evaluated at t he pair of nodes with the equ ation

(E ' _ I + E ' ) ~ . (1 5)P" = 2t x 10 a

The difference between the values calculated with equations (14) and (15) is small,

this will be seen later when some examples are solved.When the sequence described above is followed the convergence is always certain bu t

ver y slow. If, for example, th e surface compression at a ce rtain pair of nodes is analys ed

in the consecutive iterations t he change of the surface deformations can be represented

as shown in Fig. 2(g) converging to a con stan t val ue. To increase the speed of convergence,

the surface compression used to define the modulus of elasticity for the third iteration isobtained by aver aging the values from the first and second iteration at each pair of nodes.

For the f ourth iteration )~, is defined as the average of the first average and the third

iteration. Following this procedure the speed of convergence is increased many times.

4 . S P R I NG METHOD

This method is very similar to t he plate method, instead of finite plates, springs are

used at the surfaces in contact Thus, only normal stiffness is incorporated and shear

effects are neglected. Here agai n the compo nents are div ided into finite elements in an

independent form, and the only restriction is that at the contact surface the nodes must

coincide forming pairs.

The computer programs1," used are general, where a system can be formed from

rectangular and tri angular plates, prismatic elements and beams. The beams can be

presented with torsional, flexural, shear and normal stiffness. Here for the spring method,only a normal stiffness will be introduc ed and bendin g effects neglected. This normal

stiffness is defined by the modulus of elasticity which can be the same as the components ;

a lengt h and a cross-section th at is calculat ed as a funct ion of the surface finish and t he

pressure at the contact.

The general procedure for preparation of data for the first and subsequent iterationsand the convergence is the same as that described in the plate method. However, in this

case it is necessary to define the cross-section of the beam.Whe n a t a certa in pair of nodes the pressure is known, AM is the influence area and if

the mesh is fine then the force trans mitt ed across this area is given by

F , , = p . A . . (16)

If between t he t wo nodes a sprin g is introd uced w ith t he cross-section a n and length Lthen the following relationship is valid

-Pn = E)tn a.10aL (17)

o r

10aF .La n = E ~ .

When $'n from equation (16) and p. from equation (14) are substituted in equation (17)the following equatio n is obtained

I O L A . , ~ ( 1 / ~ - I )a . = c 1/'~ E (18)

Here again, when ;1~ is negati ve, the n t he s pring is in tension, and a. is equa ted tozero or as small as possible that permits the solution, depending upon the computer

program.The pressure distribution, when the last iteration is prepared, is calculated also withequa tion (14) or by t he following equ atio n

a. EA.p. = 10a A.L . (19)



1 0 0 0 N . B A C K , M . B U R D E K I N a n d A . C OW L EY

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

i s a l so v e r y s m a l l a n d c a n b e s e e n l a t e r i n t h e p a p e r .

F o r t h e s p r i n g m e t h o d , i n s te a d o f c h a n g i n g t h e c r o ss - se c t i o n o f t h e b e a m f r o m o n e

i t e r a t i o n t o a n o t h e r , t h i s c r os s -s e c ti o n c a n b e m a i n t a i n e d c o n s t a n t a n d t h e m o d u l u s o f

e l a s t i c i ty re d e f in e d i n t h e c o n s e c u t i v e i t e ra t io n s . W h a t m u s t b e o b s e r v e d i n th i s m e t h o d

i s t h a t t h e s p r i n g s h a v e n o t r a n s v e r s a l r i g i d i t y a n d , t h e r e fo r e , t h e s y s t e m c a n b e c o m eu n s t a b l e . I f i t i s n o t p o s s i b l e t o i n t r o d u c e r e s t r i c t io n s f o r s o m e n o d e s , t h e n i t i s n e c e s s a r y

t o i n c l u d e s o m e e x t r a r i g i d i t i e s o r i n t r o d u c e t h e s h e a r s t i ff n e s s o f m a c h i n e d s u r f a c e s a s

d e s c r i b e d i n r e f . ( 1) .

5 . F R I C T I O N A T T H E C O N T A C T S U R F A C E

O F M A C H I N E D C O M P O N E N T S

I n g e n e r a l , a t t h e c o n t a c t o f t h e c o m p o n e n t s r e l a t i v e d i s p l a c e m e n t s o f t h e n o d e s

t a n g e n t i a l t o t h e s u r f a c e a r e o b s e r v e d a n d , t h e r e f o r e , f r i c t i o n fo r c e s a r e d e v e l o p e d w h e n

t h e s u r f a c e s a r e a s s u m e d t o b e r i g i d in s h e ar . T h e s e f o r c e s c a n m o d i f y t h e f i n al p r e s s u r e

d i s t r i b u t io n a n d t h e w h o l e d e f o r m a t i o n s o f t h e c o m p o n e n t s .

a )

Fr i ). . / ; / / / / / / / / / / / A

F / / 2 / / / f i , / / / / / / / / / / / / / / / / / / ;F r ( i ) J v ( i ) I

F ( i ) !

\ \ \ \

b)

F r o . 3 . R e p r e s e n t a t i o n o f t h e f r i c t i o n f o r c es a t t h e c o n t a c t i n g s u r fa c e .

(a ) S p r i n g m e t h o d ; (b ) h y d r o s t a t i c m e t h o d .

I n t h e p l a t e m e t h o d s h e a r f or c es a r is e a t t h e c o n t a c t b u t t h e r e p r e s e n t a t io n o f f r ic t i o n

f o r c es i s d i ff ic u l t. F o r t h e h y d r o s t a t i c a n d s p r i n g m e t h o d t h e f r i c t i o n f o r ce s c a n b e

i n t r o d u c e d v e r y e a s il y . C o n s i d e r f o r e x a m p l e a p a r t o f a s u r fa c e i n c o n t a c t a s s h o w n i n

F i g . 3 ( a) w i t h s p r i n g s c o n n e c t i n g t h e p a i r o f n o d e s . T h e r e l a t i v e d i s p l a c e m e n t s f o r a p a i r

i s g i v e n b y v~ t h e n a s i t is s h o w n i n F i g . 3 (a ) t h e f r i c t i o n f o r c e F r i i s a p p l i e d a t t h e n o d e s .

T h e f r i c t i o n f o r ce F r t w i l l b e c a l c u l a t e d w i t h t h e f o l l o w i n g e q u a t i o n

F r i = I ~p i A t , ( 20 )

w h e r e / L i s t h e f r i c t i o n c o e f f i c i e n t .

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

s a m e a s i t i s w h e n o n l y t h e e x t e r n a l l o a d s f o r m t h e l o a d v e c t o r . I n t h e c a se o f t h e h y d r o -

s t a t i c m e t h o d t h e f r ic t i o n f o rc e s a r e a d d e d a s s h o w n i n F i g . 3( b) . A m o r e g e n e r a l a n d

s o p h i s t i c a t e d m o d e l u s e d t o r e p r e s e n t t h e s h e a r s t if f ne s s a n d f r i c t i o n f o r c es a t t h e i n t e r -

f a c e s i s d e s c r i b e d i n r e f . ( 1 ) .

6 . A P P L I C A T I O N O F T E C H N I Q U E S T O T H E S O L U T I O N

O F P R O B L E M S

P r o b l e m 1

T h e b e a m o f m i l d s t e e l s h o w n i n F i g . 4 (a ) w a s u s e d i n r e f. ( 1) . M e a s u r e m e n t s o f t h e

d e f l e x io n s a lo n g t h e s u r f a c e i n c o n t a c t w e r e m a d e w h e n t h e l o a d s w e r e a p p l i e d a t t h ec e n t r e a n d s u p p o r t e d b y a b a s e t h a t w a s a s s u m e d t o b e r i g id . T h e s u r f a c e c h a r a c t e r i s t i c s

w e r e g i v e n b y e = 0 . 2 9 a n d m = 0 .5 .

F o r t h e t h e o r e t i c a l s o l u t i o n o f t h e p r o b l e m , t h e b a s e w a s a l so c o n s i d e r e d t o b e r i g i d

a n d t h e c o n t a c t b e t w e e n t h e c o m p o n e n t s i s e la s t i c a c c o r d i n g t o e q u a t i o n ( 1 ).



