1. Dynamic Analysis of Structures by
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 Objective 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
 main idea
conventional:

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

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

12. Numerical examples Structures Normalized RMS error
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
(a) Single input load
(b) Multi-input load

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

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