Stresses Around Pin Loaded Holes in

Mechanically Fastened Joints

ByNeville A. Tomlinson, PhD

Howard UniversityWashington DC

January 2007

Abstract

• An analytical method for determining the stress distribution in pin loaded orthotropic plates is presented based on the complex stress function approach. The method assumes that the contact boundary at the pin-plate interface is unknown a priori and must be determined as part of the solution. It is further assumed that the pin is rigid, clearance exists between pin and plate, and the coefficient of friction remains constant throughout the contact zone. The boundary conditions at the pin-plate interface are specified in terms of the unknown contact angle and a trigonometric series used to represent the displacement field in the contact zone. Numerical results are presented for normal, tangential and shear stresses on the cavity for different lay-ups of graphite/epoxy laminates.

Introduction

• The increasing use of composite materials has caused engineers to increase their efforts to understand the stress fields associated with these materials.

• One application that has received much attention is the stresses associated with the mechanical joining of composites

• Mechanical joining includes bolted, riveted and pinned joints which are relatively easy to assemble and disassemble.

• These joints are however prone to high stress concentrations which occurs in the vicinity of the hole, which is undesirable, and is often the source of premature failure.

Schematic of Pin joint

pin

plate

Fig 1.

Problem Definition

loadpinP

anglecontac

disppinu

clearance

radiuspinr

radiusholer

B

p

====

==

θ

λ.0

Fig. 2

Exaggerated view of deformed hole by rigid pin

Fig 3

The contact equation

• The equation that governs an ellipse can be written as

Consider triangle BAD in Fig.3. Point B has coordinates

2 2

2 21

x y

a b+ =

(1)

( )0 cospx u rλ ψ= + + (2)

' sin By R θ= (3)

The contact equation

• From the ellipse

• Substituting (2-4) in (1) yields

• Equation (5) is the non-linear contact equation.

rb

rua p

≅

++= 0λ(4)

( ) ( ) ( ){ }( ) ( ){ } ( )

222 2

0 0

2 22 2 2 20 0

cos 1 cos *

cos 1 cos 0

B p B

B B p p

u r u

r u r r u r

λ θ λ θ

θ θ λ λ

+ + − + −

+ − + + − + + = (5)

Boundary conditions at the pin-plate interface

• The b.c. can be described as

• , •

• , • •

0u u= 0v = 0θ =

1 0u uα=Bθ θ=

0rθτ =0rσ =

Bθ θ≥ ±

0( ) cos sinu u vθ θ− = B Bθ θ θ− ≤ ≤

(6)

(7)

(8)

(10)

(9)Bθ θ≥ ±

Intrduction of Friction

• Friction is introduced into the constitutive model by assuming a Coulomb frictional relation as

• Work done by shear can be written as

• Using (10) and (11) and considering symmetry yields

rf

r

θτµσ

= − (11)

B

B

s rW rdθ

θθ

τ θ−

= ∫ (12)

0 0

B B

rr frd rdθ θ

θτ θ µ σ θ= −∫ ∫ (13)

Displacement field along hole boundary

• Assume displacement in the form

• This tree trems trig. series was chosen to facilitate the simultaneous solution of equations (8), (9) and (13)

• To determine the constants in (14) an additional condition was introduced which is described as

1 2 3

1 2 3

cos 2 cos 4 cos6

sin 2 sin 4 sin 6

u u u u

v v v v

θ θ θθ θ θ

= + += + +

(14)

2 0u uα= (15)

Coefficients of u

.

( )

4 2

1 0 0 0 0 21

0 2 0

sec 2 222 3 564 1 2

BB

B B

B BB

co Sec u u u cos u cosu

u cos u coscos

θθ α θ α θα θ θθ+ + −

= − ++

( ) ( )

4

0 1 0 0 2

2 0 0 2

0

2sec2

2 2 38 1 2cos 1 2cos 2cos 2

4

BB

B BB B B

B

u u u cosco

u u cos u cos

u cos

θ α α θθ α θ

θ θ θθ

+ − − = + − + + + +

( )

4 2

0 1 0 0

3 0 2 0 2

0

sec sec2 2

2 2 264 1 2cos cos 2 cos3

3

B BB

B BB B B

B

u u u cosco

u u cos u cos

u cos

θ θ α θα θ α θ

θ θ θθ

− − − − = + + + + + −

(16)