Pressure distributio n and deformations of machined components in contact 1001

120

10 0

,~ . 6 0

4 0

2 O

aJ

b )

c )

F

Wid th 25 .4 tu rn

[ l I I I L I I i l I I i I I , I I l I I I I I L I I I i J l

r , , ,. . . . . . . . I a - - - " , - a - - - - r l , , . . . . . . I - , - ~ - - i , . . . . . . . . . . . . . .1 I , I r . . . .I " ~ - - 1~ l l ¢ l l l¢ / l # ¢ 1 1 1 ¢ l l l # /1 1 1 1 1 ¢ l l # / 1 1 / l ll l l ll # l l 1 4 1 1 1 1 1 1 1 1 1 1 1 / l l l /h

I " 3 o o , . , ~ ~ 1

_ ~ ' 5 m m

~ 13 2 kgf l crn2

6 . 6

2"65

~ 1"32

i I I I I I I I I i J~ . l r " ~ l i i i ¢" . . I , . . L I i i i i i I i i I I

150 300 m m

-81- \ / /- 6 ' 6 5

: t . o _

I 0

9

8

?

"~ 5

3

2

I

d)

_ ~ 5 th i t e r o t i o n

~ th

. . . . . . / J S . , , , , , , . . . . . . .150 300turn

FIG. 4. (a) Fini te element division and dimensions of the beam on arigid base; (b) pressure distrib ution for different apparent interfacepressures using the hydrostatic method; (c)contact deformations for

different apparent interface pressures using the hydrostatic method;

(d) sequence of the calculated pressure distrib ution using the h ydrostaticmeth od an d app aren t i nterface pressure of 1.32 kgf/cm =.

Applyin g the hydros tatic method, the pressure distribu tion and contact deformationsfor different interface pressures are shown in Fig. 4(b) an d (c). Fig. 4(d) shows the sequenceof th e pressure dis tri but ion for t%he consecutive itera tions in the case of 1.32 kgf]em2

appar ent interface pressure. The nu mbe r of iterations for the other loads was in the sameorder.

In Fig. 5(a), the deformed beam for the interface pressure of 6 . 6 kgf]cm 2 usin g theplate meth od is shown. Similarly, the deformed beam could be drawn for the othermethods a nd loads.



1002 N. BACK, M. BURDEKIN and A. COV~'LEY

In Figs. 5(b) an d (c) the pressure dist ribu tion and the contact deformations arerepresented using the plate and spring method for the interface pressures of 1.32 and

6.6 kgf/cm =. Figs. 5(d) and (e) show the sequence of the con tact deflexions in the successiveiterations, here using the plate met hod b ut for the s pring method it is similar.

D e l o r m o h o n s c o l e

0 10 20 3 0 4 0 . u r nF = 5 10 k g f

+ • S p r l n9 m e t h o d0 P a = 6 6

3 0

~ 2o

I P a = 1 -

1 5 0 3 0 0 r n mbJ

- 6 o P o : 6 " 6 *

- 4 P a : 1 " 3 2 *

- 2 • • + ~ +

0

: J te)

- 5 4 t h i t e r a t i o n

- 4 3 r d

- 3

- 2

0 , , , ,'L3

4

\ i , t ~ " ~ '' ' -

d )

- 2 5 t h i t e r a t io n

- t- 5 4 ~3 th

- I 0 2 n d

0 , , , ,

t ~ . ~ . , . x ~ f5 ~ " I s t

1-0

1 5

e )

F I G . 5. ( a) F i n i t e e l e m e n t d i v i s i o n a n d d e f o r m a t i o n s o f t h e b e a m

using the plate me thod; (b) pressure dist ribution using the spring

and plate method for the apparent interface pressure of 6.6 andl'32k gf/cm2 ; (c)co ntac t deform ations ; (d) sequence of the contac t

deformations using the plate method and 6.6 kgf/cm 2 apparen t inter-

face pressure; (e) sequence of the contact deformatio ns using th eplate meth od and 1-32 kgf/cm s apparen t interface pressure.



Pressure distribution and deformations of machined components in contact 1003

Due to the small differences between the results using the different methods and t he

eval uati on of the pressure w ith equati ons (14), (15) and (19) the comparis on is shown in

Table 1. In this Table are also represented the results of the in tegrated pressure distribu-

tion. Only when the hydrostat ic and spring methods are used in conjunction with

equati on (19) will the in tegrated pressure be equal in magnitude. The reason for this

can be appreciated when the assumptions for each method are analysed. For the alterna-tive ways of calculating the pressure dist ribution the differences depend upon the number

of iterations performed. It must be remembered tha t the pressure is a function of the

surface deformation to the power of appro ximate ly 2 and, therefore, any difference in the

final deflexion gives a l arger difference in t he pressure.

T A B L E 1. C O ~ P A R m O N O F TH E CO N TA CT D EFO R~IA TIO N S A N D PRESSU RE D ISTRIBU T IO N

O V ER TH E H A LF LE I~G TH IN CO N TA CT

Hydros method Plate method Spring method

For ce p p p p(kgf) ~ p ~ (eqn (14)) (eqn (15)) )l (eqn (14)) (eqn (19))

102

510

0-87 9.50 0.86 8.94 8.76 0-88 9.35 9.65

0.79 7.90 0.77 7"31 7-29 0"79 7.69 7.91

0"60 4.79 0-59 4.28 4.39 0.61 4.54 4-67

0.40 2.14 0-40 1.90 2.00 0.41 2.01 2.09

0-20 0.53 0.21 0-51 0.59 0-21 0-52 0-59

0.00 0.00 0.02 0.00 0.18 0.01 0.00 0.04

Tot. F. 102.0 Tot. F. 93"83 95-65 Tot . F. 98.73 102.0

2.20 59.2 2.18 57.8 53.4 2-20 59.3 59.1

1.93 45.3 1.90 43-9 41.2 1-93 45-6 44.8

1.35 20.4 1.30 20.7 19.8 1.34 21.9 20-7

0.72 5.09 0.69 5.75 5.59 0.70 5.92 5.33

0.10 0.00 0.10 1.20 2-30 0.06 0.47 0.41

- - 0-53 0-00 -- 0.50 0.00 0.00 -- 0-59 0.00 0.00

Tot . F. 510.0 Tot . F. 509.9 485-2 Tot . F. 526.1 511.9

p, kgf/ cm ~, 2, ~m; Tot. F., tot al force.

For this parti cular prob lem the results of the three methods gave very good correlation

but this is not general; the hydrostatic and spring methods behave similarly for anyproblems where they can be applied but the plate met hod will in general differ in particular

when the relative tangential displacements of the surface are expected.

As was seen before, the numb er of iterations is in the order of four t o five but when

for a particular problem the deformations for several loads are required then the number

of iterati ons can be decreased as follows. For t he lowest load the procedur e is the same as

described before or beginning for example with a uni form pressure distribution. For t he

second load amplitude, the pressure distrib ution from the first load can be used to define

the characteristics of the finite element connexions and so on. When this procedure is

followed then, depending upon the difference between the loads, the numbe r of iterations

can be reduced to one or two.

Problem 2

This problem, which is shown in Fig. 6(a), was examined exp erimentally by Levina. a

The beam of cast iron was placed on a rigid cast iron base, and the surface finish at the

contact was hand-scraped but this was not specified in detail. As is known for hand-scraped surfaces the compliance depends upon the qua lity of the hand-scraping. For the



1 0 0 4 N . B A C K , M . B U R D E K I N a n d A . C OW T ,E Y

c a l c u l a t io n s t h e s u r f a c e c h a r a c t e r i s t i c s t h a t w e r e u s e d r e p r e s e n t a p p r o x i m a t e l y t h e r a n g e

f o r t h i s t y p e o f s u r fa c e f i n i s h3 a n d t w o v a l u e s o f c w e r e c h o s e n , 0 . 6 a n d 0 .8 , a n d m = 0 . 5.

T h e m o d u l u s o f e l a s t i c i t y f o r t h e c a s t i r o n w a s t a k e n a s 9 5 0 0 k g f / m m ~.

o)

b )

¢)

Deformohon cale

F=200 QfI

i t i "

4 - t [ ,1 - 4 - - l - - ' J - ' - ' - q - d - d - 4 - - l - + - F - - - ',-d ' , -- 4 t 20ram

n . _ L _ _ L - - J _ J _ J _ J _ j _ d _ d _ 4 _ J _1_ _1_ L L R

L . . . . j

12

tO

#

~ s

2

~ + II PIotemethodSpring method

250 500 m m

- 1 . [ +

0 ~ . , , , I ~ , , , ~ ,

~ 2

3

FL e L I L . ~i i i 1 i l

2

5

"- 332

1

i

e )

F IG . 6 . ( a) F i n i t e e l e m e n t d i v i s i o n a n d d e f o r m a t i o n s o f t h e b e a m

u s i n g t h e p l a t e m e t h o d , p a = 4 k g f / c m 2 a n d c = 0 . 8 ; ( b ) c a l c u l a t e d

p r e s su r e d i s t r i b u t i o n u s i n g t h e s p r i n g a n d p l a t o m e t h o d f o r pa = 4 kgf /

c m ~ ; (c ) c o n t a c t d e f o r m a t i o n s f o r p a = 4 k g f ] cm * ; ( d) c o n t a c t d e f o r m a -

t i o n s f o r p a = 2 k g f / c m * ; ( e) p r e s s u r e d i s t r i b u t i o n f o r ID a = 2 k g f / e m *.

