1. Motion and Stress Analysis by Vector Mechanics Edward C. Ting Professor Emeritus of Applied Mechanics Purdue University, West Lafayette, IN National Central University, ChungLi, Taiwan

2. a computer framework for the study of a multi-component structural system with component motion component interactions: connection, contact, collision, penetration geometrical changes: deformation, displacement, fragmentation, collapse stress distribution behavior and material property changes

6. a physics approach of mechanics motion analysis and VFIFE * vector mechanics ---- particle mechanics * discrete description * intrinsic finite element ---- physical structural element

7. example: a rod in plane motion Newton’s law 1. displacement is a motion

8. analytical mechanics: For motion analysis, assume 1. rigid body, 2. functional description

9. pendulum problem: hinged at end 1 motion analysis 1. general formulation: 2. complete formulation :

10. stress analysis assume: 1. deformable body, 2. Hooke’s law

11. 1. An approximation → separate motion analysis and stress analysis ► continuous bodies: motion--rigid body; stress--deformable body ► variables: motion--displacement; stress--deformation ► governing equations: motion--translation and rotation; stress--equilibrium 2. Described by continuous functions → discretization computation based on analytical mechanics

12. 1. Newton’s law 2. behavior model 3. kinematics 4. Hooke’s law 5. pendulum: constraint conditions hinged end: straight rod: vector mechanics

13. properties: 1. structure: a set of particles 2. always a dynamic process 3. always deformable advantages: 1. suitable for computation 2. a general and systematic formulation 3. explicit constraint conditions

14. development needs: 1. describe structural geometry: intrinsic finite element 2. kinematics: fictitious reversed motion 3. continuity requirements 4. mechanics requirements 5. material model: standard tests elements: plane rod, plane frame, plane solid, space rod, space frame, 3d membrane, 3d solid, 3d plate shell V-5 research group: e. c. ting, c. y. wang, t. y. wu, r. z. wang, c. j. chuang

15. motion analysis procedure: a simple rod structure

16. discrete model: mass particles and structural elements

17. vector form equation of motion

18. path element 1. element geometry remains unchanged 2. small deformation discrete path:

19. kinematics and force calculation 1 material frame: configuration at 2. variable: nodal deformation 3. fictitious reversed motion to define deformation 4. infinitesimal strain and engineering stress 5. nodal forces: use finite element 6. internal forces are in equilibrium

20. reversed motion for nodal deformation

22. governing equations

23. difference equation (symmetrical case)

24. ,

25. 4 node plane solid element

26. estimate the rigid body motion

27. fictitious reversed motion

28. nodal deformation

29. deformation coordinates to define independent variables

30. shape functions:

33. nodal forces

35. stress

51. Thank You