(17)

(18)

Coefficients of v

.1 0 0 1 2 3 2 0 0 1 2 3

2 0 0 1 2 3 2 0 1

2 3 2 0 1

1

9 7 4 2 ( 2 12 12 9 7 )cos2

3 3( 4 4 9 11 11 )cos 4 cos 7 cos

2 2 2

3 311 cos 12 cos 6 cos 2 6 cos 2

2 2

v ( 1)

B

B B B

B BB B

u u u u u u u u u u

u u u u u u u

u u u u

W

θα α

θ θ θα α

θ θα θ θ

− + + + + − − + + + + − − + + + − + +

+ − + +

=

2

3 2 0 0 1 2

3 2 0 0 1 2 3

0 1 2

8 cos 2

5 5 5 510 cos 2 2 cos 6 cos 7 cos 5 cos

2 2 2 2

55 cos 2 cos3 6 cos3 7 cos3 4 cos3 2 cos3

2

7 76 cos 5 cos 4 cos

2 2

B

B B B BB

BB B B B B

B B

u

u u u u u

u u u u u u

u u u

θ

θ θ θ θθ α

θα θ θ θ θ θ

θ θ

+

− − + + + − − + + +

− + + 3 0

1 2 3 1 2

3 2 3 3

7 73 cos 2 cos 4

2 2

9 93 cos 4 4 cos 4 4 cos 4 cos 3 cos

2 2

9 114 cos cos5 3 cos5 cos

2 2

B BB

B BB B B

B BB B

u u

u u u u u

u u u u

θ θθ

θ θθ θ θ

θ θθ θ

+ − + + + + + + + + +

(19)

Coefficients of v

.6 4 2

2

cos s s4 4 2

1

2048 1 2cos2

B B B

B

ec ec ec

W

θ θ θ

θ

− = +

Coefficients of v

.

(20)

1 0 2 0 0 1 2 3 0 1 2 3

2 0 0 1 2 3 0

1 2 3 2 0

0 1

2

4 5 6 6 ( 10 10 11 12 )cos2

3( 8 4 11 10 9 )cos 12 cos

2

3 3 312 cos 9 cos 6 cos 6 cos 2

2 2 2

6 cos 2 10 cos 2

( 2)

B

BB

B B BB

u u u u u u u u u u

u u u u u u

u u u u

u u

v W

θα α

θα θ

θ θ θ α θ

θ θ

− − + + + + − + + + + − − + + + − +

+ + − −

+=

2 3 0

1 2 3 2 0

0 1 2 3 0

1 2 3

58 cos 2 6 cos 2 8 cos

2

5 5 58 cos 7 cos 6 cos 2 cos3

2 2 2

72 cos3 5 cos3 6 cos3 6 cos3 2 cos

2

7 7 72 cos 4 cos 6 cos

2 2 2

B

B B BB

BB B B B

B B B

u u u

u u u u

u u u u u

u u u

θθ θ

θ θ θ α θ

θθ θ θ θ

θ θ θ

+ + − + + + − −

+ + + − + + +

1

2 3 2 3 3

cos 4

9 92 cos 4 4 cos 4 cos 2 cos cos5

2 2

B

B BB B B

u

u u u u u

θ

θ θθ θ θ

+ + + + + +

Coefficients of v

.6 2 2

2

cos s s4 4 2

2

512 1 2cos 1 2cos 2cos2 2

B B B

B BB

ec ec ec

W

θ θ θ

θ θθ

=

+ + +

Coefficients of v

. 1 0 2 0 0 1 2 3

2 0 0 1 2 3

2 0 0 1 2 3

2 0 0 1

3 2 3

4 7 10 8 5

( 6 14 18 15 11 )cos2

( 6 10 15 13 11 )cos

3 3 32 cos 8 cos 11 cos

2 2 2

3 3( 3) 11 cos 10 cos

2 2

B

B

B B B

B B

u u u u u u

u u u u u

u u u u u

u u u

v W u u

α αθα

α θθ θ θα

θ θ

− − + + + +

− − + + + + − − + + + −

− + + = + −

2 0

0 1 2 3

0 1 2

3 1 2 3

2 3 3

2 cos 2

4 cos 2 6 cos 2 8 cos 2 10 cos 2

5 3 32 cos 3 cos 5 cos

2 2 2

38 cos cos3 3 cos3 5 cos3

2

7 3cos 3 cos cos 4

2 2

B

B B B B

B B B

BB B B

B BB

u

u u u u

u u u

u u u u

u u u

α θ

θ θ θ θθ θ θ

θ θ θ θ

θ θ θ

−

+ + + −

+ + + + + + +

+ +

(21)

