1. Development of Curved Thermoplastic Composite Tubes Theory and Experiment Prepared by: Hamidreza Yazdani Supervised by: Dr. Suong. V. Hoa and Dr. Mehdi Hojjati

2. Outline Problem statement Literature review Toroidal Elasticity Theory Conclusions Future works 2

3. Problem Statement Aluminum landing gears have some technical issues Fabrication is difficult and expensive Failure due to the corrosion and Fatigue The objective of this research project is to investigate how a composite landing gear can be designed and manufactured using thermoplastic composite materials and automated fiber placement technology. 3

4. Literature Review Finite element approach (Redekop and Tan 1987) Analytical approach – Shell bending theory (Boyle and Spence 1992) – Membrane theory (Bushnell 1981) – Toroidal Elasticity (TE) Stress approach (Lang 1989) Displacement approach (Redekop 1991, 1993) 4

5. Toroidal Elasticity for Orthotropic Materials Displacement field for composite toroidal structures based on successive approximation method The governing equations in three toroidal coordinate system Method of successive approximation The governing equations in different orders Different order displacement components 5

6. Governing Equations R is bend radius. Angular coordinates are θ and φ, and radial coordinate is r. 6

7. Governing Equations Kinematics relations Constitutive equations Governing Navier equations in toroidal coordinate 7

8. Method of Successive Approximation The solution is a series in terms of a small parameter ε=1/R. 8

9. Governing Equations for the Different Orders The zeroth order: The first order: The second order: The third order: 9

10. General Solution Complementary solution General form solution 10

11. Boundary Conditions The boundary conditions on the curved surfaces are satisfied by each order 11

12. The Zeroth Order Solution 12

13. The First Order Solution Governing equations Displacement components 13

14. The Second Order Solution Governing equations Displacement components 14

15. The Third Order Solution Governing equations Displacement components 15

16. Conclusions Toroidal elasticity is a developing three-dimensional theory which can be used for the elastostatic analysis of thick-walled curved tubes. A displacement based toroidal elasticity has been used. The governing equations in the toroidal coordinates are much more complicated than those used in Cartesian coordinates. The successive approximation method has been employed to simplify the governing equations. 16

17. Previous and Future works Literature review Finding the appropriate theory Obtain the displacement field for isotropic materials Obtain the displacement field for a layer of orthotropic materials Achieve the displacement field for a laminate of orthotropic materials Analysis of the curved tube dynamically Comparison of the theoretical result with the experimental one 17

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