1. An Advanced Shell Theory Based Tire Model by D. Bozdog, W. W. Olson Department of Mechanical, Industrial and Manufacturing Engineering The 23 rd Annual Tire Society Meeting 2004

2. September 20, 20042Tire Society Meeting 2004 Outline Objectives Objectives Tire Model Formulation Tire Model Formulation Numerical Method Numerical Method Results Results

3. September 20, 20043Tire Society Meeting 2004 Motivation Wide variety of tire models Wide variety of tire models Simplistic (spring-damper structures or curve fits of experimental data) Simplistic (spring-damper structures or curve fits of experimental data) FEM extremely complex FEM extremely complex Potential of elasticity based shell theory tire model Potential of elasticity based shell theory tire model Provide both the benefits Provide both the benefits Complex analysis Complex analysis Fast computations Fast computations Assume material properties closed to real values Assume material properties closed to real values Requires small number of input parameters Requires small number of input parameters Can be used for all types of tire design Can be used for all types of tire design

4. September 20, 20044Tire Society Meeting 2004 Objectives Short term Short term Determine the deformed shape of tire Determine the deformed shape of tire Internal pressure Internal pressure Vertical loadings Vertical loadings Longitudinal and lateral forces Longitudinal and lateral forces Determine the stress-strain distribution in structure Determine the stress-strain distribution in structure Long term Framework for tire simulation analysis and development based on shell theory Provide a solution for tire modeling for vehicle dynamics simulation software

5. September 20, 20045Tire Society Meeting 2004 Tire model formulation General approach General approach General Linear Thin Shells Theory General Linear Thin Shells Theory Mechanics of Laminated Composite Materials Mechanics of Laminated Composite Materials Numerical Method Numerical Method

6. September 20, 20046Tire Society Meeting 2004 Tire model formulation General approach General approach General Linear Thin Shells Theory General Linear Thin Shells Theory Mechanics of Laminated Composite Materials Mechanics of Laminated Composite Materials Numerical Method Numerical Method

7. September 20, 20047Tire Society Meeting 2004 Tire model formulation General compatibility equations of plane strain thin shells: General compatibility equations of plane strain thin shells:

8. September 20, 20048Tire Society Meeting 2004 Tire model formulation Shell force and moment resultants Shell force and moment resultants

9. September 20, 20049Tire Society Meeting 2004 Tire model formulation Equilibrium equations for static shell Equilibrium equations for static shell

10. September 20, 200410Tire Society Meeting 2004 Tire model formulation Applied forces to shell element Applied forces to shell element

11. September 20, 200411Tire Society Meeting 2004 Tire model formulation General approach General approach General Linear Thin Shells Theory General Linear Thin Shells Theory Mechanics of Laminated Composite Materials Mechanics of Laminated Composite Materials Numerical Method Numerical Method

12. September 20, 200412Tire Society Meeting 2004 Tire model formulation Mechanics of Laminated Composite Materials Mechanics of Laminated Composite Materials

13. September 20, 200413Tire Society Meeting 2004 Tire model formulation Mechanics of Laminated Composite Materials Mechanics of Laminated Composite Materials Constitutive equations: Constitutive equations:

14. September 20, 200414Tire Society Meeting 2004 Tire model formulation Mechanics of Laminated Composite Materials Mechanics of Laminated Composite Materials Stress and Moment resultants Stress and Moment resultants

15. September 20, 200415Tire Society Meeting 2004 Tire model formulation Mechanics of Laminated Composite Materials Mechanics of Laminated Composite Materials

16. September 20, 200416Tire Society Meeting 2004 Tire model formulation 32x8.8 Type VII aircraft tire 32x8.8 Type VII aircraft tire

17. September 20, 200417Tire Society Meeting 2004 System of equations System of equations: System of equations: equilibrium equations equilibrium equations constitutive equations constitutive equations compatibility equations compatibility equations boundary conditions boundary conditions 17 equations 17 equations 12 first order diff. eq. 12 first order diff. eq. 6 linear eq. 6 linear eq. 17 unknowns Displacements Strains Force and Moment Resultants

