On the calculation of stresses in pin-loaded anisotropic plates

ON THE CALCULATION OF STRESSES IN

PIN-LOADED ANISOTROPIC PLATES

Th. de Jong

TR diss

A

ON THE CALCULATION OF STRESSES IN

PIN-LOADED ANISOTROPIC PLATES

ON THE CALCULATION OF STRESSES IN

PIN-LOADED ANISOTROPIC PLATES

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus, prof .dr. J.M. Dirken,

in het openbaar te verdedigen ten overstaan van een

commissie door het College van Dekanen daartoe aangewezen, op dinsdag 3 november 1987 te 14.00 uur door Theo de Jong geboren te Leeuwarden vliegtuigbouwkundig ingenieur

TRdiss ^

1581

1. De klassieke theoretische spanningsconcentratiefaktor is bij composiet­ materialen misleidend als richtlijn voor constructieve maatregelen.

2. Het streven naar een optimaal gewichts rendement is bij composietmaterialen gevaarlijker dan bij metalen.

3. Bij het ontwerp en de berekening van composietconstructies mag het principe van de Saint-Venant niet worden gebruikt.

4. Een efficiënt middel om spanningsconcentraties in composietmaterialen te verlagen is het toevoegen van lagen onder ±45° ten opzichte van de hoofd­ richting van de belasting. Het woord 'softening' waarmee deze methode wordt gekarakteriseerd is misleidend.

5. 'Partial strength produces general weakness' is voor het ontwerpen met geavanceerde composieten een zeer aktuele stelling.

Sir Robert Seppings (1767-1840), Surveyor of the Navy, I813-I8UO, aangehaald door J.E. Gordon in zijn boek Structures, Plenum Press, London, 197Ö.

6. De opmerking van Koiter dat de theorie van de complexe behandeling van elasticiteitsproblemen voor anisotrope platen in zekere zin eenvoudiger wordt dan voor isotrope platen is slechts geldig voor een beperkte klasse van problemen.

Koiter, W . T . , Complexe behandeling van het twee-dimensionale elas-ticiteitsprobleem, de Ingenieur, 72, no. 23, I960.

7. Bij gegeven randbelastingen op isotrope platen zijn de spanningsverdelingen voor de gegeneraliseerde vlakspanningstoestand en de vlakke vervormings­ toestand gelijk indien de resulterende kracht op iedere gatrand afzonderlijk nul is. Bij anisotrope platen is dat niet zo.

8. Het is opmerkelijk dat bij een scheur in X-richting in een anisotrope plaat de o -verdeling in het verlengde van de scheur onafhankelijk is van de elastische eigenschappen van het plaatmateriaal. Bij een aantal anisotrope materialen zijn alle spanningen daar onafhankelijk van de elastische eigenschappen.

Savin, G.N., Stress concentration around holes, Pergamon Press, 1961.

9. Laminaten met gelijke E-moduli in twee onderling loodrechte richtingen gedragen zich in bepaalde opzichten opvallend isotroop.

10. In het onderwijs op het gebied van construeren met composietmaterialen dient meer aandacht te worden besteed aan de relatief hoge glijdingsmodulus die met deze materialen kan worden verkregen.

11. Bij het geodetisch wikkelen van optimale, rotatie-symmetrische drukvaten hoeft het wikkeloog niet van punt tot punt over de wikkelmal te worden gestuurd. Bij de ontwikkeling van programmatuur voor de besturing van een wikkelmachine moet voor die gevallen dan ook uitsluitend aandacht worden besteed aan de juiste vulling van het maloppervlak.

12. De Nederlandse wetgeving met betrekking tot het dragen van autogordels sluit niet logisch aan bij het Europese gebruik scharnieren aan de voorkant van autodeuren te plaatsen.

ANISOTROPIC PLATES

Th. de Jong

Errata

line 17-18

loaded with .... over half

should read

assuming several expressions for the pin-loads on

line 11

The results . Agarwal [24], Results of finite element studies obtained by Agarwal [29],

line 16

Agarwal [29] Agarwal [24,29]

last three expressions

(z

k " 4

b

' "

d2)

(z£ - p*. b

2

-

a*y

c.9

cos {(n/s) V2(l + l£) cos {(n/s) 72(1 + u M )

C.ll

In this thesis a method is presented for the calculation of stresses in an anisotropic plate with a row of equally spaced, loaded holes. The pin-plate configuration represents a joint in a composite laminate where load is transferred from a row of bolts or rivets into the laminate. The method of solution is based on Lekhnitskii's theory of complex variable functions. The boundary conditions of the problem are satisfied in the approximate sense employing a numerical method. Numerical results are presented showing the effects of the various, relevant parameters of the problem.

TABLE OF CONTENTS

Summary 3

1. Introduction 5

2. The equations of Lekhnitskii 12 2.1. The general solution of plane problems in anisotropic materials 12

2.2. The boundary conditions for external forces and moments 18

2.3. Expressions for the stresses and displacements 20

2.4. The uniqueness of the functions ">, (z, ) 21 2.5. The plate with one circular opening 25

3- The problem of a single, pin-loaded hole 32 3.1. The boundary conditions for the pin-plate interface 32

3.2. The expression for the load functions f, 35 3.3» Determination of the holomorphic functions 38

3.4. Evaluation of the load coefficients 4l

3.5« Discussion 44

4. The approximation for a joint with a row of pin-loaded holes 46 4.1. The stretched plate with an infinite row of unloaded,

open holes 48 4.2. The infinite plate with a row of pin-loaded holes 54

4.3. The boundary conditions for the central hole 56

4.4. The boundary conditions at infinity 59

4.5. Discussion 63

5- Numerical evaluation 66 5.1. Choice of parameters 66 5.2. Numerical results for the plate with a row of unloaded,

open holes 69 5-3' Numerical results for the joint with a row of pin-loaded holes 74

6.1. Evaluation of the solution 101

6.2. Some general remarks 102 6.3- The concept of bearing strength 103

6.4. The influence of the various parameters 104

7. Concluding remarks 110

References 112

Appendices

A.1. A simple expression for I' A, ln(z.-ps) 119

P

A.2. The stresses at infinity 120 B. The coefficients B ^ 123

m

C. Expressions for P( k ) and -^- ( P( kM 125

m dz. l m J

k

Samenvatting 129

SUMMARY

In this thesis a method is presented for the calculation of stresses in an anisotropic plate with a row of equally spaced, loaded holes. The pin-plate configuration represents a practical joint in a composite laminate where load is transferred from a row of mechanical fasteners into the laminate. The pins fit without clearance in the holes and they are supposed to be infinitely rigid. Numerical results are presented showing various effects of geometrical and material parameters on the stress distribution.

The problem of a row of pin-loaded holes in an anisotropic plate is a com­ plicated one of the mixed-boundary type. Several solutions can be found in literature based on simplifying assumptions regarding symmetry and geometry of the problem. The present solution is more general in those respects. The solution can be considered to be an application of Lekhnitskii's theory on stress distributions in anisotropic plates with holes. With this theory the determination of the stresses can be reduced to the determination of two complex functions which must satisfy the specified boundary conditions of the problem.

After a brief discussion in Chapter 1 of the background of the problem and the method of solution the theory of Lekhnitskii is discussed in Chapter 2. Special attention is given to the general solution and to the similarity of the complex functions for anisotropic plates with a loaded or unloaded hole. In Chapter 3 the boundary conditions at the pin-plate interface are dis­ cussed and the problem of an infinitely large plate with a single, pin-loaded hole is solved. In Chapter 4 the solution is extended to a row of equally spaced, pin-loaded holes. It is shown there that the boundary condi­ tions on both sides of the row of holes are in agreement with those of a practical joint. The boundary conditions at the pin-plate interfaces are satisfied in the approximate sense employing a numerical method.

Numerical results are evaluated in Chapter 5 for combinations of parameters which specify a large number of joints in terms of geometry, material properties and loading mode of the holes. They are correlated where possible with results from literature. It is illustrated how the stresses in a joint with two or more rows of pin-loaded holes can be approximated using the

solution for a single row in combination with a by-pass factor. An evalua­ tion of the solution and a discussion of results is presented in Chapter 6.