F o r t h e t e s t s 3 a n d c a l c u l a t io n s t h e a p p l i e d l o a d s a t t h e c e n t r e w e r e e q u i v a l e n t t o2 a n d 4 k g f / c m ~ n o m i n a l i n t e r f a c e p r e s s u r e .

F i g . 6 (a ) s h o w s t h e c a l c u l a t e d d e f o r m a t i o n s o f t h e b e a m f o r 4 k g f / c m 2 i n t e r f a c e

p r e s s u r e a n d b y u s i n g t h e p l a t e m e t h o d w i t h c - 0 -8 . I n F i g s . 6 ( b) a n d ( c) t h e p r e s s u r e

d i s t r i b u t i o n a n d c o n t a c t d e f o r m a t i o n s a r e r e p r e s e n t e d f o r t w o v a l u e s o f c a n d u s i n g t h e



Pressure distributio n and deformations of machined components in contact 1005

plat e a nd sprin g methods when th e interface pressure is 4 kgf/cm 2. Similarly, results for

2 kgf/ cm 2 are shown in Figs. 6(d) an d (e).In Fig. 7• the same proble m was considered usin g the hydro stati c method.

P i q - i ~ - 4 4 - ! i - i J ! 1 4 i 1 - ~ , m ,~, ,- ~ ~ ~ . . i- " - - t 4 4 4 = ~ - ' ~ - ~ - ] - ' - ' 4 - ~ - 20L ] J J J _ J _ ~ ] . . J d - J - J - J - J - ' - - L J - ~ - -

a )

12

~08

2

b )

¢)

d )

~ 2 + • C='6C = " 8

po =4kgf crn2

i I

250 500 nm

0 " ' ' l . . . .

2

3

6 • 5 h terotlon

I ~ " ~ t T M

i L S . . . , . \ .

11 ~ T h terationtO 3 d

8 ~ 2 n d

~7

"~. 6

~5

N I i l i s t

7. \!

, | ° o

e )

Fzo. 7. (a) Fi nit e element divisi on of the be am; (b) pressure distrib utio n

using the hydrostatic metho d; (e) contact deformatious fo r/ ~ -- 2 and4 kgf/em 2 by using the hydro stati c met hod ; (d) sequence of the pressuredistr ibut ion for pa---2kgf/cm~ using the hydrost~tie metho d;

(e) sequence of the pressure d ist rib uti on for pa = 4 kgf/em =.

The comparison of the calculated contact deflexions using the three methods and the

measured deflexion8 are shown in Fig. 8. Here the correlation between the calculated

and measured contact deformations is quite good. Comparing the calculated results thehydrostatic an d spring methods give approximately the same values which are larger tha n

the plate method as can be expected.



1 0 0 6

o )

N . B a c K , M . B U RD E K IN a n d A . C O W L E r

0 L ~ , , , % . 2 5 0 . J 5 0 0 m m

2

3

b )

• Hydr os ta t ic method

~ - - . P l a t e m e th o d+ Spr i ng method

x M e a s u r e d m e t h o d

FIG. 8. Comparison of the calculated and measured contact deformationsc o n s i d e r i n g c = 0 . 8 . ( a ) F o r p a = 4 k g f / c m 2 ; ( b ) f o r p a = 2 k g f / c m 2.

P r o b l e m 3

T h e b e a m i n F i g . 9 ( a) w a s a l so u s e d b y L e v i n a 3 i n w o r k o n t h e m e a s u r e m e n t o f c o n t a c t

d e f o r m a t i o n s . T h e s u r f a ce f in i s h a n d m a t e r i a l w a s t h e s a m e a s i n p r o b l e m 2 . I n F i g s . 9 ( b )

a n d (e ) t h e p r e s s u r e d i s t r i b u t i o n a n d c o n t a c t d e f o r m a t i o n s a r e s h o w n f o r 2 a n d 4 k g f / c m 2

a p p a r e n t i n t e r f a c e p r e s s u r e . T w o v a l u e s o f t h e s u r f a c e s t i f fn e s s c o e ff i ci e n t e w e r e c o n -

s i d e re d i n c o n j u n c t i o n w i t h t h e h y d r o s t a t i c m e t h o d . F i g s. 9 ( d) a n d (e ) s h o w a g a i n g o o dc o r r e l a t io n b e t w e e n t h e c a l c u l a t e d a n d m e a s u r e d a d e f l e x i o ns .

T h e t h r e e p r o b l e m s so f a r p r e s e n t e d a r e n o t h i n g m o r e t h a n a b e a m o n a c o m p l e x

f o u n d a t i o n o r g e n e r a l ly sp e a k i n g , a n o n - l i n e a r f o u n d a t i o n . E v e n s o t h e t r e n d o f t h e

r e s u l ts c a n b e c o m p a r e d w i t h t h e b a s i c t h e o r y o f b e a m s o n a l i n e a r e l a s t ic f o u n d a t i o n . 14

F o r a n i n f i n i t e l y l o n g b e a m , t h e d e f l e x i o n s d u e t o a c o n c e n t r a t e d l o a d F a r e d e f i n e d

b y t h e f o l lo w i n g e q u a t i o n

= ( F 0 / 2 ] c ) e - ° ~ ( c o s 0 x + s i n 9 x ) , ( 2 1 )

w h e r e

0 = ~ / ( ] c / 4 E I ) .

H e r e , ]c i s a c o n s t a n t o f t h e l i n e a r f o u n d a t i o n , E t h e m o d u l u s o f e l a s t i c i t y a n d I t h e

s e c o n d m o m e n t o f a r e a . F o r t h e l i n e a r c a s e t h e l e n g t h o f t h e w a v e s o f t h e c o n t a c td e f o r m a t i o n s ( o r h e r e w i l l b e t e r m e d t h e l e n g t h o f c o n t a c t ) i s d e p e n d e n t o n l y u p o n ~ o r k

a n d E x I . T h e s a m e h a p p e n s f o r t h e n o n - l i n e a r c a se to a c e r t a i n e x t e n t , b u t n o w t h e

f o u n d a t i o n m o d u l u s i s d e p e n d e n t u p o n t h e d e f l e x io n a n d c o n s e q u e n t l y u p o n t h e l o a d .

I f 8 i n c r e a s e s t h e l e n g t h o f c o n t a c t d e c r e a s e s . T h e r e f o r e , i f / c is t h e l i n e a r m o d e l , a n d

t h e r i g i d i t y o f t h e s u r f a c e a n d t h e l o a d i n t h e n o n l i n e a r m o d e l i n c r e a s e , t h e n t h e l e n g t h

o f c o n t a c t w i l l d e c r e a s e . A n o t h e r w a y t o i n c r e a s e ~ i s t o d e c r e a s e t h e r i g i d i t y E I t h e n

a g a i n t h e l e n g t h o f c o n t a c t d e c r e a se s . T h e i n f lu e n c e o f t h e s e f a c t o r s c a n b e s e e n in t h e

f ig u re s o f t h e p r o b l e m d e s c r i b e d a b o v e .

P r o b l e m 4

T h e c o m p o n e n t s f o r t h i s p r o b l e m s i m u l a t e a s l i d in g j o i n t a n d a r e s h o w n i n F i g . 1 0 ( a) .

T h e f o r m u l a t i o n o f th e p r o b l e m i s a s fo l l ow s . T h e d e f le x i on s a n d t h e p r e s s u r e d i s t r i b u t i o na t t h e c o n t a c t o f t h e c o m p o n e n t s w i l l b e c a l c u l a t e d w h e n a l o a d o f 30 k g f is a p p l i e d i n

t h e x d i r e c t i o n as s h o w n i n F i g . 1 0 ( a) . O t h e r c o n d i t i o n s a r e t h a t f o r t h e n o d e s a l o n g A B

t h e v e r t i c a l d i s p l a c e m e n t s v a n d t h e r o t a t i o n s 0 ~ a r e z e ro . A l s o t h a t t h e n o d e s a l o n g E F

a t t h e l o w e r e l e m e n t a r e f i x e d .



