Determination of Contact Stress Distribution in Pin Loaded Orthotropic Plates

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Determination of Contact Stress Distribution in Pin Loaded Orthotropic Plates

  1. 1. Determination of Contact Stress Distribution in Pin Loaded Orthotropic Plates Neville A. Tomlinson Department of Mechanical Engineering Howard University Washington D C December 2006
  2. 2. Outline <ul><li>Introduction </li></ul><ul><ul><li>Joints </li></ul></ul><ul><ul><li>Orthotropic materials </li></ul></ul><ul><ul><li>The pin loaded plate </li></ul></ul><ul><li>Orthotropic Plate Theory </li></ul><ul><li>Problem definition </li></ul><ul><li>Mathematical Formulations For the Pin Loaded Plate with Clearance and Friction </li></ul><ul><ul><li>The contact equation </li></ul></ul><ul><ul><li>Boundary conditions at the pin-plate interface </li></ul></ul><ul><ul><li>Friction </li></ul></ul><ul><ul><li>Trigonometric displacement functions </li></ul></ul><ul><ul><li>Stress-stress functions </li></ul></ul><ul><ul><li>Stress Equations </li></ul></ul><ul><li>Results </li></ul><ul><li>Conclusions </li></ul><ul><li>Recommendations </li></ul><ul><li>Contribution </li></ul><ul><li>Acknowledgements </li></ul>
  3. 3. Introduction <ul><li>Joints </li></ul><ul><ul><li>Important form of mechanical joining of structural elements </li></ul></ul><ul><li>Reason </li></ul><ul><ul><li>Ease of assembly and disassembly </li></ul></ul><ul><li>Joint Types </li></ul><ul><ul><li>Bolted Joints, riveted joints, welded joints, pin joints </li></ul></ul><ul><li>Pin Joint </li></ul><ul><ul><li>For plane stress analysis pin joint is representative of joint analysis </li></ul></ul><ul><li>Common Joint Materials </li></ul><ul><ul><li>Iron, Steel, Aluminum (isotropic, strong, heavy and/or expensive) </li></ul></ul><ul><ul><li>Fiber reinforced Composites (orthotropic, can design strength, light, relatively inexpensive) </li></ul></ul>
  4. 4. Orthotropic Materials <ul><li>Anisotropic materials </li></ul><ul><ul><li>Material having different properties in different directions </li></ul></ul><ul><li>Orthotropic materials </li></ul><ul><ul><li>Thin anisotropic materials whose transverse stresses are considered negligible and therefore transverse properties are ignored </li></ul></ul><ul><ul><li>Considered in plane problems of elasticity </li></ul></ul><ul><ul><li>Simplifies the analysis </li></ul></ul>
  5. 5. The Pin Loaded Plate <ul><li>In studying pin loaded joints only the plate is considered in this analysis </li></ul><ul><li>Assumption: pin is rigid </li></ul>Schematic of a Pin Loaded Plate P P/2 P/2 clevis
  6. 6. Orthotropic Plate Theory <ul><li>Equilibrium Equation (no body force) </li></ul><ul><li>Constitutive Equation </li></ul><ul><li>Compatibility Equation </li></ul>
  7. 7. Orthotropic Plate Theory (Continued) <ul><li>Airy Stress Function </li></ul><ul><li>Governing Partial Differential Equation for Orthotropic Material </li></ul>
  8. 8. Orthotropic Plate Theory (continued) <ul><li>Complex Stress Function </li></ul><ul><li>Characteristic Equation </li></ul><ul><li>Roots </li></ul><ul><li>Re-writing Governing Equation </li></ul><ul><ul><ul><ul><ul><li>k = 1 to 4 </li></ul></ul></ul></ul></ul><ul><li>Solution for F (invoking rule of complex addition) </li></ul>
  9. 9. Orthotropic Plate Theory (continued) <ul><li>By defining two complex functions </li></ul><ul><li>and where </li></ul><ul><li>In terms of complex stress functions stresses become </li></ul>
  10. 10. Problem Definition 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 =
  11. 11. Mathematical Formulations For the Pin Loaded Plate with Clearance and Friction Equation of ellipse Point B has coordinates Contact Equation
  12. 12. Boundary Conditions at the Pin-plate Interface
  13. 13. Friction <ul><li>Assuming Coulomb Frictional relation, </li></ul><ul><li>-ve sign: shear opposes the direction of relative displacements between the pin and the plate </li></ul><ul><li>Introduce Friction into the model by the relation </li></ul>Constant Coefficient of Friction
  14. 14. Trigonometric displacement function y Traction conditions on the hole boundary requires that Assumed three terms displacement field along the contact region as
  15. 15. Trigonometric Displacement Function (continued)
  16. 16. Trigonometric Displacement Function (continued) Using the following relation, Evaluate at
  17. 17. Determination of the Stress Functions and Stresses Constants
  18. 18. Determination of the Stress Functions and Stresses (continued) mapping function Introduce unit circle to satisfy boundary conditions