1. INTRODUCTION

A m a j o r problem r e l a t e d t o t h e u s e of composite m a t e r i a l s i n e n g i n e e r i n g i s t h e d e s i g n o f r e l i a b l e j o i n t s between composite p a r t s and t h e remainder of a s t r u c t u r e . A d h e s i v e b o n d i n g h a s always been v e r y p o p u l a r f o r c o n v e n t i o n a l c o m p o s i t e s and t h e a d v a n t a g e s of c o n t i n u o u s load t r a n s f e r , f l u i d t i g h t n e s s a n d s m o o t h n e s s o f t h e j o i n t s u r f a c e a r e w e l l known. However, s i n c e t h e development of t h e s o - c a l l e d advanced c o m p o s i t e s t h e r e h a s b e e n a g r o w i n g i n t e r e s t i n m e c h a n i c a l l y f a s t e n e d j o i n t s . Advanced composites have p o t e n ­ t i a l l y i m p o r t a n t a p p l i c a t i o n s i n h e a v i l y l o a d e d , p r i m a r y s t r u c t u r e s and a l t h o u g h m e c h a n i c a l j o i n t s have a r e l a t i v e l y low s t r u c t u r a l e f f i c i e n c y they w i l l most l i k e l y be p r e f e r r e d t o a d h e s i v e l y bonded j o i n t s f o r u s e i n t h o s e s t r u c t u r e s . An i m p o r t a n t r e a s o n i s t h a t high f o r c e s a t t h e j u n c t u r e of a bonded j o i n t i n composites may a f f e c t t h e s t r u c t u r a l i n t e g r i t y o f t h e com­ p o s i t e m a t e r i a l i t s e l f because o f i t s low i n t e r l a m i n a r s t r e n g t h , a problem which does n o t occur i n bonded j o i n t s between m e t a l p a r t s . Hence t h e l o a d c a r r y i n g c a p a c i t y of bonded j o i n t s i n c o m p o s i t e s i s l i m i t e d . Another, more g e n e r a l r e a s o n f o r a p p l i c a t i o n of mechanical j o i n t s i s t h a t b o n d e d j o i n t s a r e o f t e n i m p r a c t i c a b l e i n view of d i s a s s e m b l y and f a b r i c a t i o n a l s i m p l i c i t y . Due t o t h e i n t e r e s t f o r many e n g i n e e r i n g a r e a s t h e p i n - p l a t e c o n f i g u r a t i o n r e c e i v e d c o n s i d e r a b l e a t t e n t i o n up t i l l now. U n l i k e i s o t r o p i c m a t e r i a l s the p i n b e a r i n g c a p a c i t y of composites i s s t r o n g l y d e p e n d e n t on t h e d e s i g n of t h e m a t e r i a l i t s e l f a n d t h e r e f o r e j o i n t o p t i m i s a t i o n i n composite p l a t e s r e q u i r e s knowledge of t h e l o c a l s t r e s s e s around t h e f a s t e n e r s . Two a s p e c t s of t h e t h e o r e t i c a l work on p i n - p l a t e c o n f i g u r a t i o n s a r e of s p e c i a l i n t e r e s t : - the modelling of the j o i n t as a pin-loaded a n i s o t r o p i c p l a t e and t h e a c ­

c u r a t e a n a l y s i s of the s t r e s s e s ;

- the p r e d i c t i o n of f a i l u r e in terms of the r e s u l t i n g s t r e s s e s .

I n t h e p r e s e n t work we w i l l emphasize on the f i r s t a s p e c t . A s o l u t i o n w i l l be presented for the problem of an a n i s o t r o p i c p l a t e w i t h a row of e q u a l l y s p a c e d and e q u a l l y loaded h o l e s . This problem r e p r e s e n t s a p r a c t i c a l j o i n t where load i s t r a n s f e r r e d from one p l a t e i n t o another by a long row of p i n s . The s o l u t i o n i s based on L e k h n i t s k i i ' s [ 1 ] continuum method of complex functions which s a t i s f y the governing a n i s o t r o p i c d i f f e r e n t i a l equation. The boundary conditions are f u l f i l l e d in the approximate sense using a numerical

method of boundary c o l l o c a t i o n . Hence t h e s o l u t i o n i s n e i t h e r p u r e l y

a n a l y t i c a l , nor purely numerical.

The analysis of an a n i s o t r o p i c p l a t e with a row of p i n - l o a d e d h o l e s can be

s i m p l i f i e d c o n s i d e r a b l y by assuming symmetry of t h e problem. I t i s then

possible to model the p l a t e as a f i n i t e width s t r i p with a s i n g l e p i n and

with known boundary conditions a t the edges. F i n i t e element methods are very

appropriate for such problems. In r e a l j o i n t s , however, t h e r e q u i r e m e n t s

regarding symmetry are not always met and we therefore w i l l not assume i t i n

advance. Hence m o d e l l i n g of t h e p l a t e as a s t r i p i s u s e l e s s s i n c e t h e

boundary c o n d i t i o n s a t t h e edges cannot be defined. Consequently we w i l l

t r e a t the problem as a p l a t e problem with a periodic s e t of e q u a l l y loaded

h o l e s .

There have been a number of t h e o r e t i c a l s t u d i e s which aimed a t t h e d e t e r ­

m i n a t i o n of t h e d e t a i l e d s t r e s s d i s t r i b u t i o n i n pin-loaded i s o t r o p i c or

anisotropic p l a t e s . The c l a s s i c a l work c o n c e r n i n g i s o t r o p i c p l a t e s was

performed by Bickly [ 2 ] . He solved the problem of an i n f i n i t e l y large p l a t e

with a single hole loaded with e i t h e r a s i n u s o i d a l or a uniform p r e s s u r e

d i s t r i b u t i o n o v e r h a l f t h e c o n t o u r . Knight [ 3 ] t r e a t e d a more r e a l i s t i c

problem by introducing a f i n i t e width. The f i r s t work known t o t h e a u t h o r

d e a l i n g with rows of p i n l o a d e d h o l e s i s due t o Schulz [ 4 ] . He presented

solutions for one and two rows of pin-loaded holes in i s o t r o p i c p l a t e s where

t h e p i n - l o a d s were r e p r e s e n t e d by s i n u s o i d a l p r e s s u r e d i s t r i b u t i o n s .

Theocaris [5] c a l c u l a t e d s t r e s s e s i n an i n f i n i t e l y l o n g s t r i p loaded by a

c e n t r a l l y p l a c e d p i n of the same material as the s t r i p . He assumed n o - s l i p

f r i c t i o n and a contact angle of 180° between t h e pin and t h e s u r r o u n d i n g

s t r i p m a t e r i a l .

All published a n a l y t i c a l solutions for pin-loaded composite l a m i n a t e s c o n ­

t a i n t h e d e t e r m i n a t i o n of L e k h n i t s k i i ' s complex functions, s a t i s f y i n g the

specified boundary conditions of t h e problem. A p r e c o n d i t i o n f o r t h e a p ­

p l i c a t i o n of t h i s method i s t h a t t h e laminates a r e idealized to homogeneous,

two-dimensional s h e e t s w i t h a n i s o t r o p i c e l a s t i c p r o p e r t i e s . Hence t h e

l a m i n a t e l a y - u p and r e l a t e d through thickness e f f e c t s are always ignored.

The p i n - p l a t e i n t e r a c t i o n i s e s s e n t i a l l y d i f f e r e n t for d i f f e r e n t l a m i n a t e s

and t h e r e f o r e t h e c o n t a c t s t r e s s e s between the pin and the p l a t e are not

known a p r i o r i . The evaluation of t h e s e s t r e s s e s as p a r t of t h e a n a l y s i s

requires the solution of a complicated mixed boundary value problem since at the contour of the hole restrictions are imposed to both the displacement and the forces. This aspect of the solution has been emphasized in the majority of the published work.

