Determination of Contact Stress Distribution in Pin Loaded Orthotropic Plates

Neville A. Tomlinson

Department of Mechanical Engineering

Howard University

Washington D C

December 2006

Outline Introduction

Joints Orthotropic materials The pin loaded plate

Orthotropic Plate Theory Problem definition Mathematical Formulations For the Pin Loaded Plate with Clearance and

Friction The contact equation Boundary conditions at the pin-plate interface Friction Trigonometric displacement functions Stress-stress functions Stress Equations

Results Conclusions Recommendations Contribution Acknowledgements

Introduction

Joints Important form of mechanical joining of structural elements

Reason Ease of assembly and disassembly

Joint Types Bolted Joints, riveted joints, welded joints, pin joints

Pin Joint For plane stress analysis pin joint is representative of joint

analysis

Common Joint Materials Iron, Steel, Aluminum (isotropic, strong, heavy and/or expensive) Fiber reinforced Composites (orthotropic, can design strength,

light, relatively inexpensive)

Orthotropic Materials

Anisotropic materials Material having different properties in different directions

Orthotropic materials Thin anisotropic materials whose transverse stresses are

considered negligible and therefore transverse properties are ignored

Considered in plane problems of elasticity

Simplifies the analysis

The Pin Loaded Plate

In studying pin loaded joints only the plate is considered in this analysis

Assumption: pin is rigid

Plate

Pin

hole

Schematic of a Pin Loaded Plate

P

P/2 P/2

clevis

Orthotropic Plate Theory

Equilibrium Equation (no body force)

Constitutive Equation

Compatibility Equation

0

0

xyx

xy y

x y

x y

11 12 16

12 22 26

16 26 66

x x

y y

xy xy

a a a

a a a

a a a

2 22

2 20y xyx

x yy x

Orthotropic Plate Theory (Continued)

Airy Stress Function

Governing Partial Differential Equation for Orthotropic Material

2

2x

F

y

2

2y

F

x

2

xy

F

x y

4 4 4 4 4

22 26 12 66 16 114 3 2 2 3 42 (2 ) 2 0

F F F F Fa a a a a a

x x y x y x y y

Orthotropic Plate Theory (continued)

Complex Stress Function

Characteristic Equation

Roots

Re-writing Governing Equation

k = 1 to 4

Solution for F (invoking rule of complex addition)

( , ) ( ) x yF x y F x y e

4 3 211 16 12 66 26 222 (2 ) 2 0a a a a a a

1 2 3 1 4 2, , ,i i i i

( 0k kF x yx y

1 1 2 22Re[ ( ) ( )]F F z F z

Orthotropic Plate Theory (continued)

By defining two complex functions

and where

In terms of complex stress functions stresses become

11 1

1

( )dF

zdz

22 2

2

( )dF

zdz

k k

k k

z x y

z x y

2 ' 2 '1 1 1 2 2 22Re ( ) ( )x z z

' '1 1 2 22Re ( ) ( )y z z

' '1 1 1 2 2 22Re ( ) ( )xy z z

Problem Definition

hr

region of no contact

region of contact

Plate thickness = unity

Hole radius =

Clearance =

Pin radius =

Pin force =

Pin displacement =

Hole center = A

Contact point = B

Contact angle =

( 0.02)hr

p hr r

P

0u

B

Mathematical Formulations For the Pin Loaded Plate with Clearance and Friction

Equation of ellipse

Point B has coordinates

2 2

2 21

x y

a b

0 cospx u r

'sin By R

222 2

0 0

2 22 2 2 20 0

cos 1 cos *

cos 1 cos 0

B p B

h B B p h p

u r u

r u r r u r

Contact Equation

Boundary Conditions at the Pin-plate Interface

1 0u u0u u0v

0 B

0r 0r B

0( ) cos sinu u v B B

Friction Assuming Coulomb Frictional relation,

-ve sign: shear opposes the direction of relative displacements between the pin and the plate

Introduce Friction into the model by the relation

r f r

0 0

B B

h hrr fr d r d

f Constant Coefficient of Friction

Trigonometric displacement

function

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

0 0

0

0

0B B

rB

r

h h Brr fr d r d

y

2 0 ,2Bu u

Traction conditions on the hole boundary requires that

Assumed three terms displacement field along the contact region as

Trigonometric Displacement Function (continued)