Coefficients of v

.6 2

2

cos s4 4

33

2048 1 cos cos 1 2cos 2cos2 2 2

B B

B B BB

ec ec

W

θ θ

θ θ θθ

− = + + + +

Determination of stress functions

• Lekhnitskii (1) has shown that if the displacements at the hole edge can be written in the form

• Then the stress functions can be written as

0

0

m mm m

m

m mm m

m

u

v

ϑ ϑ ς ϑ ς

ρ ρ ς ρ ς

−

−

= + +

= + +

∑∑

(22)

( ) ( ) ( )

( ) ( ) ( )

2 4 61 1 1 1 2 1 2 1 2 2 2 2 1 3 2 3 2 1

2 4 62 2 2 1 1 1 1 2 2 1 2 1 2 3 1 3 1 2

1( ) ln

21

( ) ln2

A u q iv p u q iv p u q iv pD

B u q iv p u q iv p u q iv pD

φ ξ ξ ξ ξ ξ

φ ξ ξ ξ ξ ξ

− − −

− − −

= + − + − + −

= − − + − + − (23)

Definition of stress function terms.where

( )( ) ( ) ( )

1 1 1 2 1 2 12 22 1 2 1 2

1 1 2 1 1 2

a aPA

i

µ µ µ µ µ µ µ µ µ µπ µ µ µ µ µ µ

+ + −=

− − −

( )( ) ( ) ( )

2 2 2 1 2 1 12 22 1 2 1 2

1 1 2 1 1 2

a aPB

i

µ µ µ µ µ µ µ µ µ µπ µ µ µ µ µ µ

+ + −=

− − −

2 2 2(1 )

(1 )k k k

kk

z z R

r i

µξ

µ± − +

=−

1,2k =

1 2 2 1D p q p q= −

221 12 1 16

1

aq a aµ

µ= + − 22

2 12 2 262

aq a aµ

µ= + −

Determination of stresses

.

Where

( ) ( )2 ' 2 '1 1 1 2 2 22x eR z zσ µ φ µ φ = +

( ) ( )' '1 1 2 22y eR z zσ φ φ = +

( ) ( )' '1 1 2 2 22xy eR z zτ µφ µ φ = − +

(24)

(25)

(26)

' k kk

k kz

φ ξφξ

∂ ∂=

∂ ∂

Stress Transformation

• Transformation relation from Cartesian to polar coordinates

2 2

2 2

2 2

cos sin 2sin cos

sin cos 2sin cos

2sin cos 2sin cos cos sin

r x

y

r xy

θ

θ

σ θ θ θ θ σσ θ θ θ θ στ θ θ θ θ θ θ τ

= −

−

(27)

