The 23rd Annual Tire Society Meeting 2004
An Advanced Shell Theory Based
Tire Model
by
D. Bozdog, W. W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
Outline

Objectives

Tire Model Formulation

Numerical Method

Results
September 20, 2004
Tire Society Meeting 2004
2
Motivation

Wide variety of tire models



Simplistic (spring-damper structures or curve fits of
experimental data)
FEM extremely complex
Potential of elasticity based shell theory tire model




Provide both the benefits
 Complex analysis
 Fast computations
Assume material properties closed to real values
Requires small number of input parameters
Can be used for all types of tire design
September 20, 2004
Tire Society Meeting 2004
3
Objectives

Short term



Determine the deformed shape
of tire
 Internal pressure
 Vertical loadings
 Longitudinal and lateral
forces
Long term

Framework for tire simulation
analysis and development based
on shell theory

Provide a solution for tire
modeling for vehicle dynamics
simulation software
Determine the stress-strain
distribution in structure
September 20, 2004
Tire Society Meeting 2004
4
Tire model formulation

General approach

General Linear Thin Shells Theory

Mechanics of Laminated Composite Materials

Numerical Method
September 20, 2004
Tire Society Meeting 2004
5
Tire model formulation

General approach

General Linear Thin Shells Theory

Mechanics of Laminated Composite Materials

Numerical Method
September 20, 2004
Tire Society Meeting 2004
6
Tire model formulation

General compatibility equations of plane strain thin shells:
1  u  1  A 
w
  
 v 
A    AB    R1
1  v  1  B 
w
 u 
     
B    AB   
R2
B    v  A    u 
         
A    B  B    A 
1   u 1 w  1 A  v 1 w 
 
 
 

K 
A   R1 A   AB   R2 B  
1   v 1 w  1 B  u 1 w 
 
 
 

K 
B   R2 B   AB   R1 A  
1  1 A w 1 B w  2 w  A   u  B   v 
 
 



 
 
AB  A   B     R1 B   A  R2 A   B 
 z   z   z  0
 
K
September 20, 2004
Tire Society Meeting 2004
7
Tire model formulation

Shell force and moment resultants
September 20, 2004
Tire Society Meeting 2004
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Tire model formulation

Equilibrium equations for static shell

N B    N A  N A  N B  Q AB  p AB  0




R1


B
A
AB
( N A) 
( N B)  N
 N
 Q
 p AB  0




R2


AB
AB
(Q B) 
(Q A)  N
 N
 p z AB  0


R1
R2


( BM  ) 
( AM  )  M 




( AM  ) 
( BM  )  M 


M  M 
N  N 

0
R1
R2
September 20, 2004
A
 M 

B
 M 

B
 Q AB  0

A
 Q AB  0

Tire Society Meeting 2004
9
Tire model formulation

Applied forces to shell element
v
p   h 2 r  ,   cos   sign (u ) Fl  ,   cos[tan 1  ]
u
 1  v 
p   sign (v) Fl  ,   sin  tan  
 u 

p z  pi  h 2 r  ,  sin   Fl  ,  
September 20, 2004
Tire Society Meeting 2004
10
Tire model formulation

General approach

General Linear Thin Shells Theory

Mechanics of Laminated Composite Materials

Numerical Method
September 20, 2004
Tire Society Meeting 2004
11
Tire model formulation

Mechanics of Laminated Composite Materials
2
 1   cos 
    sin 2 
 2 
 12   sin  cos
sin 2 
cos 2 
sin  cos
2 sin  cos    
 
 2 sin  cos    
cos 2   sin 2     
   Q11 Q12 Q16     
 
  
    Q12 Q 22 Q 26     
   Q16 Q 26 Q 66    
  
 
September 20, 2004
Tire Society Meeting 2004
12
Tire model formulation

Mechanics of Laminated Composite Materials
    0  zK
    0  zK
    0  zK

Constitutive equations:
Q11 Q12 Q16    0 
Q11 Q12 Q16   K 
 

 0 


 



Q
Q
Q


z
Q
Q
Q
K
 
22
26    
22
26  
 12
 12
 
Q
  0 


  

Q
Q
26
66  
 [ k ]  16
 k  Q16 Q 26 Q 66   K  k 
September 20, 2004
Tire Society Meeting 2004
13
Tire model formulation

Mechanics of Laminated Composite Materials

Stress and Moment resultants
 N 


N
 
 N 


  
 M 
  
h/2 
h/2 




     dz ;  M        zdz
h / 2
h / 2
  
 M  
  
  k 

 k 
  k 
k 
 N 
  
 M 
  
n
n
zk 
zk 






N


dz
;
M


    zk 1   
    zk 1    zdz
 N  k 1
  
 M   k 1
  


 


 
September 20, 2004
Tire Society Meeting 2004
14
Tire model formulation

Mechanics of Laminated Composite Materials
 N   A11
N  
    A12
 N   A16


M
    B11
 M    B12

 
 M    B16
n
A16
B11
B12
A22
A26
A26
A66
B12
B16
B22
B26
B12
B16
D11
D12
B22
B26
D12
D22
B26
B66
D16
D26
 
A   Q ij
k 1
A12
n
k 
k 1
 
1 n
B   Q ij
2 k 1
k 
 
1 n
D   Q ij
3 k 1
September 20, 2004
 
(hk  hk 1 )   Q ij
n
2
k 1
k 1
t
[k ] [k ]
 