Oplinger and Gandhi [6] considered single rigid pins in infinite and finite width plates as well as periodic array configurations of rigid pins. They solved the mixed boundary value problem by using series expansions for Lekhnitskii's complex functions. The unknown coefficients of these series were determined by least squares boundary collocation. An iterative proce­ dure was employed to determine the separation zone between the pin and the plate. In following work Oplinger and Gandhi [7] provided a study on the effects of friction at the pin-hole interface and showed significant influences. Later on Ogonowski [8] applied the least squares boundary col­ location method to finite geometry plates and among others he calculated the stresses in isotropic lugs with round and square ends.

Muller [9] analysed the stress distribution in an infinitely large aniso-tropic plate with a single, pin-loaded hole. He represented the pin load by a sinusoidal pressure distribution, a uniform distribution and a concen­ trated force respectively. He did not employ displacement boundary condi­ tions. The present author [10] considered the same problem, but treated the mixed boundary value problem as a problem with prescribed forces on the contour of the hole by assuming a Fourier series expansion for the unknown radial stresses. So he could deal with the pin-hole interface as one of prescribed forces and generate complex functions representing an arbitrary load distribution on a part of the contour and zero loads on the remaining part. The unknown Fourier coefficients were then evaluated from displacement boundary conditions in a limited number of points on the loaded part of the contour. In later work De Jong and Beukers [11,12] used modified infinite plate results to estimate the stresses in finite width plates. De Jong and Vuil [13] introduced an arbitrary load direction and in [14,15] the present author showed the simultaneous influence of friction and load direction on the stresses in orthotropic plates with a single pin-loaded hole.

The papers discussed up till now either considered a rigid, perfectly fit­ ting pin, or assumed a given pressure distribution at the pin-hole inter­ face . As a result the analyses reduced to boundary value problems involving

a single body, namely the plate, only. Klang [16] and Hyer and Klang [17,18, 19] were the first who explicitely modeled the pin and its interaction with the hole by including pin-elasticity. They showed that pin-elasticity is rather unimportant in stress predictions compared to clearance, friction and elastic properties of the plate material.

Zhang and Ueng [20] presented a compact solution for a rigid pin loading an infinite plate. They adopted certain displacement expressions for the edge of the hole which were supposed to satisfy the physical displacement requirements. Lekhnitskii's complex functions were evaluated using these expressions as boundary conditions. Mangalgiri [21] studied the stresses around a frictionless pin loading an infinite plate and emphasized on the effects of the type of fit. His solution was based on an inverse formulation of the problem. This implies that he assumed radial stresses on the entire contour of the hole. The radial stresses were represented by a Fourier series and the unknown coefficients were solved from a given configuration of contact and separation between the pin and the surrounding plate. This method is opposite to the method of Oplinger and Gandhi and the present author who obtained the extent of the contact area as a part of the solu­ tion. Naidu et al. [22] used Mangalgiri's method of inverse formulation for the prediction of the stresses in finite geometry plates loaded by a frictionless pin. They also assumed radial stresses on the entire contour of the hole represented by a Fourier series. Powers series with positive and negative powers were adopted for Lekhnitskii's complex functions. Half of the set of unknown coefficients of these power series was eliminated with the load condition of zero friction at the edge of the hole. The other coefficients and the Fourier coefficients were evaluated from the remaining boundary conditions of the hole and the boundary conditions of the outer contour of the plate. This last step in the solution involved the generation of a set of algebraic equations in terms of the unknown coefficients by minimizing the boundary errors. An important feature of the work of Mangalgiri and Naidu et el. is that an essentially non-linear problem as a pin-loaded hole with clearance can be treated with a linear analysis by considering successively various configurations of contact and separation. A drawback of the method is the use of symmetry relations in the solution which confine the applicability to symmetric problems only.

The method proposed by Naidu e t a l . i s an a c c u r a t e , a n a l y t i c a l method f o r

t h e p r e d i c t i o n of s t r e s s e s i n f i n i t e geometry p l a t e s . The f i n i t e element

method, however, i s o b v i o u s l y the most p r e f e r r e d f o r t h o s e p l a t e s . A

p a r t i c u l a r d i f f i c u l t y r e l a t e d with t h e use of n u m e r i c a l methods i s the

formulation of the boundary conditions. In order t o overcome t h i s d i f f i c u l t y

f i n i t e element i n v e s t i g a t i o n s are always r e s t r i c t e d to symmetric problems

with s i m p l i f i e d conditions for the edge of the h o l e . Waszcak and Cruse [ 2 3 ]

d i s c u s s e d the a p p l i c a t i o n of f i n i t e element methods t o pin-loaded composite

s t r i p s . The e f f e c t of t h e p i n was r e p r e s e n t e d by a c o s i n u s o i d a l r a d i a l

p r e s s u r e d i s t r i b u t i o n and two other, s i g n i f i c a n t l y d i f f e r e n t d i s t r i b u t i o n s .

The r e s u l t s of f i n i t e element s t u d i e s o b t a i n e d by Agarwal [ 2 4 ] , Wong and

Matthews [25] and Chang e t a l . [26,27,28] were a l s o based on assumptions for

t h e r a d i a l p r e s s u r e d i s t r i b u t i o n . A more r e a l i s t i c assumption f o r t h e

i n t e r a c t i o n between the pin and the p l a t e i s t h a t of a displacement boundary

condition a t the edge of the hole corresponding to a r i g i d d i s p l a c e m e n t of

t h e p i n . This assumption was used by Chang e t a l . [ 2 7 ] , Agarwal [29] and

Ramkumar [30] together with the condition of f r i c t i o n l e s s c o n t a c t . The same

assumption was used by Wong and Matthews [25] in connection with a condition

of n o - s l i p of the p l a t e m a t e r i a l over t h e p i n , which i s e q u i v a l e n t t o a

uniform d i s p l a c e m e n t of t h e contour of the h o l e . Studies which model both

the p l a t e and t h e p i n a r e provided by Hong [ 3 1 ] . Crews e t a l . [ 3 2 ] and

S p e r l i n g [ 3 3 ] f o r t h e f r i c t i o n l e s s c a s e and by Matthews e t a l . [ 3 ^ ] .

Sperling [33] and Maekawa e t a l . [35] for n o - s l i p c o n t a c t . Matthews e t a l .

[ 3 ^ ] and S p e r l i n g [33] a l s o i n c l u d e d t h r e e - d i m e n s i o n a l e f f e c t s in t h e i r

c a l c u l a t i o n s . Naik and Crews [36] used M a n g a l g i r i ' s method of i n v e r s e

formulation i n order to study the effects of clearance.

The c l a s s i c a l and well known work on the experimental determination of f u l l

-f i e l d s t r e s s e s around pin-loaded holes i n i s o t r o p i c m a t e r i a l s was per-formed

by Frocht and H i l l [ 3 7 ] , using c o n v e n t i o n a l p h o t o - e l a s t i c i t y . S i n c e then

t h e r e have been many other experimental i n v e s t i g a t i o n s of pin-loaded holes

in i s o t r o p i c p l a t e s . We w i l l only mention t h e work of T h e o c a r i s [ 3 8 ] and

Ghosh e t a l . [ 3 9 ] who used c o n v e n t i o n a l p h o t o - e l a s t i c i t y t o g e t h e r with

a u x i l i a r y methods, and the work of Nisida and S a i t o [40] who combined

photo-e l a s t i c i t y and i n t photo-e r f photo-e r o m photo-e t r y . Comparphoto-ed to thphoto-e numbphoto-er of s t u d i photo-e s of i s o ­

t r o p i c m a t e r i a l s there have been very few e x p e r i m e n t a l s t u d i e s i n v o l v i n g

a n i s o t r o p i c m a t e r i a l and p u b l i s h e d r e s u l t s of f u l l - f i e l d s t r e s s e s a r e

s c a r c e . Moiré techniques and holography were used by Wilkinson e t a l . [ 4 l ]

and Wilkinson and Rowlands [42] to study the stresses around mechanical joints in wood. The same techniques were used by Rowlands et al. [43] to analyse the stresses in other anisotropic media. Prabhakaran [44] and Hyer and Liu [45,46,47] showed that transmission photo-elastic techniques in combination with birefringent composite materials can be usefully employed in the investigation of bolted joints in composites. This technique requires the material to be transparant and therefore the method is confined to glass fiber reinforced resins only. Nevertheless the results can be useful to test theoretical analyses.