Complex stresses

.1 2 1 2 1 1 1 1 1 1 2 1 1 2

1 2 1 2 1 1

1 1 2 1 1 2 2 1 1 2 1 11 2

1 2 1 2 1 1 1 1

1 2 1 2 1

( ) 1sin

2

( ) 1cos

2

1

2

2 Rer

u q iv p u q iv p u q v pA B

iu q v pir iDr

u q i v p q u i v pA B

iu q v p iq u v piR iDr

u q iv p u q

iDr

µ µθ

µ µ

µ µ µ µµ µ θ

σ

− − + − − + + + + − − + − + + + + + − −

− + ++

=

1 1 1 1 2 1 1 1 2 1 1 2

2 1 1 2 2 2 2

2 1 2 1 1 2 2 1 2 2 2 1 2 1 2

1 1 2 1 1 2 1 1 2 1 1 2 1 2 1 2 1 1

1 1 1 2 2 1 2 2

2 2 1 2 2

2 2 sin 3

2 2 2 2 2 2

12 2

22 2

iv p iu q v p iu q

v p u q iv p

u q iv p i u q p v iu q v p

u q i v p u q iv p iu q v p iu q

v p q u i p viDr

u q i v p

µ µ µµ θ

µ µ µµ µ µ µ

µ µµ µ

− − − + + + − − + − − + +

− − + − − ++ + + −

− + 1 2 2 2 2 2 1 2 1

2 2 2 2 1 2 2 1 1 2 2 2 1 2

2 1 2 2 1 2 3 2 3 2 3 1 3 1

1 3 2 1 3 2 2 3 1 2 1 1

2 2 1

cos3

2 2 2 2

2 2 2 2 2 21

2 2 3 3 3 3 sin 52

3 3 3 3

2 21

2

iu q v p iu q v p

u q iv p q u iv p i q u v p

iu q v p u q iv p u q iv piDr

i u q v p i u q v p

u q i

iDr

θ

µ µµ µ θ

µ µ µ µµ µ

+ + − − − + + − − −

+ + + + − − + − − + +

−+

1 2 2 2 1 2 2 2 1 2 1 2

2 2 2 2 2 1 2 1 1 3 2 1 3 2

2 3 1 2 3 1 3 2 3 2 3 1 3 1

3 2 3 2 3 1 3 1 1 3 2 1 3 2

2 3 1 2 3 1

2 2 2

2 2 2 2 3 3 cos5

3 3 3 3 3 3

3 3 3 3 3 31

3 32

v p q u i v p i p v

iu q v p iu q v p u q i v p

u q i v p iu q v p iu q v p

u q iv p u q iv p i u q v p

i u q v piDr

µ µ µµ µ θ

µ µµ µ

µ µ

− + + − − + + + − − + + + − − − + + − − −

+ + +

1 3 2 1 3 2 2 3 1 2 3 1 3 2

3 2 3 1 3 1

sin 7

3 3 3 3 31cos 7

3 3 32

u q i v p u q i v p iu q

v p iu q v piDr

θ

µ µ µ µθ

− − + − + − + +

(28)

Complex stresses

. 1 2 1 2 1 1 1 1 1 1 21 2

1 1 2 1 2 1 2 1 1

1 2 1 2 1 1 1 1 1 2 1

1 2 1 1 1 2 1 1 2

1 2 1 2

( ) 1sin

2

( ) 1cos

2

1

2

2 Rer

iu q v p iu q v p u qA B

i v p iu q i v pir iDr

u q iv p q u iv p iu qA B

v p iq u v pir iDr

iu q v p i

iDr

θ

µµ µ θµ µ µ

µθ

µ µ µ

τ

+ − − + + + − − + − + + − + + + + + − −

+ −+

=

1 1 1 1 1 2 1 1 2 1

1 1 2 2 1 1 2 2 2 2 2 1

2 1 1 2 2 1 2 2 2 1 2 2 1 2

1 1 2 1 2 1 1 1 1 1 2 1 1 2 1

1 1 2 1 1 2 2 2 2 2 2 1 2 1

2 2 2 sin 3

2 2 2 2 2

12 2 2 2

2

u q v p u q iv p

u q i v p iu q v p iu q

v p u q i p v u q iv p

u q iv p u q iv p iu q v p

iu q v p q u ip v u q iv piDr

µ µµ µ θ

µ µ µ µµ µ

µ µ

− − + + − + + + − + − − − − + + − − − +

+ + − + + −+ 2 2 1 2 2 1 2 1 2 2 1 2

2 2 2 2 1 2 2 1 1 2 2

2 1 2 2 1 2 2 1 2 1 3 2

1 3 2 2 3 1 2 3 1 3 2 3 2

3 1 3 1

cos3

2 2 2 2

2 2 2 2 2

2 2 2 31sin 5

3 3 3 3 32

3 3

2

1

2

iu q v p iu q v p

iu q v p iq u v p q u

iv p u q iv p u q

i v p u q i v p iu q v piDr

iu q v p

iDr

θµ µ µ µ

µµ µ µ µ

θµ µ µ

+ − −

+ − − − + + − + − + − + + + − − −

+

2 2 2 2 1 2 2 1 2 2 1

1 2 2 2 2 2 2 1 2 3 2 3 2

3 1 3 1 1 3 2 1 3 2 2 3 1

2 3 1

1 3 2 1 3 2 2 3 1 2 3 1

3 2 3 2 3 1

2 2 2 2

2 2 2 3 3cos5

3 3 3 3 3

3

3 3 3 31

3 3 3 32

u q iv p q u iv p iu q

p v iu q v p u q iv p

u q iv p i u q v p i u q

v p

u q i v p u q i v p

iu q v p iu q viDr

µµ µ µ

θµ µ µ

µµ µ µ µ

+ + − − − + + − + + − + + − − − + + − +

++ − − 3 1

3 2 3 2 3 1 3 1 1 3 2

1 3 2 2 3 1 2 3 1

sin 7

3 3 3 3 31cos7

3 3 32

p

u q iv p u q iv p i u q

v p i u q v piDr

θ

µθ

µ µ µ

− + + − − + − + +

(29)