(h  h )   Q ij
2
k
B16     


B26     
B66     


D16   K 
D26   K 

 
D66   K 
~
t
h
[k ] [k ] [k ]
 
 ~ 2 t[3k ] 

(h  h )   Q ij [ k ]  t[ k ]h[ k ] 

12 
k 1
k 

n
3
k
3
k 1
Tire Society Meeting 2004
15
Tire model formulation

32x8.8 Type VII aircraft tire
 N  2003.95 4035.6
0
0
0
52.48     
N  
  
4035
.
6
9350
.
85
0
0
0
124
.
91
   
   
 N   0
0
3854.21 52.48 124.91
0     




M
K
0
0
52
.
48
11
.
12
22
.
39
0
  
  
 M   0
0
124.91 22.39 51.87
0   K 

 

 
M
K
124.91
0
0
0
21.38    
    52.48
September 20, 2004
Tire Society Meeting 2004
16
System of equations

System of equations:

17 unknowns





equilibrium equations
constitutive equations
compatibility equations
boundary conditions
Displacements


Strains


17 equations


12 first order diff. eq.
6 linear eq.
September 20, 2004

u , v, w
 ,  ,   , K , K , K
Force and Moment Resultants

N , N , N , Q , Q

M  , M  , M 
Tire Society Meeting 2004
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Tire model formulation

General approach

General Linear Thin Shells Theory

Mechanics of Laminated Composite Materials

Numerical Method
September 20, 2004
Tire Society Meeting 2004
18
Discrete structure

Uniform grid
September 20, 2004
Tire Society Meeting 2004
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Numerical solution

Transform system of 17 equations to eight-order system
of 3 governing partial differential equations
    2  2  2 
f( , , 2, 2,
,)  p ( ,  )
    
    2  2  2 
g( , , 2 , 2 ,
, )  p ( ,  )
    
    2  2  2 
h( , , 2 , 2 ,
,)  p z ( ,  )
    
September 20, 2004
Tire Society Meeting 2004
20
Numerical solution

Apply finite difference method for partial derivatives
f
f (i  1, j )  f (i  1, j )


2
f
f (i, j  1)  f (i, j  1)


2
2 f
f (i  1, j )  2 f (i, j )  f (i  1, j )

2

 2
2 f
f (i, j  1)  2 f (i, j )  f (i, j  1)

2

 2
2 f
f (i  1, j  1)  f (i  1, j  1)  f (i  1, j  1)  f (i  1, j  1)


4
September 20, 2004
Tire Society Meeting 2004
21
Numerical solution

Fit tire profile in  and  direction with continous
functions
nmax
s( )  a 0   ak cos[ k ] in 
direction
t (  )  a 0   ak cos[ k ] in 
direction
k 1
mmax
k 1
September 20, 2004
Tire Society Meeting 2004
22
Setting parameters

Radii of curvature & Lamé parameters
( s 2 ( )  [ s( )]2 )3 / 2
R1 ( ,  )  2
s ( )  2[ s( )]2  s( ) s( )
 s( ) sin    s  cos 
  arctan 

 s sin    s  cos  
September 20, 2004
Tire Society Meeting 2004
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Iterations

Successive iterations for internal and external loads


p,, p, , pz,
Compute after each iteration


Radii of curvature R1, R2 and A, B parameters
Deformed profile
September 20, 2004
Tire Society Meeting 2004
24
Results

Computer code developed using MathematicaTM software



Strong performance in symbolic computation
Advanced mathematical tools
Fast execution

Iteration performed for pressure load by setting p with 5psi
increment for 32x8.8 Type VII aircraft tire

Results are determined for all 17 variables



Displacements
Strains & Change of Curvature
Forces and Moment Resultants
September 20, 2004
Tire Society Meeting 2004
25
Displacements

Cross-section tangential displacements 0-95psi for 32x8.8
Type VII aircraft tire
September 20, 2004
Tire Society Meeting 2004
26
Displacements

Cross-section normal displacements 0-95psi for 32x8.8
Type VII aircraft tire
September 20, 2004
Tire Society Meeting 2004
27
Displacements

Cross-section displacements 0-95psi for 32x8.8 Type VII
aircraft tire
September 20, 2004
Tire Society Meeting 2004
28
Displacements



Initial
Cross-section displacements for 95psi
Previous results by Brewer
September 20, 2004
Tire Society Meeting 2004
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Conclusion

Tire Model





Confirm previous results
Computer code (still in work)
 Will have ability to perform complete analysis for pressure,
longitudinal and lateral external forces
Theoretical model can incorporate
 Variable thickness of cross-section
 Variable stiffness matrix for tread, sidewall and bead regions
 Variable cord path can be incorporated
Accuracy of solutions is highly dependent on size of shell grid
and CPU
Code can be customize for specific tires
September 20, 2004
Tire Society Meeting 2004
30
Questions ?
September 20, 2004
Tire Society Meeting 2004
31