The literature as it is reviewed in the previous paragraphs covers almost the whole field of parameters which are of interest for mechanical joints. All discussed work, however, concerns either infinitely large plates with only one pin-loaded hole or symmetric, finite geometry problems. As already referred to before, the last case is generally not representative for practical joints in composites where symmetry is not always assured. In the present thesis the theory of pin-loaded plates is therefore generalized to the problem of a row of pin-loaded holes in a composite plate where the centerline of the holes and the direction of the pin-loads are not neces­ sarily coinciding with one of the material symmetry axes. A numerical method of solution of this more general problem is not possible because of con­ siderable problems in formulating the boundary conditions, both at the contour of the holes and at the outer contour of the considered domain.

The solution of the problem is based on the consideration that in a practi­ cal joint the load transfer consists of two basic load systems:

- a plate with a row of open, unloaded holes with a self-equilibrating load system at the outer contour;

- a plate with a row of pin-loaded holes, transferring the load from the pins into the plate. The load applied at the pins is reacted at the outer contour of the plate.

Lekhnitskii's complex functions corresponding to these two basic systems are derived separately and in a following step of the solution the functions together are made to satisfy the boundary conditions of the problem, using a simple collocation technique. This results in a set of linear equations in terms of the unknown coefficients of Lekhnitskii's functions. Finally these

coefficients are evaluated and the stresses are calculated from the dif­ ferentiated functions. Numerical results are presented showing the effects of the various, relevant parameters of the problem. A computer program is available for the numerical calculations.

2. THE EQUATIONS OF LEKHNITSKII

The complex representation of the general solution of isotropic, plane strain or generalized plane stress problems was introduced by Kolossov and further developed by Muskhelishvilii. Lekhnitskii [1] extended the method to anisotropic elastic materials. His general solution of anisotropic plate problems contains two analytic functions $k(z. ) (k = 1, 2) satisfying the

boundary equations of the plate region. Lekhnitskii's book presents many explicit solutions of $. (z, ) . They are of great relevance for various important technical problems related to the use of modern composites as structural materials. This chapter is devoted to the general equations of Lekhnitskii's theory.

2.1. The general solution of plane problems in anisotropic materials

In the basic formulation of anisotropic elasticity a composite laminate is considered as an elastic, homogeneous flat plate of uniform thickness. The plate is in equilibrium as a result of the forces distributed on its edges and the body forces. The middle plane is taken as the coordinate plane X-Y. It is assumed that at each point of the plate there is a plane of elastic symmetry which is parallel to the middle plane. As a result the shear stresses T and x and the shear strains T and T in the generalized

yz zx yz zx ° Hooke's law are independent of the other stresses and strains. It is further assumed that the deformations of the plate are small.

If a plate satisfies these conditions and if the forces applied on its edges:

- act within planes parallel to the middle plane;

- are distributed symmetrically with respect to that plane; - change only slightly over the plate thickness,

the state of stress in the plate is called a state of generalized plane stress, according to Lekhnitskii. In this state of stress the middle plane of the plate does not bend and remains flat.

In what follows only anisotropic materials will be considered with three mutually perpendicular planes of elastic symmetry passing through every point. They are called orthogonal anisotropic or simply orthotropic

m a t e r i a l s . Hooke's law for these m a t e r i a l s in a s t a t e of g e n e r a l i z e d p l a n e

s t r e s s w i l l be w r i t t e n i n the form

e

±

= S

±i 0 j

( i . j = 1, 2, 6) (2.1)

The stresses (and strains) in (2.1) are the average values of the stress (and strain) components with respect to the thickness. S, . are the trans-formed material compliances in the directions of the coordinate axes. <P is the angle between these axes and the axes of elastic symmetry in the middle plane.

If the previously described plate is in a state of plane strain parallel to the X-Y-plane the displacements perpendicular to that plane are zero and the displacements parallel to the X-Y-plane are functions of x and y only. This situation is defined by

e = 0 z

T = T = 0 yz xz

which results in expressions for Hooke's law analogous to those of (2.1). In a plane strain situation, however, the stresses (strains) in Hooke's law are the real stress (strain) components instead of the average values and the material compliances S. . have to be exchanged with the reduced compliances

S., - S._

S*. - S . . - % « (2.2)

1 J

*

1 J

*

S

3 3

I t i s n o t e d t h a t a p l a n e s t r a i n s i t u a t i o n i n a homogeneous, a n i s o t r o p i c

p l a t e y i e l d s constant s t r e s s e s with respect t o the p l a t e t h i c k n e s s . In the

c l a s s i c a l lamination theory, however, an a n i s o t r o p i c material i s considered

as a l a m i n a t e c o n s i s t i n g of v a r i o u s homogeneous l a y e r s w i t h d i f f e r e n t

e l a s t i c p r o p e r t i e s . In terms of t h a t t h e o r y t h e c o n s t a n t p l a n e s t r a i n

s t r e s s e s are average s t r e s s e s as w e l l . The s t r e s s e s i n the v a r i o u s l a y e r s

can d i f f e r considerably from the average v a l u e s , depending on the respective

e l a s t i c p r o p e r t i e s and o r i e n t a t i o n s .

In absence of body forces the equilibrium equations are satisfied by

stresses given by Airy's stress function U

a*u 3MJ iiy_ /

P

„

°x ~ ay* °y ~ 3x* xy ~ " 3x3y

K

*'*'

where no distinction has been made between the average stresses in the

generalized plane stress situation and the real stresses in the plain strain

situation.

After substitution of (2.3) into (2.1) the compatibility equation

3

2

e 3

2

e 3*Y

5

+

X xy_

3y* 3x

2

3x3y

can be written as

e O _ 2 S - ^ - + <2S *S ) 3*U 2 S ~ ^ -*22 3x* * ^26 3 x ' 3 y U b1 2 b6 6 ' 3x* 3y* ^ bl 6 3x3yJ <fi <D *P *P *P

or, with the use of four linear differential operators

ki k -

k

]

{y

2 k - k

]

(p

3 k - k

]

K k -

k

] u =

°

(2

-

5)

Comparison of (2.4) and (2.5) yields a set of four equations showing that u,

(k = 1, 2, 3, 4) are the roots of

u< - 2u' S

l 6

/ S

u

♦ y»(2S

12

♦ S

6 6

J / S

n

- 2u S ^ / S

n

0 0 0 0 0 0 0

* S

2 2

/ S

n

= 0 (2.6)

u, cannot easily be solved from (2.6). However, by using the transformation

formulae for S. . the equation can be simplified considerably. Expressed in

S. . it becomes

2 s + S S m cos <p + s i n <pi* 12 66 t\i cos <P + s i n <f\* 22 _ _ 0 7 ) *cos <p - u s i n 9 ' S ^ *cos <P - u s i n <p' S1 1 v • / ;

The f o u r r o o t s u. cos <p + s i n <p s, = — : (k = 1 , 2 , 3 . 4) k cos <p - u. s i n <p ' ' - " ' of ( 2 . 7 ) a r e e a s y t o s o l v e . With 1 2 Q + 6 6 = 2 a and ^ = r ' ( 2 . 8 ) bl l bl l t h e y a r e Ut c o s <P + s i n <P / _ \ /

— =—: = N / ^ V

5 +

i X /

5 1

^ -

5

= s , (2.9a)

c o s <P - y1 s i n <p V 2 V 2 1 VU c o s <t> + s i n <p \ / r - a cos <t> - u_ s i n <p

-V

J

r + 2 r + 2 a a sl S2 ♦ i W ^ - T ^ - s , (2.9b)

"3

C

°

S

* * "" V jUr^.Mr^

(2

.