P r e s s u re d i s t r i b u t i o n a n d d e f o r m a t i o n s o f m a c h i n e d c o m p o n e n t s i n c o n t a c t

D e f o r m a t i o n s c a l e

~=2oo g, b ;o 2'o Jo ~'o~,mI

I--i ~--.-"i--= ,~-~---~-i--!-- .--~2J - i - i - H - + T i - " ~ - ~ ' - T ~ - ' - - ! - i - - i - ; - - : - -. - _ . - , - , - - , , - - , - - . - - , , - - . . - . - - , - . . - , - - , - . - - , - - , - , - -

t _ F i _ i_ i _ i _ i _ i _ i _ i _ i _ ~ . j _ j_ ! _ ~ j _ + ~ _ _ l _ _ oo .,.l - - r- r ~ - -r . . . . . . ~ - -- - i . . . . 7 -- i - '~ - - - P-. _ , _ , _ _ . _ , _ , _ , _ _ , _ , _ _ . _ , _ _ . , _ _ . _ . _ . _ _ , - , - , . . . .

L O m m . Ia )

b)

7 ~ ~o • + C= ' 6

/ \ - M e a s u r e d

5

/ . + . \ . /po=4*gf /cm2

2

1

I I i i i ! ! i ! ~

250 500 mm

1 0 0 7

o ~ . ; . , . ' i . . . . .

c)

. o | . . . . , . . . . .

"ot f

d )

"0 [ j , , , I ~ ~ ' ' I~

I~ "5 ~

~ I'0

7 .5 k g f I c m 2

e )

F I G . 9 . ( a) F i n i t e e l e m e n t d i v i s i o n a n d d e f o r m a t i o n s o f t h e b e a m u s i n g

t h e h y d r o s t a t i c m e t h o d , P a = 4 k g f / e m 2 a n d c = 0 . 8 ; ( b) p r e s s u r e

d i s t r i b u t i o n f o r p a = 2 a n d 4 k g f / c m 2 a n d c = 0 . 8 ; (c ) p r e s s u r e d i s t r i b u -

t i o n f o r c o n t a c t d e f o r m a t i o n s ; (d ) c o m p a r i s o n o f t h e c a l c u l a t e d

a n d m e a s u r e d c o n t a c t d e f o r m a t i o n s f o r p~ = 4 k g f / c m 2 ; (e ) c o m -

p a r i s o n o f t h e c a l c u l a t ed a n d m e a s u r e d c o n t a c t d e f o r m a t i o n s f o r

p ~ = 2 k g f / e m 2.

I t c a n b e s e e n t h a t w h e n t h e c o m p o n e n t s a r c u n d e r l o a d t h e n c o n t a c t w i l l o n l y o c cu r

a l o n g t h e s u r fa c e 01). B y a p p l y i n g t h e h y d r o s t a t i c m e t h o d w i t h e - - 0 . 6 , m = 0 .5 ,

E = 9 5 00 k g f / c m * a n d u n i f o r m t h i c k n e s s o f t h e j o i n t o f 1 0 m m t h e d e f o r m e d c o m p o n e n t s

a r e r e p r e s e n t e d i n F i g s . 1 0 ( b ) a n d ( e) .

W h e n t h e p l a t e m e t h o d i s a p p l i e d t h e n t h e c o n n e c t i n g f in i te p l a t e i s o n l y i n t ro d u c e da l o n g 01). F o r t h i s c as e , a n d u s i n g t h e c h a r a c t e r i s t i c s g i v e n a b o v e , t h e c a l c u l a t e d

d e f o r m a t i o n s a r e r e p r e s e n t e d i n F i g . 1 0 (d ) .

F o r t h e e a se o f t h e h y d r o s t a t i c m e t h o d , k n o w i n g t h e p r e s su r e d i s t r i b u t i o n t h e n t h e

s u r f ac e d e f o rm a t i o n s c a n b e c a l c u l a t e d a t t h e c o n t a c t a n d w i t h c o m p o n e n t d e f o r m a t i o n



a)

N. BACK, M. BURDEKIN an d A. COWLEY

t h e n t h e w h o l e d e f o r m e d j o i n t c a n b e r e b u i lt . T h i s c o u l d b e d o n e b u t w i ll g i v e a p p r o x i -m a t e l y t he s am e a s F i g . 10 ( d) . Th e t o t a l de f l ex i on o f 24 .0 ~m s how n i n F i g . 10 (d ) w a sa l so o b t a i n e d w i t h t h e h y d r o s t a t i c m e t h o d . F i g . 1 0 ( e ) s h o w s t h e c o m p a r i s o n o f t h ep r e s s u r e d i s t r i b u t i o n c a l c u l a t e d w i t h t h e s e t w o m e t h o d s .

L . . . . . . . . . |

A

i . . . . . . "1

,,

L I

3 0 - - H y d r o s t o t ic m e t h o d

2 5 X P l a t e m e t h o d

u 2 0" ~ t 5 K

t

5 tO 15 20 2 5 $flmm

1008

D e f o r m o t i o n s c a l e, .,,,,h

b ;o 2 o ~ o .m

D i m e n s ~ o n o l s c a l e

d)

Fz6~ . 10. ( a) F i n i t e e l em en t d i v i s i on o f t he com po nen t ; ( b) and ( c)d e f o r m a t io n s o f t h e c o m p o n e n t s u s i n g th e h y d r o s t a t i c m e t h o d ;(d ) d e f o r m a t i o n s o f t h e p a r t u s i n g t h e p l a t e m e t h o d ; (e ) c o m p a r i s o n

o f t h e p r e s s u re d i s t r ib u t i o n u s i n g t h e p l a t e a n d h y d r o s t a t i c m e t h o d .

7 . C O N C L U S I O N S

T h e f i rs t a s p e c t t o b e d i s c u s s e d i n t h e p r e c i s i o n o f t h e r e s u l t s o b t a i n e d f r o m

t h e m e t h o d s . O b s e r v i n g t h e t h e o r e t i c a l a n d m e a s u r e d d e f l ex i o n s f o r p r o b l e m s 2

a n d 3 i t c a n b e s e e n t h a t t h e c o r r e l a t i o n is g o o d b u t a s w a s m e n t i o n e d t h e

e x p e r i m e n t a l d a t a w e r e n o t s p e c if ie d s u c h t h a t t h e a c t u a l s u r f a c e c h a r a c t e r is t i c s

c o u l d b e u s e d f o r t h e c a l c u l a t io n s . I f t h e v a l u e s o f c = 0 .8 a n d m = 0 -5 r e p r e s e n t

t h e h a n d - s c r a p e d s u r fa c e f i n i s h m o s t f r e q u e n t l y u s e d i n p r a c t i c e t h e n p r o b a b l y

t h e d i f f e re n c e s c a n n o t b e a n a l y s e d f r o m t h i s p o i n t o f v i e w . I f t h e s h a p e o f t h e

d i f f e re n c e s b e t w e e n t h e c a l c u l a t e d a n d m e a s u r e d d e f le x i o n s i s c o n s id e r e d i t c a n

b e c o n c l u d e d t h a t t h e s u r f a c e c o m p l i a n c e u s e d i n t h e c a l c u l a t i o n s a r e v e r y c l o se

t o t h e o n e i n t h e t e s t . 8 W h e n a t e s t is u s e d t o c o m p a r e w i t h t h e t h e o r e t i c a l

r e s u lt s , a s is t h e c a s e h e r e , o n e m u s t b e c e r t a i n t h a t t h e p r e c i s i o n o f t h e

m a c h i n e d s u r f a c e s i s g o o d . F o r s m a l l d e f o r m a t i o n s a t t h e s u r f a c e i n c o n t a c t



Op tim um fibre-reinforced sheets for two alte rna tive loadings 1017

(al, a2) are of opposite sign the n (%, a~) are also of opposite sign. If the fibres are fully

strained, the nac, = - - a ~

andI o ~ , I = E e l .

v~ an d v~ are th en der ived fr om (11), giving

[ al [ + [ a~ I I ta n a + ta nv ---- E~! tan ~ (16)

from (2). In this case it ma y be noted t ha t v is depe nden t on (a, fl). Also, since

tan ~ tan l = - a2/ax > 0

(cf. equat ion (10)) bo th ~ and fl lie in the same qu adr ant. It ma y be seen, from {10) an d

(16), t ha t v becomes infinite wheneve r

t a n ~ = t a n ~ = 4 ( - ~ , /~ ) -

Minimi zatio n of the f unct iona l form of v given in equ atio n (16) gives least weight when

(a,/g) ---- (0 °, 90 °) wi th[o'1 [+[a~ [

Vmin ~ E~ I

There is thus a cont rast wi th the case, in Section 3.1.1, when the pr incipa l stresses

are of the same sign. In the present case an infinity of fibre arrangeme nts is again possible,

their weight bein g given by e quation (16); a un ique min im um occurs, however, when the

fibre directions coincide with the princ ipal stress directions.

