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

5. Influence matrix singular nonsingular 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 Governing equation: RodBeam Fundamental solution

8. Boundary integral equations Rod Beam

9. Degenerate kernels RodBeam

10. Degenerate kernels for rod problem Kernels Domain

11. Degenerate kernels for beam problem Kernel Domain Kernel Domain Kernel Domain Kernel Domain

12. Influence matrices By approachingtoandinto the boundary integral equations RodBeam

13. Translation matrix RodBeam

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 Eqs.

16. Singular value decomposition Thematrix can be expressed as Thedenoted bysatisfies 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 RodBeam formulation

20. Influence matrix RodBeam Whenand, When 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 Spurious mode (generalized force) Rigid body mode (generalized displacement) RodBeam

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!