9c) cos <P - p_ s i n <p V 2 V 2 3

^ cos o ♦ sin o r~r^ .

\AT77

,

3 0

. .

cos * - u4 s i n o = V~2~ " X \ T ~ 2 ~ " 8* ( 2'9 d )

yielding the roots of (2.6)

s, cos <p - sin <p

u, = — : (k = 1, 2, 3, 4) (2.10) Mk cos <p + s. sin <P \ t . ->t /

k

Spies [48] showed that r + a always has a positive value. The parameters s. in (2.9) are therefore either complex or imaginary and

if r > a s_ = s_, s^, = s. (2.11a)

The bars in (2.11a) denote that the conjugate complex values of the barred functions have to be taken.

If r £ a s_ = s., S j = s- (2.11b)

s, cos <t> - s i n <p ^ " cos * * B. s i n , { k " *• 2) <2-1 2 a> k

s, cos <p - sin <p

Üj, = — Z (k = 1, 2) (2.12b)

cos <p + s. sin <p

k ■ .

P

k

are called the complex material parameters of the first order. They

characterize the degree of anisotropy of the material. After Spies [48] the

quantities a and r in (2.8) will be called the angularity factor and the

directionality factor respectively. They are real parameters characterizing

the degree of anisotropy as well. Spies [49,50] derived several simple real

variable expressions for stress distributions in anisotropic plates using r

and a instead of u. .

k

With (2.11) equation (2.5) is now written as

K

aï

-

a? ^2 ai

"

Jï

]

K

aï

"

a^ ^2 aï

" ^

U =

°

and since

the expression for U must be

U = F

1

(x +

U;L

y) + F

2

(x + p

2

y) + F

3

(x + j^y) + F^x + j^y) (2.13)

The case r = a yields pairwise equal parameters s. and hence pairwise equal

parameters u

k <

The expression for U in this case is somewhat different from

(2.13)

U = F

1

(x +

U;L

y) + (x + ^ y ) F

2

(x + p

i y

)

+ F

3

(x + y

x

y) + (x + ^ y ) F ^ x + ^ y )

This equation plays an important role in the theory of elasticity of

iso-tropic materials. It will not be considered here.

z

k

= X +

V (k = 1, 2) (2.11a)

*k

= x +

^k

y

(k = 1. 2) (2.14b)

U itself is a real function of the real variables x and y. Therefore F, must

k

satisfy the conditions

F

k + 2

= F

k

(k = 1. 2) (2.15)

With (2.13), (2.14) and (2.15) the following, well known expression for the

general solution of the plane problem in anisotropic materials is obtained:

U = 2Re[F

1

(z

1

) + F

2

(z

2

)] (2.16)

The equations given here imply that any, arbitrary analytical function of z,

represents a stress field fulfilling the equilibrium conditions and the

compatibility of the deformations. Boundary conditions, however, are not

considered and arbitrary functions F, (z. ) may yield unrealistic tractions or

displacements at the boundary of the region considered. The plane problem of

anisotropic materials therefore is a problem of determining the functions

under certain conditions which they must satisfy on the boundary.

In the next section expressions will be derived for the boundary conditions

for given external forces. They are essential for the solution of the

problem of elastic equilibrium of a body if the external stresses or forces

are given. This problem is called the first fundamental boundary value

problem. The problem of elastic equilibrium of a body if the displacements

of the points of its boundary are given is called the second fundamental

boundary value problem. Expressions for the displacements will be derived in

a subsequent section. The contact problem of a pin-loaded hole belongs to a

third group of problems. They are called the mixed fundamental boundary

value problems. A mixed boundary value problem is characterized by known

displacements on one part of the boundary and known stresses or forces on

the remaining part.

2.2. The boundary conditions for external forces and moments

The region of the plate is defined by an external contour and by the inter­ nal contours of the openings in the plate. The expression for a contour can be written in parametric form as

x = x(s) and y = y(s)

where s is the arc length of the contour starting from an arbitrary initial point and

d_ = 1_ dx 3_ dy ,2 1 ? )

ds 3x ds 3y ds l*«-WJ

The positive d i r e c t i o n of s i s defined so that the p l a t e region always

remains at the left-hand s i d e of s . In t h i s s i t u a t i o n and with n as the

external direction normal to both the external and the internal contour the

direction cosines are

cos (n,x) - fa , cos (n,y) = - ^- (2.18)

The equilibrium equations at the contours of the region become with (2.18) and the stresses expressed in Airy's stress function

Ü U dy. + 3'U dx _

3y* ds 3x3y ds " n 3'U dy 3*U dx _ 3x3y ds 3x* ds r n

or, shortly

where X and Y are external forces per unit area in X- and Y-direction n n

respectively.

The important functions \ ( \ ) a r e n o w defined as

d Fk( zk)

W " "ST"

(2

'

20)

Y

n

=

' ds" t*l

(z

l

) +

* 2

( z

2

) +

W

+

* 2

( Z

2 ^

( 2

-

2 1 a )

X

n

=

di [ V l

( z

l

) + u

2 * 2

( z

2

) +

*

,

l*l

(z

l

) + u

2 * 2

( z

2 ^

( 2

*

2 1 b )

The boundary conditions for the external forces are obtained by integration

of (2.21) over an arbitrary arc length AB

r

- j Y

n

ds = 2Re [♦

1

(z

1

) + *

2

(z

2

)] (2.22a)

A

A

B

B

J X

n

ds = 2Re [p

1

0

1

(z

1

) + p

2

*

2

(z

2

)] (2.22b)

A

A

B

[ ] denotes the increase of the expression between the brackets as z.. and

A

z_ pass along the arc AB. Expressions (2.22) are often written in the partly

indefinite form

s

2Re [ ^ ( z ^ + ♦

2

(z

2

)] = - J Y

n

ds + K

t

(2.23a)

2Re [y

1

*

1

(z

1

) + y

2

*(z

2

)] = J X

n

ds + K

2

(2.23b)

where K.. and K

?

are integration constants. Their values depend on the place

of the starting point of the integration path.

The moment AM of X and Y about the origin of the coordinate system is

n n

equal to

A M v v d i3U» d faUi

AM = x Y - y X = -x -r- — I - y -j- \r-\

_n

J

n ds

l

3x' ds

v

3y'

ds

l

* 3x

+ y

3y

J

3x ds 3y ds

or, with (2.17)

.

M

d r au alii dU

AM = - 3- lx —- + y — I +

-r-ds

l

3x

J

3y' ds

The resultant moment of the external forces along arc AB about the origin of

the coordinate system is

B

B

M = ƒ (x Y

n

- y X

n

) ds - [-(x |j[

+

y B) ♦ u] (2.24)

A

A

With ( 2 . 1 6 ) , ( 2 . 2 0 ) and F

k

( z

k

) = ^ k ^ k *

d z

k

e x

P

r e s s i o n

(2-2

1

*) can be

transformed i n t o

B B

M = 2Re [ J ♦

1

(z

1

) dzj_ + J *

2

( z

2

) dz,, - ( z ^ f z ^ * z^

2

(z

2

)} ] (2.25)

I f a r c AB i s a c l o s e d c o n t o u r and 4> (z, ) are single valued functions the

expression

B

( z

1

*

1

( z

1

) + z

2

*

2

( z

2

) )

disappears from (2.25). To exclude resultant moments the single-valuedness

of 0 b,(z.) dz, would also be necessary.

2.3. Expressions for the stresses and displacements

With (2.16) expressions for the stresses are now easily obtained from (2.3).

