1. Frequency-Domain Analysis of An Elevated Rail Bridge Using A Periodic Method Y.S. Yang (speaker) National Center for Research on Earthquake Engineering Y.J. Lee National I-Lan Institute of Technology T.W. Lin National Taiwan University The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering

2. Objective Ground vibration response Induced by an elevated rail bridge The elevated rail bridge Consists of hundreds of spans The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering

3. Analysis methods (1/2) N-span time-domain analysis Advantages : Complicated structure configuration Nonlinear response 1-span frequency-domain analysis (periodic method) Advantage: Needs fewer degrees of freedom The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering

4. Analysis methods (2/2) N-span time-domain analysis Periodic method Limitation: Linear analysis (freq. domain) Identical spans A large number of spans The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering

5. Periodic method (1/4) Time phase of response The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering t ulul t urur t’= S/V S: span length V: train speed u r (t)= u l (t - t’) p r (t)= - p l (t - t’) Fourier transform U r (w)= U l (w) P r (w)= - P l (w)

6. Periodic method (2/4) Transfer everything to frequency domain The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering U r (w)= U l (w) P r (w)= - P l (w)

7. Periodic method (3/4) Using Lagrange’s method The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering Original equations Constraint equations Lagrange multiplier

8. Periodic method (4/4) Unit moving load The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering Fourier transform

9. Finite element model The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering Track: BC elements Horizontal springs Girder: BC elements Periodic constraint nodes Foundation: 6x6 stiffness matrix (by FE/BE method)

10. Foundation model The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering

11. Periodic analysis result (1/2) Frequency range: 0.02 Hz ~ 15 Hz, df = 0.02 Hz Ground response (horizontal X direction) The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering Inverse Fourier transform

12. Time history analysis result (1/2) N-span finite element model N=6 to 40 Dynamic analysis method HHT dynamic time integration Using ABAQUS Time interval=0.005 sec. The 9 th International Conference on Computing in Civil and Building Engineering April 3-5, 2002 Taipei, Taiwan National Center for Research on Earthquake Engineering

13. Comparison of analysis results N=6 N=40

14. Summary and possible future work Summary A periodic method for an elevated rail bridge Frequency domain A large number of identical spans Linear analysis Compare to a time-domain dynamic analysis (N spans) The results tend to consistent when the N is larger Vibration of higher frequency differs FE model for the periodic method is much smaller

15. Summary and possible future work Possible future work The foundation can be modeled: Foundation-foundation interaction can be considered

16. Thank you very much