Complex stresses

.2 2

1 2

1 2

( sin cos ) ( sin cos )2 Re

sin cos sin cos

A Bi

rθµ θ θ µ θ θ

σθ µ θ θ µ θ

+ +−= + − −

( ) ( )( ) ( )( ) ( )

1 2 1 2

2 2 2 2

3 2 3 2

( 2) cos 2 sin 21

( 4) cos 4 sin 42

( 6) cos 6 sin 6

u q iv p i

A A u q iv p iD

u q iv p i

θ θ

θ θ

θ θ

− − −

= + + − − − + − − −

( ) ( )( ) ( )( ) ( )

1 1 1 1

2 1 2 1

3 1 3 1

( 2) cos 2 sin 21

( 4) cos 4 sin 42

( 6) cos 6 sin 6

u q iv p i

B B u q iv p iD

u q iv p i

θ θ

θ θ

θ θ

− − −

= − + − − − + − − −

(30)

Real stresses

• By defining two real parameters

• And by defining

221 2

11

ak

aµµ=− = (31)

661 2 12

11

( ) 2( )a

n i ka

µ µ υ= − + = − + (32)

12 21 22 66(1 )g a a kυ υ= − + (33)

Real stresses

.( )1 2 3 4 5cos cos3 cos5 cos 7r H H H H Hσ θ θ θ θ= + + + +

1 2 3 4 5( )sin sin 3 sin 5 sin 7r I I I I Iθτ θ θ θ θ= + + + +

1 2 3

4 5 61

cos cos cos 2 cos cos 42

cos cos 6 cos cos8 cos cos10

E

rEθ

θ

θ θ θ θ θσ

θ θ θ θ θ θΓ + Γ + Γ +

= Γ + Γ + Γ

Stress coefficients

.( )

( )

1 2 22 1 12 1 1 11 111

22 1 2 12 1 2 1 23

11 11 2 11 1

22 12 2 22 12 22 3

24 12 11

11

1(1 ) ( ) ( )

2 2

( ( 1) 2 ( 1)) 2 21

2 2 ( ) ( )

(2 2 ) ( 3 3 3 )1

( 2 2

PH H a u n ka u v ka v n k

r a grk

a u n u n ka u u v vH

kga r ka v n k ka v n k

a a k n u a a k a n u

H a k a kkga r

π+ =− + + − + − +

− + + − − − − =−

+ + − − + − + − − −

= + − −

( )

( ) ( ) ( )( )

( ) ( )( ) ( )

11 2

212 11 11 3

25 22 12 22 3 12 11 11 3

11

1 2 22 1 12 1 1 11 111

22 1 2 12 1 2 1 2

311 11

2 )

( 3 3 3 )

1(3 3 3 ) ( 3 3 3 )

11

2 2

1 2 1 2 21

2 2

a kn v

a k a k a kn v

H a a a n u a k a k a kn vkga r

PI I a u n ka u v a v k n k

r kga r

a u n u n a k u u v vI

kga r a k

π

+ + − − −

= + − + − − +

+ =− − + + + + +

− − + − + − +=

− ( ) ( )( )

( )

2 1

22 12 22 2 22 12 22 3

4 2 211 12 11 11 2 12 11 11 3

25 22 12 22 3 12 11 11 3

11

( 2 2 2 ) ( 3 3 3 )1

(2 2 2 ) ( 3 3 3 )

1( 3 3 3 ) (3 3 3 )

v n k v n k

a a a n u a a k a n uI

kga r a k a k a kn v a k a k a kn v

I a a k a n u a k a k a kn vkga r

+ + − − − + + − − − +

= + − + − − −

= − − + + + −

Stress coefficients

.

