Derivation of stiffness and flexibility for rods and beams by using dual integral equations

1. Derivation of stiffness and flexibility for rods and beams by using dual integral equations 海洋大學河海工程學系 報 告 者：謝正昌 指導教授：陳正宗 特聘教授 日期：2006/04/01 中工論文競賽(土木工程組)

2. Outlines • Introduction • Dual boundary integral formulation for rod and beam problems • Discussion of the rigid body mode and spurious mode • Conclusions

3. Outlines • Introduction • Dual boundary integral formulation for rod and beam problems • Discussion of the rigid body mode and spurious mode • Conclusions

4. Introduction For undergraduate students, it is well-known in mechanics of material. stiffness For graduate students, they revisited it in the finite element course. flexibility Dual boundary integral equations were employed to derive the stiffness and flexibility of the rod and beam .

5. Influence matrix nonsingular Influence matrix singular Degenerate scale problem Fredholm theorem and SVD updating technique Rigid body mode Spurious mode

6. Outlines • Introduction • Dual boundary integral formulation for rod and beam problems • Discussion of the rigid body mode and spurious mode • Conclusions

7. Rod and beam problems Rod Beam Governing equation: Governing equation: Fundamental solution Fundamental solution

8. Boundary integral equations Rod Beam

9. Degenerate kernels Rod Beam

10. Degenerate kernels for rod problem

11. Degenerate kernels for beam problem

12. Influence matrices By approaching to and into the boundary integral equations Rod Beam

13. Translation matrix Rod Beam

14. The stiffness matrix of rods Stiffness matrix for rod problems using dual BEM

15. Stiffness matrix for the beam by using the direct method

16. Singular value decomposition The matrix can be expressed as The denoted by satisfies the four Penrose conditions. The pseudo-inverse is identified as

17. The flexibility matrix of rods Flexibility matrix for rod problems using dual BEM

18. Outlines • Introduction • Dual boundary integral formulation for rod and beam problems • Discussion of the rigid body mode and spurious mode • Conclusions

19. Fundamental solution Rod Beam formulation

20. Influence matrix Rod Beam , , When When and and

21. Mathematical SVD structures of the influence matrices According to Fredholm alternative theorem

22. Spurious modes and the rigid body modes for a rod and a beam in BEM Rod Beam

23. Conclusion • Dual boundary integral equations were employed to derive the stiffness and flexibility of the rod and beam which match well with those of FEM. • Both direct and indirect methods were used. • The displacement-slope and displacement-moment formulations in the direct method can construct the stiffness matrix. • The single-double layer approach and single-triple layer approach work for the constructing of stiffness matrix in the indirect method.

24. Conclusion • The rigid body mode and spurious mode are imbedded in the right and left unitary vectors of the influence matrices through SVD.

25. The end Thank you for your kind attention!