3.2. T h r e e f i b r e d i r e c t io n s

Since a three-fibre system is statically determinate it is possible to calculate by static

means alone the strains arising from a given applied stress; interest may be restricted

therefore to fully strained systems since overstraining in an y direction may be counteredby increasing the amo unt of fibre in that direction, with a corresponding increase in

weight.

3.2.1. P r i n c i p a l s tr e ss e s o f s a m e s i g n . First, consider a fully strained lamin ate in which

the str ain is isotropic in either tensio n or compression, with

and

Then

v = = [ a , l + l a , f

E e I (17)

from equation (1).The other possible fully strained system is one in which one fibre direction (~, say)

is stra ined in the opposite sense to the o ther two. In this case

[ a l + a 2 [v ~ - v # + v y = .E ez

from (1), an d the we ight is thus greater by 2v~ th an t ha t given by (17).

The optimum is thus given by (17) and corresponds to a state of isotropic tensile or

compressive strain, whichever is appropriate to the applied loading. There is an infinity

of choices for (~, fl, 7) a nd hence an inf ini ty of op tim a ; the choice of op ti mu m (c~, l, ~) is,

however, n ot completely uncons traine d since the only admissible angles are those which

give rise to sim ulta neous tens ion or compression in all three fibre directions. Such admis-sible angles are deriv ed from co nsidera tion of equati on (8). The angles ~ an d ~ may be

regarded as given, but it will always be true tha t these angles and their conjugates (~', fl ')

lie in the sequence

, ~ , , 8 , ~ ' , ~ ' , ~ ;

67



1 0 1 8 P . B A RT H OL O M EW a n d G . Z . H A R R I S

f r o m e q u a t i o n (8 ) i t m a y b e s h o w n t h a t p r o v i d e d 7 l ie s b e t w e e n a ' a n d / 3 ' , f ib r e s i n a l l

t h r e e d i r e c t i o n s w il l b e s t r a i n e d i n t h e s a m e s e n s e . A t t h e e x t r e m e s o f t h e r a n g e ( a ', f l' ) ,

Y w i ll f o r m a c o n j u g a t e p a i r w i t h e i t h e r a o r ]3, f ib r e s i n t h e o t h e r d i r e c t i o n c h a n g i n g t h r o u g h

z e r o to t h e o p p o s i t e s t r a i n s t a t e .

A s a n i l l u s t ra t i o n , F i g . 2 ( a ) s h o w s t h e v a r i a t i o n o f t h e t e t a l w e i g h t w h e n ~2 = 2 ~

a n d t h e a n g l e s a r e c h o s e n s o t h a t a = 5 0 ° , /3 - - 1 6 0 ° ( g i v i n g a ' = 1 2 0 . 2 ° , ]~ ' = 7 9 . 7 ° ) ; t h er e g i o n o f c o n s t a n t o p t i m u m w e i g h t , f i ' < 7 < a ' i s e v i d e n t . F i g . 2( b) s h o w s t h e v a r i a t i o n

o f s t r e s s r e s u l t a n t s w h i c h d e t e r m i n e t h e c o n t r i b u t i o n s v~, v~, v v t o t h e t o t a l f i b r e w e i g h t .

1

IIIIIIIII : :II : :' < : < :

0 ° 1 8 o i f

' : :It?

I l t ' I

5 L F e ~ S / I • i I

r e s u l t a n t l I ~ : ] I !

< . . , . = , . b ( ' , // I ~ I \ ] } t l i / " l

t , / \

l { / I i 7 / i ~ j/ i t l < < - , , , o : I _ , ' _ A

F I G . 2 . ( a) V a r i a t i o n o f t o t a l w e i g h t w i t h 7 f o r % = 2 a 1 w h e n a a n d ]3 a r e s p e c if ie d .

( b ) D i r e c t s t r e s s r e s u l t a n t i n t h e f i b r e d i r e c t i o n f o r e a c h s y s t e m o f p l i es .

3 . 2 . 2 . Principal stresses of opposite sign. W h e n 0.1 a n d 0.2 a r e o f d i f f e r e n t s i g n , i t i s

e v i d e n t f r o m e q u a t i o n ( 1) t h a t 0.~, 0.~, 0.v c a n n o t a l l b e o f t h e s a m e s i g n . I f t h e f i b r e

d i r e c t io n s a r e s o n a m e d t h a t t h e f~ d i r e c t i o n is t h e o n e i n w h i c h t h e f i b r e s a r e s t r a i n e d i n

t h e o p p o s i t e s e n se to t h e o t h e r t w o d i r e c ti o n s , t h e n

0.1 + 0 . 2

v~ -v ~+ v v : ++

f r o m (7 ) ; t h e u p p e r s i g n c o r r e s p o n d s t o c o m p r e s s i v e s t r a i n i n t h e t7 d i r e c t i o n . T h e v o l u m e

f r a c t i o n o f t h e w h o l e s h e e t i s t h u s g i v e n b y

± ( Y l + 0 " 2v: zvt~T -- t lS)

- E ~ s

f r o m ( 2) . T h u s , t o m i n i m i z e t h e w e i g h t o f t h e s h e e t i t i s s u f f ic i e n t t o m i n i m i z e v~, s u b j e c t

t o t h e c o n s t r a i n t t h a t t h e / 3 f ib r e s c o n t i n u e t o c a r r y a s t r e s s o f o p p o s i t e s i g n t o t h a t c a r r i e d

b y t h e ( a , 7 ) fi b r e s .

T h e o p t i m u m i n th i s c a s e i s f o u n d t o c o r r e s p o n d t o t w o - f ib r e d i r e c t io n s o n l y , r t ~ t he r

t h a n t h r e e ; t h e s e t w o d i r e c t i o n s c o in c i d e w i t h t h e p r i n c i p a l s t r es s d i r e c t i o n s . A n o u t l i n e

p r o o f i s n e x t g i v e n o f th i s s t a t e m e n t .

I n i t i a l l y i t is c o n v e n i e n t t o r e g a r d a a n d Y a s g i v e n w h i l e ]5 v a r i e s . T h e c o n s t r a i n t o n fi

i s t h a t t h e f l d i r e c t i o n s h o u l d b e s t r e s s e d i n t h e o p p o s i t e s e n s e to t h e ( a , 7 ) d i r e c t i o n s . I f

]5 i s c o n j u g a t e t o a , t h e n e y = 0 (e L S e c t i o n 2 . 2) . T h u s a c o n s t r a i n t e x i s t s a t ]3 = c ( s i n c e

s v c h a n g e s s i g n i f f~ p a s s e s t h r o u g h a ' ; e v i d e n t l y a s i m i l a r c o n s t r a i n t a l so e x i s t s a t / 7 = y ' .A m i n i m u m c a n b e f o u n d w h e n e i t h e r o f t h e s e c o n s t r a i n t s b e c o m e s a c t i v e , b u t t h i s r e su l t s

i n t h e t h i r d f i b r e d i r e c t i o n b e i n g u n s t r a i n e d . T h i s m a y b e d i s r e g a r d e d s i n ce t h e a r r a n g e -

m e n t s o d e r i v e d i s a t w o - f ib r e s y s t e m i n w h i c h t h e d i r e c t i o n s a r e c o n j u g a t e , w h i c h i s

p r e c i s e l y t h e c a s e c o n s i d e r e d a l r e a d y i n S e c t i o n 3 . 1 .2 .



O p t i m u m f i b re - r ei n f o rc e d s h e e t s f o r t w o a l t e r n a t i v e l o a d i n g s 1 0 19

A n u n c o n s t r a i n e d o p t i m u m i s t h u s s o u g h t. S i n ce

v~ a ~ s i n ( ~ - 8 ) s i n ( ~ - 8 ) = a l s i n y s i n a + a~ c o s y c o s a

f r o m ( 8b ) , t h e a n g l e 8 w h i c h m i n i m i z e s v~ i s t h a t w h i c h m a x i m i z e s [ s in ( y - 8 ) s i n ( a - 8 ) [.

T h i s g iv e s a f re e m i n i m u m w h e n

8 ½(a-l-y) } (19)o r 8 = ½ ( a + Y ) + 9 0 ° .

