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

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

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

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

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)

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)

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

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

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)

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

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

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

Comparison of analysis results N=6 N=40

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

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

Thank you very much