Using (2.20) they become

a

x

a

2Re [ y * * ^ ) + u

2

*

2

(z

2

)] (2.26a)

2Re [ *'(

2l

) + 4>

2

(z

2

)] (2.26b)

x

xy

= -2Re [ ^ ( z ^ + u

2

*

2

(z

2

)] (2.26c)

where

W dz,

Substitution of (2.26) in (2.1) and integration of the first two equations results in expressions for displacements u and v in X- and Y-direction respectively

u = 2Re [ u ^ f z j ) + u2*2(z2)] + f^y) + K^ (2.27a)

v = 2Re [ v ^ t z j ) + v2*2(z2)] + f2(y) + ^ (2.27a)

where

u

k

= S

ll 4

+ S

12 "

S

16 \ (k = 1, 2) (2.28a)

<P <P <P

V

k

= ( S

12 K

+ S

22 "

S

26 \

) / p

k (k = 1. 2) (2.28b)

From the t h i r d equation of (2.1) and

Y

in + ix

xy " 3y 3x

can be evaluated that fx(y) = K 3 y

f2(x) = -K3 x

K„, KK and K_ are constants which characterize the rotation and the transla-tional displacement of the plate as a rigid body. Since these displacements do no yield deformations the constants are not relevant for the solution of the stress problem.

2.4. The uniqueness of the functions ♦. (z. )

Muskhelishvilii [51] and Meijers [52] discussed the uniqueness of the complex functions in the isotropic case if either the state of stress or the displacements are given. Their approach will be adopted here for the study of the uniqueness of the functions in the orthotropic case.

First suppose that two sets of two functions, ^(Zj.) a°d t|>, (z. ) , k = 1, 2,

define the same stress distribution. The principal material axes will be chosen as coordinate axes, hence u, = s, .

From expression (2.26b) for the stress o it can be concluded that the difference between the two sets of functions can be

\ <zk > - V2k > - ± ckzk+ ck <2-29>

Constants C* are possibly complex and C, may be arbitrary real constants or complex constants if lm C. = -Im C?. However, from (2.26a) and (2.26c) it is

concluded that V ^ z , ) give the same stress distribution as <t, (z.) only if

s — s

Ck = k s g C £ = 3~k (2.30)

where C is an arbitrary real constant.

C* substituted in (2.27) yield rigid body displacements which do not affect the stress distribution. C£ and C, do not affect equations (2.22) either. So, except for one real constant C and two possibly complex constants C* the functions *k(z.) are completely determined for given stresses and for given

external forces on the contours of the plate.

The displacement of the points of the plate completely determine the stresses. This implies that the same functions *k(zk) and

\ <zk > = V « k > + ± ck2k+ ck

must be assumed f o r t h e s t u d y of t h e a r b i t r a r i n e s s of the functions for given boundary displacements.

S u b s t i t u t i o n of i C. z, i n ( 2 . 2 7 ) r e s u l t s i n a r i g i d body r o t a t i o n with displacement components

u = 2 S n C y i ( s2 - s1) ( s ^ - s^) (2.31a)

The possibly complex constants C* yield a rigid body translation with com­ ponents

u* = 2Re [UlC* + u2C*] (2.32a)

v* = 2Re [VlC* + v2C*] (2.32b)

<&, (z, ) and tp, (z, ) give the same displacements only if the rigid body rota­ tion (2.31) and the rigid body translation (2.32) are eliminated. It then follows from (2.31) that C must be zero and from (2.32) it can be concluded that only one of the two constants C* may be chosen arbitrarily. So the complex functions $k(zk) are completely determined except for one possibly

complex constant.

We now consider an infinitely large plate with a homogeneous stress dis­ tribution a = p , a = p and T = p The

x *x' y *y xy *xy yielding such a distribution must be of the form

tribution a = p , a = p and T = p The complex functions 4>. (z, )

x x y y xy xy k* k'

g( k )z * C* go zk ^k

for which expressions (2.26) become

px = 2Re [s^ g^1* ♦ s2 g£2 )] (2.33a)

py = 2Re [ go 1 } ♦ g<2 )] (2.33b)

px y = -2Re [Sl e{o1] + s2 g<2 )] (2.33c)

(k) -(k)

The four coefficients gv ' and g (k = 1, 2) cannot be solved from these

°o o

equations. One of them remains indeterminate and can be arbitrarily chosen. All other coefficients will then be expressed by means of it. From the previous discussion on the uniqueness of the functions *k( zk) it is known,

however, that these functions are completely determined only if a condition is imposed on the rigid body rotation of the plate. In the present case of a homogeneous stress distribution the displacement components are

12

J P

xy

2 S

n

y Re [ s j g^ + s!, g

Q 2 )

] + u*

u = S

l l

x P

x

+ S

12

x

P

y

"

S

i o y P

(2.34a)

v = S

1 0

y p + S__ y p - S

1 0

x p

12 x 22 *y 12 xy

'1~2

v

"2

é

o

+ 2S

1;1

x Re [ s ^ f s , , g^' + s

1

g ^ ' ) l

+

v*

'1 *o

(2.34b)

They include a r i g i d body r o t a t i o n

l È - g ) = S

1 1

R e [

(

s - s

i

) (.

l g ( ( 1 )

- s g

( 2 )

) l

o

s

2

g

o '

J

Elimination of t h i s r o t a t i o n r e s u l t s in a f o u r t h e q u a t i o n f o r t h e c o e f f i ­

c i e n t s g and g* '

Re [ ( s | - s^) ( s

1

g

( ( 1 )

- s , g

( 2

> ) ] = 0

o 2

6

o

/ J

(2.35)

Solution of (2.33) and (2.35) yields the now fully determinate coefficients

g.

(k)

P„ " P„ s 2(s^ - s|)

Z

£ lx£

±_

" * \ k = 1, 2 £ = 3 - k (2.36)

and t h e i r conjugate values.

I t i s emphasized t h a t t h e r i g i d body r o t a t i o n may be given an a r b i t r a r y

v a l u e , s i n c e i t does n o t a f f e c t s t r e s s e s ( 2 . 3 3 ) - Iff f °

r

i n s t a n c e , the

r o t a t i o n i s represented by the expression

-z S ^ ( s

i

+ s_)* p

2 11 1 2 xy

the coefficients become

P- ~ P„ s'

g.

(k) _ "x *y °£ _ _{2(s^ - a») 2(s}_xy_ _{k - 8£)}

For the present work coefficients (2.36), corresponding with the zero rota­ tion, will be used. The rigid body translation of the plate is eliminated by putting C* = 0, k = 1, 2.

It is noted that the expressions for the rigid body rotation in the previous paragraphs are symmetric with respect to the complex material parameters s1

and s_. Consequently all equations (2.33) and (2.35) for the coefficients (k) -(k)

g and g are symmetric with respect to s1 and s_, resulting in a solu­

tion which can be written in the form

So*

0 = g

o < V

S

3-k>

k

"

X

'

2

This i m p l i e s t h a t a g e n e r a l , homogenous s t r e s s d i s t r i b u t i o n with properly

fixed boundary conditions i s represented by two s i m i l a r complex functions

V

2

k > ■ <

g

o

(

V

s

3-k

) } z

k

k = 1

'

2 (2

'

37)

The similarity of $..(z..) and $_(z_) will be discussed in a more general context in the next section.

In the case of the pin-loaded hole problem the plate contains one or more openings where the resultant forces on the contours of the holes are not zero. The complex functions ^^.(\) in that case are therefore multi-valued.

The problem of multi-valuedness will be treated in the next section as well.

2.5» The plate with one circular opening

The basic problem is considered where the external forces are given. The plate of thickness one is assumed as being infinite. It contains an opening with dimensions which are large compared to the plate thickness. The contour of the opening is loaded by forces with resultants R and R . The loads are reacted at infinity where the stresses are zero.

At this stage of the discussion of Lekhnitskii's theory it is not yet necessary to assume a circular opening and we will treat the circle as a special case of the more general ellipse. The contour of the ellipse is given in parameteric form:

x = d cos 6, y = b sin 6 (2.38)

cos e = SL-p

sin

e , °_=_ö

{239)

d and b are the length of the semi-axes of the ellipse along the X- and Y-axis respectively, o is the unit-circle

a = e = cos 9 + i sin 9 (2.40)

If the hole is circular, d = b = R is to be applied where R is the radius of the circle.