18. September 20, 200418Tire Society Meeting 2004 Tire model formulation General approach General approach General Linear Thin Shells Theory General Linear Thin Shells Theory Mechanics of Laminated Composite Materials Mechanics of Laminated Composite Materials Numerical Method Numerical Method

19. September 20, 200419Tire Society Meeting 2004 Discrete structure Uniform grid Uniform grid

20. September 20, 200420Tire Society Meeting 2004 Numerical solution Transform system of 17 equations to eight-order system of 3 governing partial differential equations Transform system of 17 equations to eight-order system of 3 governing partial differential equations

21. September 20, 200421Tire Society Meeting 2004 Numerical solution Apply finite difference method for partial derivatives Apply finite difference method for partial derivatives

22. September 20, 200422Tire Society Meeting 2004 Numerical solution Fit tire profile in and direction with continous functions Fit tire profile in and direction with continous functions

23. September 20, 200423Tire Society Meeting 2004 Setting parameters Radii of curvature & Lamé parameters Radii of curvature & Lamé parameters

24. September 20, 200424Tire Society Meeting 2004 Iterations Successive iterations for internal and external loads Successive iterations for internal and external loads p, p p z p, p p z Compute after each iteration Compute after each iteration Radii of curvature R1, R2 and A, B parameters Radii of curvature R1, R2 and A, B parameters Deformed profile Deformed profile

25. September 20, 200425Tire Society Meeting 2004 Results Computer code developed using Mathematica TM software Computer code developed using Mathematica TM software Strong performance in symbolic computation Strong performance in symbolic computation Advanced mathematical tools Advanced mathematical tools Fast execution Fast execution Iteration performed for pressure load by setting p with 5psi increment for 32x8.8 Type VII aircraft tire 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 Results are determined for all 17 variables Displacements Displacements Strains & Change of Curvature Strains & Change of Curvature Forces and Moment Resultants Forces and Moment Resultants

26. September 20, 200426Tire Society Meeting 2004 Displacements Cross-section tangential displacements 0-95psi for 32x8.8 Type VII aircraft tire Cross-section tangential displacements 0-95psi for 32x8.8 Type VII aircraft tire

27. September 20, 200427Tire Society Meeting 2004 Displacements Cross-section normal displacements 0-95psi for 32x8.8 Type VII aircraft tire Cross-section normal displacements 0-95psi for 32x8.8 Type VII aircraft tire

28. September 20, 200428Tire Society Meeting 2004 Displacements Cross-section displacements 0-95psi for 32x8.8 Type VII aircraft tire Cross-section displacements 0-95psi for 32x8.8 Type VII aircraft tire

29. September 20, 200429Tire Society Meeting 2004 Displacements Initial Initial Cross-section displacements for 95psi Cross-section displacements for 95psi Previous results by Brewer Previous results by Brewer

30. September 20, 200430Tire Society Meeting 2004 Conclusion Tire Model Tire Model Confirm previous results Confirm previous results Computer code (still in work) Computer code (still in work) Will have ability to perform complete analysis for pressure, longitudinal and lateral external forces Will have ability to perform complete analysis for pressure, longitudinal and lateral external forces Theoretical model can incorporate Theoretical model can incorporate Variable thickness of cross-section Variable thickness of cross-section Variable stiffness matrix for tread, sidewall and bead regions Variable stiffness matrix for tread, sidewall and bead regions Variable cord path can be incorporated Variable cord path can be incorporated Accuracy of solutions is highly dependent on size of shell grid and CPU Accuracy of solutions is highly dependent on size of shell grid and CPU Code can be customize for specific tires Code can be customize for specific tires

31. September 20, 200431Tire Society Meeting 2004 Questions ?