C o n s i d e r i n g t h e f i r s t o f t h e s e e q u a l i t i e s

a 1 s i n s ½(a + y) + a s cos s ½(a + y) (20)v ~ E e f = ~ ( a l + a s ) ± s i n 2 ½ ( y - a )

f r o m ( Sb ) . A l t h o u g h t h e l o w e s t v a l u e o f v ~ i s r e q u i r e d , t h i s d o e s n o t n e c e s s a r i l y c o r r e s p o n d

t o t h e fr e e m i n i m u m o f (2 0) . T h e t w o c o n s t r a i n t s t h a t 8 = ½ ( a + y ) s h o u l d b e b e t w e e n

a " a n d y " e a c h g i v e

~< [ a l s i n s ½ ( a + y ) + a s c o s s ½ (a + y ) ] s

B y t a k i n g s i n ½ ( y - - a) a s la r g e as p o ss i b le , t h e m a g n i t u d e o f t h e s e c o n d t e r m o n t h e r i g h t

o f ( 20 ) m a y b e m i n i m i z e d w i t h r e s p e c t t o ( y - a ) . T h u s a c o n s t r a i n t b e c o m e s a c ti v e , a n d

a tw o - f i b re s y s t e m i s a g a i n d e r i v e d . E q u a t i o n (1 8) t h e n b e c o m e s

4a~ a~r e e l = _+ ( a l+ a s ) ~+

( a l + a ~) + ( a s - a l ) c os ( a + y ) "

T h i s fo r m o f v d e p e n d s o n l y o n t h e a n g l e ( a + y ) a n d i t m a y b e s h o w n t h a t t h e m i n i m u m

w e i g h t i s g i v e n b y

l a l ] + las lvEs~,

w i t h

o r

a = y = 9 o o, 8 = 0 °

a = y = 0 ° , 8 = 90°"

A s i m i l a r p r o o f m a y b e g i v e n if 8 = ½ ( a + y ) + 9 0 ° (ef. e q u a t i o n ( 1 9 )) . T h u s t h e t w o - f i b re

s y s t e m u s i n g t h e p r i n c i p a l s tr e ss d i r e c t i o n s i s l i g h t e r t h a n a n y t h r e e - f i b r e s y s t e m .

3 .3 . F o u r f ib r e d i r e c t i o n s

T h e f i r s t ca s e c o n s i d e r e d is t h a t i n w h i c h t h e p r i n c i p a l s tr e s s es h a v e t h e s a m e s i g n .I n t h i s c a se , s y s t e m s w i t h f o u r ( or m o r e) f i b re d i r e c ti o n s a r e o p t i m u m o n l y i f t h e s t r a i n

s t a t e c a u s e d b y t h e l o a d i s o n e o f i s o tr o p i c t e n s i o n o r c o m p r e s s io n . I f i n e q u a t i o n (1 ) t h e

d i r e c t i o n ~ a n d v o l u m e f r a c t i o n v 8 a r e r e g a r d e d a s g i v e n , t h e n t h e s t re s s c a r r ie d b y t h e

8 f ib r es m a y b e s u b t r a c t e d f r o m t h e a p p l i e d s t re s s ( al , a s, 0 ) l e a v i n g t h e p r o b l e m o f

d e t e r m i n i n g t h e f i b r e d i r e c t i o n s ( a, f l, y ) a n d v o l u m e f r a c t i o u s ( v, , v ~, v0 t o c a r r y t h e

m o d i f i e d l o a d

(a l + va e l cos ~ 8, as -+ v~ e l s i n g 8 , + v8 e l cos 8 s i n 8) . (21)

I f t h i s m o d i f i e d l o a d h a s p r i n c i p a l s t r e s se s o f t h e s a m e s i g n a s ( a l, a s ), i . e. i f 8 a n d v~ a r e

s u c h t h a t

v ~ < e I( al s i n 2 - ~ a s c o s2 8 ) I

t h e n t h e p r o b l e m r e d u c e s to t h a t o f d e t e r m i n i n g a n o p t i m u m t h re e - f ib r e sy s t e m , w h i c h

h a s a l r e a d y b e e n d i s c u s se d i n S e c t i o n 3 .2 .1 . T h u s t h e r e i s a n i n f i n i t y o f p o s s i b le c h o ic e s



1 0 2 0 P . B A R TH O LO M E W a n d G . Z . H A R R I S

f o r ~ a n d v a a n d e a c h s u c h c h o i c e c o m b i n e s w i t h a n i n f i n i t y o f c h o i c es o f ( a, 8 , ~ '),

(v ~, v ~, v v) a n d g i v e s ri s e to t h e m i n i m u m w e i g h t a l r e a d y g i v e n i n e q u a t i o n ( 17 ).

A s i n S e c t i o n 3 . 2 .2 , i f t h e p r i n c i p a l s tr e s s e s a r e o f o p p o s i t e s i g n , t h e n t h e o p t i m u m

i s t h e t w o - f i b r e s y s t e m i n t h e p r i n c i p a l s tr e s s d i r e c t i o n s .

4 . O P T I M U M A R R A N G E M E N T S F O R T W O A L T E R N A T I V EL O A D I N G S

4 . 1 . General

T h e l a y - u p c o n s i d e r e d is t h a t r e q u i r e d t o c a r r y e i t h e r o f th e l o a d i n g s a ' = ( ~ , a ~ , ~ v )

o r a " = ( a~ , a ~ , T ~ ,) . I f a t h r e e - f i b r e l a y - u p ( ~, 8 , ~ ) i s t a k e n , w i t h t h e f i b r e s t r e s s e s d u e

t o e a c h l o a d i n g b e i n g r e s p e c t i v e l y ( a'~ ,a ~ , a ~) a n d ( a : , a ~ , a '~ ), t h e n ( f o r e x a m p l e ) t h e

m i n i m u m f or v~ i s d e t e r m i n e d b y

m a x ( ! a~ I , l a~ t ) (22)v~ = E~

B u t

m a x ( l a ~ I , ] a : t ) = 1 5 ~ [ + [ a ~ [, (2 3)

w h e r e 5 ~, a ~ a r e t h e a - f ib r e s tr e s s e s d u e t o a p p l i e d l o a d i n g s .

= ½ ( - ' + ~ " ) , Ia = ½ ( ~ ' - - a " ) . ) ( 24 )

5 = (5 ~, 5 v , e ~v ) a n d a = (a ~, a ~ , T ~ ) r e p r e s e n t " m e a n " a n d " d i f f e r e n c e " l o a d i n g s . T h u s

1v ~ = ~ - ( 1 ~ 1 + I ~ 1 ) ( 2 5)

f r o m ( 22 ) a n d ( 23 ). S i m i l a r e q u a t i o n s f o l lo w a ls o fo r v~ a n d v r ; o n s u m m a t i o n

1E---~(I La, l+ ia ~ l+ la v l) . ( 26 )= v~+v~+ v~ = ~ -e ( i ~ l + ] ~ i + l ~ . L ) +

T h e t e r m s o n t h e r i g h t - h a n d s i d e o f (2 6) a r e o b v i o u s l y th e i n d i v i d u a l w e i g h t s o f th e

t w o ( a, 8 , ~ ) s y s t e m s w h i c h s e p a r a t e l y e q u i l i b r a t e 5 a n d a . T h u s t o o b t a i n m i n i m u m

w e i g h t i n ( 26 ) i t w i l l b e s u f f i c ie n t t o m i n i m i z e w e i g h t f o r e a c h o f 5 a n d a s e p a r a t e l y ,

p r o v i d e d t h a t l a y - u p s (a , 8 , ) ') c a n b e f o u n d w h i c h s i m u l t a n e o u s l y d o t h is . I n t h e t w o -

a n d t h r e e - fi b r e s y s te m s c o n s i d e r e d in S e c t io n s 3 .1 a n d 3 .2 t h e m i n i m u m w e i g h t f o r a

s i n g l e a p p l i e d l o a d a w a s , i n a l l ca s e s, f o u n d i n t e r m s o f t h e p r i n c i p a l s t r e ss e s ( a l, a2 ) a s

E E !

T h u s , t h e l o w e s t p o s s i b l e w e i g h t t o b e e x p e c t e d i s

1v = ~ --e t ( [ a l I + [ a 2 ] + 1 51 [ + [ ~ 2 [ ) ( 27 )

f r o m (2 6) , a l w a y s p r o v i d e d t h a t a p p r o p r i a t e l a y - u p s c a n b e f o u n d .

A n a r g u m e n t s i m i l a r t o t h e f o r e g o i n g al s o h o l d s fo r t w o f i b r e d i r e c t i o n s ( a, 8 ) .