Using (2.38) the complex variables z, can be written on the contour of the ellipse as

d ■

zk =

- i ^ b d + iv^b

2 u ' 2 CT (2.41)

With the important property of the unit-circle

o = o "1 (2.42)

equation (2.41) can be considered as being quadratic in a with roots

z ± Vz* - u*b* - d*

° = d - i u ^ b k = 1 ' 2 ( 2'4 3 )

For r e a s o n s which become c l e a r l a t e r in t h i s s e c t i o n the complex v a r i a b l e s are introduced

z, ± Ul ~ tób2 ~ d*

c = — . K . * k = 1, 2 (2.44)

k d - i y , b ' These variables have two interesting features. First, on the contour of the

ellipse apparantly

^ = q2 = o (2.45)

d " i ukb , d + i J 1kb 1 ,

2

k p - <

k

l

1 +

d-^b ^1

(2

-"

6

>

It is noted that outside the contour of the ellipse <;.. * <;_.

The stresses in the plate at infinity are zero. Therefore the differentiated

complex functions *il(

zk

) i

n

the stress formulae (2.26) can consist of terms

with negative powers of z. only. Hence the general form of *i.(

z

iJ i

n

the

considered case must be

CD

0, (z, ) = A. In z.+ 2 a

( k )

z"

n

+ a

{ k )

(2.47)

k

v

k

K

k „ n k o

n=l

where A, I n z, are multi-valued. Their presence i n the complex functions i s

required since the loads on the contour of t h e e l l i p s e have r e s u l t a n t s R

and R .

y

With (2.46) the multi-valued functions can be transformed as follows:

» d + iu, b

0

d - iu, b

A

k

in z

k

= A

k

(in ^ - I 1 ^ ] f^/n ♦ In ( ^-)) (2.48)

n=l Tc

Using the same expression (2.46) the series in (2.47) is replaced by a

series of negative powers of C,

n=l n=l

Combining t h e s e r i e s of (2.48) and (2.49) and t h e constant p a r t s of (2.47)

and (2.48) the functions ^^.(z^.) &

r

e now completely expressed i n C, i n s t e a d

of z

k

,

V

z

k > ■

A

k

l n

Sc

+

*k<Sc>

+ A

o

k )

<

2

'50>

where

W " \

A

n

k)

C

{2

-™

*•((;,) will be called the holomorphic part of *k(zi.). according to

Muskhe-lishvili [51].

We now consider the boundary conditions (2.23) for the external forces on the contour of the ellipse, where C = o. After substitution of (2.50) they become, with the integrals written as powerseries of o

2 Re I [A. In o + ♦Ma)] = Z (a om + a o~m) (2.52a)

k

it K 4 in in

m=l

2 Re Z [ukAk In o + Pk^(o)] = Z (?>maa * £ o"B) (2.52b)

k m=l

These equations clearly show the advantage of using the complex variables C, . The boundary conditions for the external forces are completely defined in terms of the unit-circle a instead of the different complex variables z, and the reals x(s) and y(s). This important feature simplifies considerably

(k)

the evaluation of the constants A, and A and hence the analytical solu­ tion of the function $, (z. ) for stress problems in orthotropic plates with an elliptical hole. It is noted that the integration constants of (2.23) and

(k)

constants Av of (2.50) have been omitted in (2.52) since they are not

relevant for the solution of the stress problem.

The coefficients A. of the multi-valued parts A, In (;. can be evaluated rather quickly from the boundary conditions of the elliptical hole consider­ ing that (2.23) must yield the resultant forces R and R for a complete cycle along the contour of the ellipse. An additional condition is that the displacements of the points of the plate are single-valued. Hence (2.27) must result in identical displacements for the point on the contour of the ellipse represented by 8 and for the (same) point represented by 8 + 2k n, where k is an integer. These conditions determine four equations for A, and A, . With the positive direction of 8 opposite to the positive direction of s along the ellipse they follow directly from (2.27) and (2.22)

U1A1 " U1A1 + U2A2 " U2^2 = ° (2«53a)

V

1

A

1 " ^ 1

A

1

+ V

2

k

2 " V = 0 (2.53b)

y

l

A

l " ^ A

+ y

2

A

2 " ^2

A

2

=

" V

2

"

1

(2.54b)

In [ 1 ] e q u a t i o n s ( 2 . 5 3 ) and (2.5^) a r e presented in a somewhat different

form. E x p l i c i t expressions for A, can be taken from [13] or [ 1 ^ ] . They are

\ " 2*i (u* - u

k

) t |

( R

x

+

V y >

+

8r\l + l ) \

S

l l > * ^ '

U

'

HR

x

+

^*y

)] {2

*

55)

k = 1, 2 £ = 3 ~ k

With e q u a t i o n s (2.5*0 and In a = i 8 boundary c o n d i t i o n s ( 2 . 5 2 ) can be

evaluated i n t o

P R 6

2Re 2 oMa) = - Y ds - r

2

- (2.56a)

k J n 2tt

P R 8

2Re Z l y ^ f c ) = J X

R

ds + ^ T (2.56b)

The l e f t - h a n d s i d e s of (2.56) c o n t a i n h o l o m o r p h i c , hence s i n g l e - v a l u e d

f u n c t i o n s o n l y . In s p i t e of t h e m u l t i - v a l u e d p a r t s R G/2n and R 6/2n the

r i g h t - h a n d s i d e s of (2.56) must be s i n g l e - v a l u e d t o o . In t h e n e x t c h a p t e r

i t w i l l be shown t h a t t h e i n t e g r a l s i n (2.56) contain multi-valued p a r t s

e l i m i n a t i n g the multi-valued expressions R 9/2n and R 6/2n.

y * By introducing the designations

R 0

f

2 " " J

Y

n

ds

" 2f" <

2

-57a)

P R

e

\ = J

X

n

ds +

2T

(2

'

57b)

the boundary equations f o r t h e e x t e r n a l f o r c e s a r e now w r i t t e n i n t h e i r

f i n a l form

(Ufi - Pk) *'k(o) * (ufi - ük) ^(cr) ♦ (ug - ü£) ♦• (a) = ugf2 - ^ (2.58)

k = 1, 2 £ = 3 - k

In this form they are used for the evaluation of the still unknown, holomor-phic parts of 4>, (z ), or, what is equivalent, the still unknown coefficients A( k ) in (2.5D.

n

0?(a) are the boundary values of functions ♦ ' ( ! ; , ) , holomorphic o u t s i d e t h e

c o n t o u r of t h e h o l e and c o n t i n u o u s o u t s i d e and on the edge of the h o l e ,

while ♦?(

0

°) = 0. Applying Cauchy's i n t e g r a l s

r

*£»

da - *• (C )

2 n i

_{a - q, k}

_v

_^k'

1 0

. 0

r

do = 0

2tti J a - S

k

t o e q u a t i o n ( 2 . 5 8 ) r e s u l t s i n e x p r e s s i o n s for t h e holomorphic p a r t s of

1 I

y

C

f

2 "

f

l

♦ • ( c ) = , • , r 0

r d o

(

2

-59)

k

v

k' 2ni (u

g

- p

k

) J a - $

k

Except for f.. and f_ the complex functions (2.50) a r e now known. They a r e

solved as functions of C • Before s u b s t i t u t i o n of these functions in expres­

sions (2.26) for the s t r e s s e s they must be d i f f e r e n t i a t e d with respect to z.

dz ( V

z

k »

=

dT ( W J dT

=

♦k

( , :

k

) k

(2-60)

a z

k

K K

ac

k K K

az

k K K

^ _

u

^

b

2 _

d

*

The s t r e s s problem of a loaded hole in an i n f i n i t e p l a t e now i s reduced t o

t h e problem of d e t e r m i n i n g t h e two functions f.. and f_ from the load d i s ­

t r i b u t i o n on t h e c o n t o u r . I n t h e n e x t c h a p t e r t h e s e f u n c t i o n s w i l l be

evaluated for the case of a pin-loaded hole of c i r c u l a r shape.

The solution of the basic problem where the displacements of the c o n t o u r of

t h e h o l e a r e given i s not e s s e n t i a l l y d i f f e r e n t from the s o l u t i o n for given

loads. Instead of functions f.. and f_ for t h e load d i s t r i b u t i o n f u n c t i o n s

for the displacements must be derived. This problem will not be discussed here.