0 0 1 2 3

1 0 1 2 3

2 0 1 2 3

cos 2 cos 4 cos 6

cos cos 2 cos3B B B

B B B

u u u u

u u u u

u u u u

B

/ 2B

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

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

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

Trigonometric Displacement Function (continued)

1 1 0 1 2

2 2 0 1 2

3 3 0 1 2

( , , , )

( , , , )

( , , , )

B

B

B

v v u

v v u

v v u

0( ) cos sinu u v

0 1 2 3

1 2 3

( ( cos 2 cos 4 cos 6 ))cos

( sin 2 sin 4 sin 6 )sin

u u u u

v v v

Using the following relation,

Evaluate at , ,2 4B

B

Determination of the Stress Functions

and Stresses

0

0

m mm m

m

m mm m

m

u

v

0 0

, tan

,

cons ts

rigid body displacement

unit hole radius

1 1 0 1 1 2 1 2 2 2 2 2 121

1 1 1( ) ln ( )

2m

m mm

z A A q p ibq ap q pD D

2 2 0 2 1 1 1 1 1 1 1 1 222

1 1 1( ) ln ( )

2m

m mm

z B B q p ibq ap q pD D

0 0,A B Constants

Determination of the Stress Functions and Stresses (continued)

k

2 2 2(1 )

(1 )k k h k

kh k

z z r

r i

cos sinie i

1 1

cos2

1 1

sin2i

mapping function

Introduce unit circle to satisfy boundary conditions

Determination of the Stress Functions and Stresses (continued)

2 2 4 4 6 6

1 2 32 2 2u u u u

2 2 4 4 6 6

1 2 32 2 2v v v v

i i i

0

0

m mm m

m

m mm m

m

u

v

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

Comparing coefficients yields

Determination of the Stress Functions and Stresses (continued)Stress stress function relationship

Stress transformation from x,y system to polar system

2 ' 2 '1 1 1 2 2 2

' '1 1 2 2

' '1 1 2 2 2

2Re

2Re

2Re

x

y

xy

z z

z z

z z

2 2

2 2

2 2

cos sin 2sin cos

sin cos 2sin cos

sin cos sin cos cos sin

r x

y

r xy

2 2' '1 1 1 2 2 2

2 2' '1 1 1 2 2 2

'1 1 1 1

'2 2 2 2

2Re sin cos sin cos

2Re sin cos sin cos

sin cos cos sin2Re

sin cos cos sin

r

r

z z

z z

z

z

Determination of the Stress Functions and Stresses (continued)

1 2 3 4

1 2 3 411

5 6 7

1 2 3 4

cos cos3 cos5 cos 7

cos cos cos 2 cos cos 42

cos cos 6 cos cos8 cos cos10

sin sin 3 sin 5 sin 7

r

h

r

H H H H

a E

r

I I I I

221 2 1 2

11

661 2 1 2 12

11

( ) ( ) 2( )

ak

a

an i i k

a

, , ,H I E Functions of material parameters, the displacement coefficients and the hole radius

Determination of

Displacement Parameters obtained as functions of

0 1 2, ,u

0 0

0

0

B B

r B

r B

h hrr f

at

at

r d r d

1 2 3 4

1 2 3 4

1 2 3 4

0

1 2 3 4

0

cos cos3 cos5 cos7 0

sin sin 3 sin 5 sin 7 0

sin sin 3 sin 5 sin 7

cos cos3 cos5 cos7

B

B

B B B B

B B B B

h

f h

H H H H

I I I I

I I I I r d

H H H H r d

, , , ,ij B fa P

Determination of B

222 2

0 0

2 22 2 2 20 0

cos 1 cos *

cos 1 cos 0

B p B

h B B p h p

u r u

r u r r u r

0 , , ,p hu r r

, , ,ij fa P Given

Substitute

Re-stated as

Determination of Determination of allows the determination of the

displacement coefficients

0 1 2, , ,B u

1 2 3 1 2 3, , , ,u u u v v and v

4 2

1 0 0 0 0 21

0 2 0

4

0 1 0 0 2

2 0 0 2

0

3

sec 2 222 3 564 1 2

2sec2

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

4

sec2

BB

B B

B BB

BB

B BB B B

B

B

co Sec u u u cos u cosu

u cos u coscos

u u u cosco

u u cos u cos

u cos

co

u

4 2

0 1 0 0

0 2 0 2

0

sec2

2 2 264 1 2cos cos 2 cos3

3

BB

B BB B B

B

u u u cos

u cos u cos

u cos

1 1 0 1 2

2 2 0 1 2

3 3 0 1 2

( , , , )

