Dynamic Analysis of Structures by Superposition of Modified Lanczos Vectors

1. 2003 한국전산구조공학회 봄 학술발표회 2003년 4월 12일 Dynamic Analysis of Structures by Superposition of Modified Lanczos Vectors Byoung-Wan Kim1), Hyung-Jo Jung2), Woon-Hak Kim3) and In-Won Lee4) 1) Ph.D. Candidate, Dept. of Civil and Environmental Eng., KAIST 2) Assist. Professor, Dept. of Civil and Environmental Eng., Sejong Univ. 3) Professor, Dept. of Civil Engineering, Hankyong National Univ. 4) Professor, Dept. of Civil and Environmental Eng., KAIST

2. Contents • Introduction • Proposed method • Numerical examples • Conclusions

3. Introduction • Background • Dynamic analysis of structures • - Direct integration method • - Vector superposition method • Vector superposition method • - Eigenvector superposition method • - Ritz vector superposition method • - Lanczos vector superposition method Eigenvalue analysis No eigenvalue analysis • The Lanczos vector superposition method is very efficient.

4. Literature review • Nour-Omid(1984) first proposed. • Nour-Omid(1995): Unsymmetric nonclassically damped system • Chen(1990): Symmetric nonclassically damped system • Ibrahimbegovic(1990), Mehai(1995): Dam-foundation system • Drawback: The method is costly when multi-input-loaded • structures are analyzed. • Objective • Improvement of the Lanczos vector superposition method • to overcome the shortcoming.

5. Proposed method • Conventional method • Dynamic equation of motion of structures mass matrix(n n) stiffness matrix(n n) displacements vector(n 1) force vector(n 1) Rayleigh damping coefficients

6. Lanczos algorithm ith Lanczos vector

7. Reduced tridiagonal equation of motion premultiply

8. Single input loads spatial load distribution vector(n 1) time variation function(scalar) • Multi-input loads spatial load distribution matrix(nk) time variation function vector(k 1) the number of input loads

9. Proposed method • Modified Lanczos algorithm conventional:  main idea

10. Reduced tridiagonal equation of motion premultiply • Single input load • Multi-input loads

11. Summary • Single input loads - conventional - proposed • Multi-input loads - conventional - proposed

12. Numerical examples • Structures • Simple span beam(Pan and Li, 2002) • Multi-span continuous bridge(Park et al., 2002) • Normalized RMS error results by the direct integration method time duration

13. Simple span beam • Geometry and material properties

14. Loading configurations - Single input load(concentrated sinusoidal force) - Multi-input load(moving load)

15. Error (b) Multi-input load (a) Single input load

16. Computing time (b) Multi-input load (a) Single input load

17. Multi-span continuous bridge • Geometry and material properties Dongjin bridge(PSC box girder type)

19. Loading configurations - Single input load(El Centro earthquake) - Multi-input load(moving load)

20. Error (b) Multi-input load (a) Single input load

21. Computing time (b) Multi-input load (a) Single input load

22. Conclusions • The Lanczos and Ritz vector superposition methods have almost the same accuracy. • For the single input loading case, the Lanczos and Ritz vector superposition methods have better accuracy than the eigenvector superposition and mode acceleration methods. • For the multi-input loading case, the eigenvector superposition and mode acceleration methods have better accuracy than the Lanczos and Ritz vector superposition methods.

23. The Lanczos and Ritz vector superposition methods have better computing efficiency than the eigenvector superposition and mode acceleration methods. • For the single input loading case, proposed and conventional Lanczos vector superposition methods have almost the same computing efficiency. • For the multi-input loading case, proposed method has better computing efficiency than the conventional Lanczos vector superposition method.