In the previous section was shown that a homogeneous stress distribution in (k) an anisotropic plate can be represented by two similar functions g* z, . Expressions (2.55) and (2.59) show that the functions

f^ In qk ♦ 0Jc(qk) k = 1, 2

representing a loaded hole in an otherwise unloaded anisotropic plate are also similar. The superimposed functions

4

M z

k

+ A

k

ln

Sc

+

*k<Sc> = V

Z

k >

k

■

X

'

2

represent a great number of boundary value problems and obviously are simi­ lar as well. Therefore these boundary value problems are reduced to the determination of only one function ♦. (z, ), k = 1 or 2, from the boundary conditions. This is an important result which simplifies stress calculations in anisotropic plates considerably.

N.B. The variables C are multi-valued and the choice of the + and - sign in (2.44) needs extra attention in numerical calculations. In a non-pub­ lished document Vuil [53] derived simple criteria for the choice of the sign, based on the continuity of t;, as functions of z, . With u, = a, + i{J, they can be summarized as follows:

The - sign must be chosen - on the Y-axis if

y Im (zfc - u^b* - d*) < 0

- for other points if

CLxy >. 0 and x Re (z, - u.b2 - d* ) < 0

or, if

3. THE PROBLEM OF A SINGLE, PIN-LOADED HOLE

The solution of the stress problem of a single, pin-loaded circular hole in an infinitely large orthotropic plate is presented in this chapter. It is an essential element in the solution of the problem of a row of pin-loaded holes, as will be shown in the next chapter.

In the first section of this chapter the various regions of the pin-plate interface where different boundary conditions exist are discussed. In the subsequent sections explicit expressions for the loads on the contour of the hole and for the complex functions ^^(2.) are derived. The numerical evalua­

tion of the problem requires a collocational procedure for the contact region between pin and plate. This procedure will be briefly discussed in the fourth section of this chapter.

3.1. The boundary conditions for the pin-plate interface

Figure 3-1 shows the different regions of the pin-plate interface. The contour of the hole is divided into:

- the region of separation, called the no-contact area, where the edge of the hole is free of tractions;

- the regions of slip, which are parts of the contact area between pin and plate. The difference in radial displacements of these parts of the edge of the hole with respect to the edge of the pin must be zero. In addition the friction force must obey a friction law;

- the region of no-slip, which is the rest of the contact area. Here the displacements of the edge of the hole are equal to the displacements of the corresponding part of the pin.

The extent of each of the regions of the interface between pin and plate depends on a number of parameters:

- the elastic properties of the pin and the plate, including the angle between principal material axes and load direction;

- the clearance between the pin and the edge of the hole;

- the pin displacement (the pin-loaded hole problem with clearance is non­ linear) ;

no-slip

Figure 3.1: The different regions of the pin-plate interface. V. is the pin displacement.

laminae directions

Figure 3.2: Positive directions of forces, angles, shear stress and re­ lative displacement of the plate material with respect to the pin.

- the loading situation (there is an essential difference between loading, unloading and possibly static loading).

The actual extent of the regions must be determined iteratively by the requirement that tensile tractions are physically excluded and by the requirement that the friction force never can exceed the value dictated by the friction law. The mathematical formulation of the boundary conditions of the various regions will be given in a subsequent section.

The first detailed work discussing the effects of pin-flexibility, clearance and friction simultaneously was presented in [16]. The variation of these parameters was studied for a single, pin-loaded hole in an infinite plate. As already indicated in the literature review of Chapter 1 the results showed neglectible differences between steel, aluminium and infinitely rigid pins. Since pin-flexibility complicates the solution of the stress problem greatly and since the present study emphasizes on finite geometry effects we will consider infinitely rigid pins only.

In [16] clearance between pin and plate was shown to be very important. Clearance introduces non-linearity since the contact area will grow with the load. As a result the stress distribution is load-dependent. In the no-clearance or push-fit situation as it was also studied in [16] the contact area appeared to be load-independent which resulted in a linear relation between the load and the pin-displacement. This was confirmed experimentally in [54,55]- The complex functions of the present study can be used for the calculations of stresses in the non-linear clearance cases as well, utiliz­ ing Mangalgiri's [21] method of inverse formulation. As already is discussed in Chapter 1 Mangalgiri treats the non-linear problem by applying succes­ sively an essentially linear analysis to various configurations of contact and separation, in this way simulating the growing contact area during loading. We will restrict the calculations to no-clearance cases only; for more details on the effects of clearance the reader is referred to the work of Hyer and Klang [16,17,18,19], Mangalgiri [21] and Naidu et al. [22~\.

Because of the presence of friction between the pin and the edge of the hole a distinction must be made between the solutions for

- the loading situation; the load increases,

- the unloading situation; the load decreases with reversed friction forces, - the static situation as it may occur at the end of the loading situation.

In the static situation the points of the edge of the hole do not displace with respect to the pin. Nevertheless friction forces will be present between the pin and the edge. The magnitude and the direction of these forces are unknown and therefore the boundary conditions cannot simply be defined. For that reason the static case will be excluded. It is noticed, however, that experimental values of stresses, which are measured during static situations, cannot agree exactly with theoretical stresses in the loading or unloading situation.

3.2. The expressions for the load functions f.

Transformation (2.44) for a circular opening |z*| = R becomes

Z

*/R ± n<m* - vi - 1

<k = 1 - i u , ( 3'1 }

In (3-1) the complex coordinates z* correspond with points of the region outside and on the circle with radius R. The dimensionless complex coor­ dinates z?/R, however, correspond with points outside and on the unit circle

|z| V I . Therefore it can be concluded from (3-1) that instead of solving the stress problem around a circular hole with radius R the problem can be solved for a hole with the radius taken unity, using the dimensionless coordinates

z = z*/R (3.2a)

and zk = z*/R (3-2b)

Hence in what follows the hole radius is assumed to be the unit and

Th e hole boundary is loaded radially by N(s) and tangentially by T(s). The positive directions of loads, shear stresses and relative displacement are shown in Figure 3«2. Since the pin-loaded hole problem is treated as a problem of linear elasticity the complex functions *i.(zk) corresponding to

N(s) and T(s) can be derived separately and simply superposed.

The problem concerning the radial load N(s) will be treated in more detail. It is assumed that pin and hole have contact for 0 _< 9 _< n only.

The radial load is a function of 9, so the expression for N (s) has to obey the next requirements:

k

N(s) = - p Z a sin n9 0 < 9 < n (3.4a) n=l,2

N(s) = 0 n < 9 < 2n (3-4b) p has the dimensions of a stress. In special cases it is the classical

bearing stress, as will be explained later. The constant 4/n has been added to provide a resultant load per unit thickness of magnitude 2p in those cases. The coefficients a will be called load coefficients.

n

An expression satisfying conditions (3-4) can be found by multiplying a sine series, continuous on the whole contour, by a step function

. - » . „ 1 f o r 0 < 9 < n 1 2 .. s i n m9 P + - Z = " m=l,2 m 0 f o r n < 9 < 2n r e s u l t i n g i n 4 r l " N(s) = - p 7 Z a s i n n9 n = l , 2 + - f Z — + Z ' a ( — + — ) cos m6}] 0 < 9 < 2n ( 3 . 5 ) n *■ A _ n n vn-m n+m' ,J - - \J ->i n = l , 3 ni.n

The dash i n (3«5) i n d i c a t e s t h a t i n t h e d o u b l e s e r i e s Z o n l y odd c o m b i n a -m,n

t i o n s of m a n d n h a v e t o b e t a k e n . E x p r e s s i o n (3.5) i s c o n t i n u o u s on t h e e n t i r e c o n t o u r of t h e h o l e . The t e r m s w i t h odd v a l u e s o f n r e p r e s e n t t h e symmetric p a r t o f N ( s ) , t h e terms w i t h even n t h e asymmetric p a r t .