I n v i e w o f t h e f o r m o f e q u a t i o n (2 4) , t h e p a i r s o f l o a d in g s

( a ' , ~ " ) , ( - a ' , a " ) , ( - ~ ' , - a " ) , ( ~ ' , - a " )

a l l d e t e r m i n e t h e s a m e p a i r (5 , a ) o f " m e a n " a n d " d i f fe r e n c e " lo a d i n g s . T h u s a n o p t i m u m

d e r i v e d f o r a n y o n e o f th e s e ( t h e p a i r ( a ', a " ) in t h e p r e s e n t c a se ) w i ll a l so p r o v i d e a n

o p t i m u m f or t h e o t h e r t h r e e p a i rs .

I n t h e r e m a i n d e r o f t h i s s e c t io n , a n d i n S e c t i o n 4 , t h e o p t i m u m l a y - u p s w i l l b e c l as s if ie d

a c c o r d i n g t o t h e s i g n s o f t h e p r i n c i p a l s tr e s se s (5 ~, 5 ~) a n d ( a p a2 ). T h e t h r e e s e p a r a t e c a s es

c o n s i d e r e d w i l l b e

(i ) 515 2> 0, (~l a2>O ,

( i i ) 51 5~ > 0, a~ a~ < 0 o r 5x 52 < 0, o* a 2 > 0,

( i i i ) a l 5~ < 0 , a l a z < 0 .



Opt imu m fibre-reinforced sheets for two altern ative loadings 1021

4.2. T w o f ib r e d i r e c t i o n s

4.2.1. 51 5s > 0, al as> 0. For a two-fibre lay-u p (a, fl) to supporg each of the loadings5 and a, the angles (a, fl) mus t be chosen to form a co njugat e pair with respect to each

load separat ely (cf. Section 3.1.1.). Th at is

5~t an~ tan 8+5 ~- fx ~( t ana +ta nS) -- 0, / (28)

ax tan a ta n 8 + a~ -- ~( ta n a + ta n 8) = 0 )

from (9) ; when 51 5s > 0, al as > 0 it may be s hown th at these equati ons always determi ne

a un iq ue pair of direct ions (a, 8)" Fr om Section 3.1.1, this pa ir (a, 8) necessarily corresponds

in each case to t he opt imu m weight ( 151 ]+ 15s ] )~Eel an d ( [ a 1 ]+ ] a s I )~Eel . In view ofthe r emarks of Section 4.1 a true opt imu m is thus derived, v being given by equ ation (27).

It may be noted that this arrangement allows a third fibre direction to be added

without changing the statically determinate natur e of the lay-up. This additional direc-

tion could be used to accommodate a further load, and it would then be possible to

consider fibre arran gemen ts for three al tern ativ e loadings. This topic is not p ursue d any

further in the present paper

4.2.2. 5 1 5 s < 0 o r a la~<0. I f

a l a s < 0 w i t h a l a s > 0

o r

515s>0 wi th a 10 s<0

then it may be shown that the equations (28) again define a unique pair of directions

(a, 8) and a two-fibre system exists which equilib rates bot h 5 an d a. However, one of

5 1 5 s an d a l a s is negati ve and the corresponding opti mum sol ution (cf. Section 3.1.2)

must then have orthogonal fibre directions parallel to the relevant principal axes; in

view of the remarks of Section 4.1, only if the solution of (28) coincides with these ortho-

gonal directions will an o ptim um be derived.

I f515s<0, a las<0,

then equations (28) do not always have a solution; furthermore, even when a solution

exists i t is op ti mu m only when th e pr inci pal stress directions of (51, 5~) an d (al, as) coincide.

Thus, when 51 52 <: 0 or al as < 0 the opt imu m does no t in general correspond to two

fibre directions only.

4.3. T h r e e f i b r e d i r e c ti o n s

4.3.1. 5 1 5 s > 0 , al as >0 . Any optimum solution must have the property that theangles (a,8, y ) give an op ti mu m for each of 5, a. As shown in Section 3.2.1 there is an

infinite nu mb er of choices in such cases, a nd several ranges of angles in fact give rise to

an optimum.(i) C o m b i n a t i o n o f a p a i r o f t w o - f i b r e s y s t e m s . The simplest system to construct is

obt ain ed by selecting an angle a together with its conj ugate angles 8, Y with respect to

the loads 5 and a. Fro m Section 3.1.1, the pai r (a, 8) will necessarily correspond to the

optimum weight ( 151 [+ [52 I )lEe1 while the pair (a, y) will correspond to t he op tim um( [ al [+ [ as [ ) / E e p Since a is arbitrary , an infini ty of opti ma is thu s derived, the weight

in each case being given by equat ion (27).(ii) Combi na t i on o f t wo ge ne ra l t hre e - f i bre s y s t e ms . If t he two angles (a, 8) are take n

as given , suppose (a', 8') an d (~", 8") to be the conjugate s of (~, 8) with respect t o 5 a nd a,

respectively. In view of the re marks of Section 3.2.1, if the ranges ( a',8 ') an d (a",8")overlap th en any angle ~ in the intersecting range will give an o ptim um solution.

If only a is regarded as given, the n the condition tha t the two restrictions on y should

overlap may be used to define a con straint on 8- In this manner , an infinite num ber ofoptima can be derived for an y given a.

4.3.2. 5 1 5s > O , a l a s < O o r 5 1 5 s < O , a l as > O . Suppose 515s > 0, al a~ <0 . In thi s casewe seek to use the optimum two-fibre solution for a which (cf. Section 3.1.2) is found

when the fibre directions (a,8) and the principal stress directions (al as) coincide. The



1 0 2 2 P . B A R TH O L O M E W a n d G . Z . H A R R I S

a n g l e s (a , ]3) s o d e r i v e d m a y b e u s e d i n a t h r ee - f i b r e s y s t e m t o g i v e a n o p t i m u m f o r 5

p r o v i d e d t h e t h i r d a n g l e ~ li es b e t w e e n a ' a n d f f ( cf . S e c t i o n 3 . 2 .1 ) . S i n c e t h e l a y - u p (~ , f l)

i s o p t i m u m f o r a a n d t h e s y s t e m (a , f~, y ) is o p t i m u m f o r 5 , t h e d e r i v e d s y s t e m i s o p t i m u m

w i t h w e i g h t g i v e n b y e q u a t i o n (2 7).

I f 5 1 5~ < 0 , ~1 a~ > 0 a s i m i l a r a r g u m e n t h o l d s , e x c e p t t h a t i n t h i s c a s e t h e t w o - f i b r e

o p t i m u m i s c h o s e n t o c o r re s p o n d t o 5 .4 . 3. 3. 5 1 5 2 < 0 , a 1 ( 7 2 < 0 . T h e o p t i m u m f o r e a c h o f 5 a n d (7 i s t h e t w o - f i b r e s y s t e m

h a v i n g , i n e a c h c a s e , t h e a p p r o p r i a t e p r i n c i p a l a x e s a s f i b r e d i r e c t i o n s ( c f . S e c t i o n 3 . 1 . 2 ) .

S i n c e t h e s e d i r e c t io n s w i l l n o t i n g e n e r a l c o i n c id e , a n y t h r e e - f ib r e s y s t e m w i l l b e n o n -

o p t i m u m , i ts w e i g h t e x c e e d in g t h a t g i v e n b y e q u a t i o n (2 7) .

4 . 4 . F o u r f i b r e d i r e c ti o n s

F o r s y s t e m s w h i c h a r e n o t s t a t i c a l l y d e t e r m i n a t e t h e a n a l y s i s o f S e c t i o n 4 .1 d o es n o t

h o l d , a n d i t c a n n o l o n g e r b e as s u m e d t h a t o p t i m a c a n b e o b t a i n e d b y s u p e r p o s i ti o n , U s e

m a y b e m a d e , h o w e v e r , o f t h e s l a c k f ib r e d i r e c t i o n s e s t a b l i s h e d i n S e c t i o n 2 . 4 w h e n

o r t h o g o n a l d i r e c t io n s a r e f u l l y s t r a i n e d i n o p p o s i t e s e n se s .