All constants not shown can be obtained from [2] Appendix A

( )( )( )

2 4 2 2 2 212 22 12 22

1 2 2 422 12 22

cos 2 cos sincos

2 2 2 sin

a k a a k a kP

a a k a k n

θ θ θθ

π θ

+ + − Γ = + + + −

( ) ( )( )( )22 22 1 3 2 3

11

124

8a k n u k

a gkΓ = Β + − + + Β + Β

( )( )( )( )( )33 22 1 2 11 1 3 3

11

13

16a k a k v v

a gkΓ = Μ + Μ + − − + Μ

( ) ( )( )( )( )3 34 1 11 2 1 3 2 3

11

12 3

8k a k v n n v v

a gk

−Γ = ϒ + − + − + ϒ + ϒ

Stress coefficients

.

All constants not shown can be obtained from [2] Appendix A

( )( )5 1 2 311

1

16k

a gkΓ = Φ + Φ − Φ

( )( )( )6 22 1 2 311

1

8a k

a gkΓ = Ζ + Ζ +Ζ

( ) ( )( ) ( )( )

22 32 27

11 11 3 12 3 3

113 1 2

16

a n uk k n

a gk k a k n v a u v

− + + Γ = + + − − + − +

Determination of

0 0

B B

rr frd rdθ θ

θτ θ µ σ θ= −∫ ∫

21,0 , ααu

0rσ =0rθτ =

Bθ θ=

),(

),(

),(

22

11

00

fB

fB

fBuu

µθααµθααµθ

=

=

=

Determination of

),(00 fBuu µθ=

Bθ

( ) ( ) ( ){ }( ) ( ){ } ( )

222 2

0 0

2 22 2 2 20 0

cos 1 cos *

cos 1 cos 0

B p B

B B p p

u r u

r u r r u r

λ θ λ θ

θ θ λ λ

+ + − + −

+ − + + − + + =

Determination of Stresses

Substituting the values of and into

(16-21) yields

These values completely determines

21,0 , ααu

),,,(

),,,(

),,,(

),,,(

),,,(

),,,(

21033

21022

21011

21033

21022

21011

B

B

B

B

B

B

uvv

uvv

uvv

uuu

uuu

uuu

θααθααθααθααθααθαα

======

Bθ

),,,,,,(

),,,,,,(

),,,,,,(

321321

321321

321321

θττθσσθσσ

θθ

θθ

vvvuuu

vvvuuu

vvvuuu

rr

rr

===

Results

Results

.

Radial stress for plate A (±45s )

Results

Shear stress for plate A (±45s )

Results

.

Tangential or hoop stress for plate A (±45s )

Results

.

Results

.

Radial stress for plate E ( [02/±45]s )

Results

.

Shear stress for plate E ( [02/±45]s )

Results

.

Tangential or hoop stress for plate E ( [02/±45]s )

Results

.

Conclusion

. A method has been presented for determining the stresses in pin loaded orthotropic plates.

. The method can be used to predict the stresses in joints with varying degrees of clearances including the case of perfectly fitting pins where clearance is zero.

. Although developed for use with orthotropic plates, the method can be used to evaluate the stresses in isotropic plates as well.

Recommendations

. Better prediction of contact angle

. Further investigation into the no slip zone and its effect on stresses

. Investigation into the development and use of a non-Colulombic frictional model

. Use of non-trigonometric displacement functions

. Experimental inquiry

References

• Lekhnitskii, S. G.,”Anisotropic Plates, English Edition (Translated by S. W. Tsai and . Cheron), Gordon and Beach, London (1968).

• Tomlinson, N. A. “ Stresses Around Pin Loaded Holes in Mechanically Fastened Joints” Thesis Howard University, Washington, DC.

• Zhang, Kai-Da and Ueng, Charles E. S., “Stresses Around a Pin-Loaded Hole In Orthotropic Plates”, Journal of Composite Materials, Vol. 18, Sept. 1984 pp. 432-446.

• de Jong, Th., “Stresses around Pin Loaded Holes in Orthotropic Materials”, Mechanics of Composite Materials Recent Advances, Pergamon Press, pp. 339-353, 1982.

• Hyer, H. W., Klang, E. C., “Contact Stresses in Pin-Loaded Orthotropic Plates”, Int. Journal of Solids and Structures, Vol. 21, 9, pp.957-975, 1985.