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Stresses Around Pin Loaded Holes in
Mechanically Fastened Joints
ByNeville A. Tomlinson, PhD
Howard UniversityWashington DC
January 2007
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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.
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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.
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Schematic of Pin joint
pin
plate
Fig 1.
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Problem Definition
loadpinP
anglecontac
disppinu
clearance
radiuspinr
radiusholer
B
p
====
==
θ
λ.0
Fig. 2
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Exaggerated view of deformed hole by rigid pin
Fig 3
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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)
![Page 8: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/8.jpg)
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)
![Page 9: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/9.jpg)
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≥ ±
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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)
![Page 11: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/11.jpg)
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)
![Page 12: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/12.jpg)
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)
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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)
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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
− = +
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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
+ + + + + +
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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
=
+ + +
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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)
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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
− = + + + +
![Page 19: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/19.jpg)
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)
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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= + −
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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
∂ ∂=
∂ ∂
![Page 22: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/22.jpg)
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)
![Page 23: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/23.jpg)
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)
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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)
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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)
![Page 26: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/26.jpg)
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)
![Page 27: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/27.jpg)
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
![Page 28: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/28.jpg)
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
+ + − − − + + − − − +
= + − + − − −
= − − + + + −
![Page 29: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/29.jpg)
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
![Page 30: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/30.jpg)
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 = + + − − + − +
![Page 31: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/31.jpg)
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
=
=
=
![Page 32: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/32.jpg)
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
+ + − + −
+ − + + − + + =
![Page 33: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/33.jpg)
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
===
![Page 34: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/34.jpg)
Results
![Page 35: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/35.jpg)
Results
.
Radial stress for plate A (±45s )
![Page 36: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/36.jpg)
Results
Shear stress for plate A (±45s )
![Page 37: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/37.jpg)
Results
.
Tangential or hoop stress for plate A (±45s )
![Page 38: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/38.jpg)
Results
.
![Page 39: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/39.jpg)
Results
.
Radial stress for plate E ( [02/±45]s )
![Page 40: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/40.jpg)
Results
.
Shear stress for plate E ( [02/±45]s )
![Page 41: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/41.jpg)
Results
.
Tangential or hoop stress for plate E ( [02/±45]s )
![Page 42: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/42.jpg)
Results
.
![Page 43: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/43.jpg)
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.
![Page 44: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/44.jpg)
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
![Page 45: Utech Presentation](https://reader031.fdocuments.in/reader031/viewer/2022031903/55a37f301a28abe6158b4833/html5/thumbnails/45.jpg)
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.