4 .4 .1 . 51 52 > 0 or (71 (72 • 0 . W h e n 5 1 5 2 > 0 a n d (71 (72 > 0 , a n i n f i n i t e n u m b e r o f o p t i m m n

f o u r - fi b r e s y s t e m s e x i s ts . S i n c e o p t i m u m t w o - a n d t h r e e - f ib r e s y s t e m s h a v e a l r e a d y b e e nd e t e r m i n e d f o r t h i s c a s e in S e c t i o n s 4 . 2 .1 a n d 4 .3 .1 t h e r e i s l i t t l e t o b e g a i n e d b y t h e

a d d i t i o n a l d e t e r m i n a t i o n o f o p t i m u m f o u r - f i b re s y s t e m s . F o r s u c h a s y s t e m , t h e c h o i c e

o f a n g l e s m u s t b e s u c h t h a t w h e n t h e l o a d s 5 a n d (7 a c t s e p a r a t e l y ~ ,, e~ , % , Sa a r e s t r a i n s

h a v i n g t h e s a m e s e n s e . A l s o , i n o n e o f t h e l o a d i n g s ( 5 + (7 ), (O - (7) t h e s t r a i n s t a t e s h o u l d

b e o n e o f i so t r o p i c l i m i t i n g s t r a i n , w i t h

W h e n 51 5 2 > 0 a n d 0 1 u2 < 0 , s y s t e m s h a v i n g f o u r f i b r e d i r e c t i o n s a r e g e n e r a l l y l e ss

e f fi c ie n t t h a n a n y o f t h e o p t i m u m t h r e e - fi b r e s y s t e m s . F r o m S e c t i o n 3 .1 .2 , t h e o p t i m u m

f o r t h e l o a d (7 r e q u i r e s t w o f i b r e d i r e c t i o n s (f~ , f l + 9 0 ° ), s t r e s s e d i n o p p o s i t e s e n s e s , c o i n c i d e n t

w i t h t h e p r i n c i p a l s t r e s s d i r e c t i o n s ((71, (72). I f a t h i r d f i b r e d i r e c t i o n a i s a d d e d , e q u i l i b r i u m

c o n s i d e r a t i o n s s h o w i t t o b e sl a ck . A n o t h e r s l a c k d i r e c t i o n y e x i s t s, a n d m a y b e t a k e n a s

t h e f o u r t h f i b r e d i r e c t i o n ; s t r a i n c o m p a t i b i l i t y c o n s i d e r a t i o n s (c f. S e c t i o n 2 .4 ) s h o w t h a t

e i t h e r

° a n d ]o r ( 2 9 )

° a n d t

N o t e t h a t a a n d y a r e n o t n e c e s s a r i ly o r t h o g o n a l . T h e l o a d 5 w i ll d e t e r m i n e a , y , v~ a n d % .

N o w t h e l o a A - c a r r y i n g c a p a c i t y o f t h e a a n d y f i br e s i s o n l y f u l l y u s e d i f, w h e n 5 a c t s a l o n e ,

A l s o

e a c h o f ~ , eB+90 h a v i n g t h e s a m e s i g n a s a~ ; t h e l a t t e r e x p r e s s i o n f o l l o w s s i n c e o n e o f

5 + ( 7 , 5 - ( 7 m u s t f u l l y s t r a i n t h e (fi, r i d - 9 0 ° ) f i b re s , w h e n c e 5 a l o n e c a n n o t f u l l y s t r a i n

t h e m . F o r t h e a n g u l a r v a r i a t i o n o f s t r a in r e p r e s en t e d b y t h e f o r e g o i n g e x p r es s io n s t o

s a t i s f y s t r a i n c o m p a t i b i l i t y c o n s i d e r a t i o n s t h e n e i t h e r

f l < a < f ~ d - 9 0 ° a n d f l < y < f l d - 9 0 0 ]

o r a n d f - 9 0 °< Y t ( 3 0

T h e i n e q u a l i t i e s o f (2 9) a n d (3 0) c l e a r ly co n f l ic t , w h e n c e t h e o p t i m u m c a n n o t b e d e r i v e d

o n t h e a s s u m p t i o n o f f o u r f i b r e d i r e c ti o n s .

A s i m i l a r a r g u m e n t t o t h e f o r e g o i n g h o l d s w h e n 5 1 5 2 < 0 a n d (71 a2 > 0 .4 . 4 .2 . 5 1 5 2 < 0 , (Tx (72 < 0 . F r o m S e c t i o n 3 . 1 .2 , t h e o p t i m u m f o r t h e l o a d (7 g i v e s r i s e

t o f u l l y s t r a i n e d f i b r e d i r e c t i o n s (fl, f l d - 9 0 ° ) c o i n c i d e n t w i t h t h e p r i n c i p a l s t r e s s d i r e c t i o n s

((71, (72), t h e w e i g h t o f t h e s y s t e m b e i n g t h e o p t i m u m v a l u e ( [(71 [ 4 -1 % [ ) / E e t ; s i m i l a r l y ,

t h e o p t i m u m f o r 5 g i v e s fu l l y s t r a i n e d d i r e c t i o n s ( a, a 4 - 9 0 °) c o i n c i d in g w i t h t h e p r i n c i p a l



Opt imu m fibre-reinforced sheets for two altern ative loadings 1023

stress directions (51, 5~), the opt imu m weight being ( ] 5 1 I + ] 5 2 ] ) / E @ . Since each fibrepair (a ,~ +9 0 °) an d (f~,f~+90°) equilibrates its corresponding load, the existence

of pairs of slack directions, derived in Section 2.4, may in this case be invoked to

superpose the two solut ions ..4, unique opt imum is derived havin g the fibre directions

(a, f~, a + 90 °, fl+ 90°), the opt imu m weight being given by equa tion (27).

5. YIEL D ENVE LOPE S

In this paper, it is assumed to be the fibre strength which limits the load-carryingcapabi lity of the sheet. This assum ptio n leads to a yield envelope expressed in terms of

Cartesian strains, with each fibre direction intro duci ng a cons train t of the form

- @ ~< e~ cos ~ a + e~ sin2 a + ~x~ sin a cos a ~<@

(eL equat ion (3)). Thus, tak ing (~, e~, ~) as co-ordinates, failure of a fibre system in

tensi on is represented by a pla ne passing thro ugh the poi nt (@, @, 0), there being a p arallel

plan e on the opposite side of the origin which passes throu gh ( - @, - @, 0) an d represents

failure in compression. Each of these planes is dista nt @ from the origin.Thus, i n the general case of n fibre directions, the yield envelope will be a solid bou nde d

by n p airs of parallel planes ; a set of n planes, each passing thro ugh (@, @, 0) will delineat e

tensile failure while a corresponding set passing throug h ( -@, -e l, 0) will correspond to

compressive failure. Unif orml y dist ribu ted fibres in all directions (n-~oo) would

correspond to the double cone shown in Fig. 3; with a finite number of fibre directions

the boun ding planes of the yield envelope are all tangential to this double cone.

q~zy

. . . . . . / . _ - - " . . . . . . , ' f . . \I ' L - -/ k - - -

\ \ \ I " ' . . ; " ' - . X .

~ (o, o,-a E~)FtG. 3. Double cone yield envelope for fibres distri buted thro ugh all angles.

In the case of two fibre directions, the resultant strain envelope is simply a tube

bounde d by two pairs of parallel planes. Fihre-strain assumptions by themselves supply

no constraint for the "scissoring" strain, although clearly a yield envelope having theform of an infin ite tub e is not a practical prop osition ; in practice failure would occur in

the ma trix or in the matrix-fibre bond.

The addition of a third fibre direction introduces an extra pair of parallel planes,

giving a box-qke yield envelope whose six corner points represent fully strained states.The addition of a fourth fibre direction introduces an additional pair of parallel planes

thr oug h (st, el, 0) an d ( - ~ I , - - @, 0), and so on.

By expressing the failure surfaces in terms of stresses rather than strains, loadenvelopes may be derived and compared with the specified restrictions on load. Thecondition that the lay-u p should withst and the alt ernative loads ~" and ~ merely requiresth at the envelope should enclose a parallelog ram (which is a two-dimens ional load

envelope) wi th corners defined by _+ ~', _+ ~". Wh en three fibre directions are used, for





Optimum fibre-reinforced sheets for two alternative loadings 1025

of the principal stresses (51, 59) and (al, as). In all cases, opti ma for alte rnat ive

loads are derived by superposing the separate opt ima derived for the individual

loads 5 and a, the optimum weight in all cases being the sum of the optimum

weights for these loadings tak en separately. The general solutions derived are

summarized in the Table 1.

REFERENCES

1. G. Z. HARRIS,Aeronaut Q. 18, 273 (1967).2. H. L. Cox, Br. J. appZ. Phys. 3, 72 (1952).3. H. L. Cox, The Design of Structures of Least Weight. Pergamon, Oxford (1965).4. H. S. Y. CH ~, Q. appZ. Math. 25, 470 (1968).5. D. C. DRUCKERand R. T. SHIELD,Q. appl. Math. 15, 269 (1957).6. C. T. MORLEY, nt. J. mech. Sci. 8, 305 (1966).

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1973 Back Pressure Distribution