  19. 19. Determination of the Stress Functions and Stresses (continued) Comparing coefficients yields
  20. 20. Determination of the Stress Functions and Stresses (continued) <ul><li>Stress stress function relationship </li></ul><ul><li>Stress transformation from x,y system to polar system </li></ul>
  21. 21. Determination of the Stress Functions and Stresses (continued) Functions of material parameters, the displacement coefficients and the hole radius
  22. 22. Determination of <ul><li>Displacement Parameters obtained as functions of </li></ul>
  23. 23. Determination of Given Substitute Re-stated as
  24. 24. Determination of <ul><li>Determination of allows the determination of the </li></ul><ul><li>displacement coefficients </li></ul>
  25. 25. Determination of stresses <ul><li>The determination of allows the complete determination of stresses </li></ul><ul><li>Where are functions of </li></ul>
  26. 26. Results <ul><li>Consider three orthotropic materials </li></ul>0.121 59.17 -5.93 8.95 49.02 A 0.667 45.18 -78.01 38.88 11.69 C 0.310 45.32 -53.52 17.27 17.27 B laminate a66 (TPa)-1 a12 (TPa)-1 a22 (TPa)-1 a11 (TPa)-1 Plate
  27. 27. Contact angle analysis Fixed clearance and varying friction (Plate A) 72.19 17.0 0.4 71.05 14.0 0.2 0.02 76.58 20.6 0.4 76.18 15.05 0.2 0.01 88.80 21.0 0.4 88.52 15.6 0.2 0.0 (degrees) (GN)
  28. 28. Contact angle analysis Fixed friction and varying clearance (Plate A) 72.19 17.0 0.02 76.58 20.6 0.01 88.80 21.0 0.0 0.4 71.05 14.0 0.02 76.18 15.05 0.01 88.52 15.6 0.0 0.2 (degrees) (GN)
  29. 29. Fixed Friction and Clearance (Plate C) 76.07 0.05 17.0 71.8 0.035 12.0 63.1 0.02 7.0 (degrees) (GN)
  30. 30. Radial Streses for Plate A for Fixed Clearance
  31. 31. Radial Streses for Plate A for Fixed Friction
  32. 32. Shear Stress for Plate A for Fixed Clearance
  33. 33. Shear Stress for Plate A for Fixed Friction
  34. 34. Hoop Stress for Plate A for Fixed Clearance
  35. 35. Hoop Stress for Plate A for Fixed Friction
  36. 36. Radial Stress for Plate B for Fixed Clearance
  37. 37. Shear Stress for Plate B for Fixed Clearance
  38. 38. Shear Stress for Plate B for Fixed Friction
  39. 39. Hoop Stress for Plate B for Fixed Clearance
  40. 40. Hoop Stress for Plate B for Fixed Friction
  41. 41. Stresses for Plate C for varying Pin displacement
  42. 42. Conclusion <ul><li>Contact angle is not significantly affected with increasing friction for fixed clearance. </li></ul><ul><li>Contact angle decreases with increasing clearance for fixed friction. </li></ul><ul><li>Friction and clearance strongly affects contact stress distribution. </li></ul><ul><li>Maximum radial stress decrease as friction increase for fixed clearance. </li></ul><ul><li>Maximum radial stress increase as clearance increase for fixed friction </li></ul><ul><li>Maximum shear and hoop stress increase with increasing friction for fixed clearance and with increasing clearance for fixed friction. </li></ul><ul><li>Maximum radial, shear and hoop stress increase with increasing pin displacements </li></ul><ul><li>Increasing pin displacement does not appear to affect hoop stress along load axis. </li></ul>
  43. 43. Recommendations <ul><li>Lateral deformation of ellipse : Account for lateral deformation of ellipse </li></ul><ul><li>Pin : Consider effect of pin elasticity </li></ul><ul><li>Plate: Consider finite plate </li></ul><ul><li>Constant Friction : Consider non linear friction </li></ul><ul><li>Friction Mode : Consider non-Coulombic friction model eg. </li></ul><ul><li>No-slip: Analysis assumed slip throughout the contact region . Consider the influence of no-slip zone on stresses. </li></ul>
  44. 44. Contributions <ul><li>Simpler method of analyzing contact region in pin loaded joints. </li></ul><ul><li>Simpler method for analyzing stresses in joints with or without clearance. </li></ul><ul><li>Method is capable of analyzing interference fitted pin joints. </li></ul><ul><li>Very little computer time is required for analysis. </li></ul><ul><li>Solution can be implemented using any computer and any symbolic mathematical software. </li></ul><ul><li>Simpler method of optimizing joint design which should prove friendly to designers. </li></ul><ul><li>Provides a simpler approach to the non-linear problem of pin loaded joints with clearance and friction. </li></ul>
  45. 45. Acknowledgements <ul><li>The thesis committee </li></ul><ul><ul><li>Dr. Lewis Thigpen </li></ul></ul><ul><ul><li>Dr. Marcus A. Alfred </li></ul></ul><ul><ul><li>Dr. Mrinal C. Saha </li></ul></ul><ul><ul><li>Dr. Mohsen Mosleh </li></ul></ul><ul><ul><li>Dr. Horace A. Whitworth </li></ul></ul>

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