( , , , )

( , , , )

B

B

B

v v u

v v u

v v u

Determination of stresses

The determination of allows the complete determination of stresses

Where are functions of

1 2, 3 1 2, 3, , ,u u u v v and v

1 2 3 4

1 2 3 411

5 6 7

1 2 3 4

cos cos3 cos5 cos 7

cos cos cos 2 cos cos 42

cos cos 6 cos cos8 cos cos10

sin sin 3 sin 5 sin 7

r

h

r

H H H H

a E

r

I I I I

,H I and 1 2, 3 1 2, 3, , ,u u u v v and v

Results

Consider three orthotropic materials

Plate a11

(TPa)-1

a22

(TPa)-1

a12

(TPa)-1

a66

(TPa)-1 laminate

A 49.02 8.95 -5.93 59.17 0.121

B 17.27 17.27 -53.52 45.32 0.310

C11.69 38.88

-78.01 45.18 0.667

12

0 0490 / 45

S

0 0 00 / 45 / 90S

0 0

20 / 45S

Contact angle analysis

Fixed clearance and varying friction (Plate A)

f P B(GN) (degrees)

0.0 0.2 15.6 88.52

0.4 21.0 88.80

0.01 0.2 15.05 76.18

0.4 20.6 76.58

0.02 0.2 14.0 71.05

0.4 17.0 72.19

Contact angle analysis

Fixed friction and varying clearance (Plate A)

f P B(GN) (degrees)

0.2

0.0 15.6 88.52

0.01 15.05 76.18

0.02 14.0 71.05

0.4

0.0 21.0 88.80

0.01 20.6 76.58

0.02 17.0 72.19

Fixed Friction and Clearance (Plate C)

P0u B

(GN)(degrees)

7.0 0.02 63.1

12.0 0.035 71.8

17.0 0.05 76.07

Radial Streses for Plate A for Fixed Clearance0 0.05u

Radial Streses for Plate A for Fixed Friction

0 0.05u

Shear Stress for Plate A for Fixed Clearance

Shear Stress for Plate A for Fixed Friction

0 0.05u

Hoop Stress for Plate A for Fixed Clearance0 0.05u

Hoop Stress for Plate A for Fixed Friction

0 0.05u

Radial Stress for Plate B for Fixed Clearance0 0.035u

Shear Stress for Plate B for Fixed Clearance0 0.035u

Shear Stress for Plate B for Fixed Friction0 0.035u

Hoop Stress for Plate B for Fixed Clearance 0 0.035u

Hoop Stress for Plate B for Fixed Friction0 0.035u

Stresses for Plate C for varying Pin displacement

Conclusion Contact angle is not significantly affected with increasing friction for fixed clearance.

Contact angle decreases with increasing clearance for fixed friction. Friction and clearance strongly affects contact stress distribution.

Maximum radial stress decrease as friction increase for fixed clearance.

Maximum radial stress increase as clearance increase for fixed friction

Maximum shear and hoop stress increase with increasing friction for fixed clearance and with increasing clearance for fixed friction.

Maximum radial, shear and hoop stress increase with increasing pin displacements

Increasing pin displacement does not appear to affect hoop stress along load axis.

Recommendations

Lateral deformation of ellipse: Account for lateral deformation of ellipse

Pin: Consider effect of pin elasticity

Plate: Consider finite plate

Constant Friction: Consider non linear friction

Friction Mode: Consider non-Coulombic friction model eg.

No-slip: Analysis assumed slip throughout the contact region. Consider the influence of no-slip zone on stresses.

Contributions

Simpler method of analyzing contact region in pin loaded joints.

Simpler method for analyzing stresses in joints with or without clearance.

Method is capable of analyzing interference fitted pin joints.

Very little computer time is required for analysis.

Solution can be implemented using any computer and any symbolic mathematical software.

Simpler method of optimizing joint design which should prove friendly to designers.

Provides a simpler approach to the non-linear problem of pin loaded joints with clearance and friction.

Acknowledgements

The thesis committee Dr. Lewis Thigpen Dr. Marcus A. Alfred Dr. Mrinal C. Saha Dr. Mohsen Mosleh Dr. Horace A. Whitworth