Bayesian estimation of instantaneous frequency reduction on cracked concrete railway bridges under high-speed train passage

Abstract When a train passes along prestressed concrete (PC) bridges, crack opening reduces the girder stiffness, providing important indicators for the practical evaluation of bridge performance. However, this phenomenon is difficult to detect in the free vibrations induced by trains or via hammer tests as these vibrations have very small amplitudes. Moreover, although the bridge under a passing train vibrates with large amplitude, the vibrational state is forced, and the girder stiffness induced by crack opening occurs only in the downward displacement. To estimate this complex but valuable system, this study establishes a time-varying autoregressive with eXogenous (TV-ARX) model and a hierarchical Bayesian estimation (Bayesian TV-ARX) method. The exogenous variables in the TV-ARX model are the modal-transformed moving loads, which constitute the main excitation frequency of the running train. The variation of the autoregressive (AR) coefficient includes the modal characteristic as a random walk process. When a train passes, the Bayesian TV-ARX estimates the instantaneous amplitude-dependent drop of the bridge frequency from the displacement response. After formulating the Bayesian TV-ARX, the effects of the vehicle–bridge interaction (VBI), train speed, and track irregularity on the accuracy of the instantaneous stiffness decrease estimated via the Bayesian TV-ARX were verified using VBI simulations of a nonlinear beam, in which the bending stiffness decreased only during a downward displacement. Even when the VBI effects overlapped, the model accurately estimated the reduced bridge stiffness due to crack opening. Next, the Bayesian TV-ARX was applied to two PC bridges on a real high-speed railway. On one of the bridges, the estimated stiffness decreased by ~ 12% when the bridge was downward-displaced by crack opening during the train passage. Such results on a real bridge under operation have not been previously reported. Four months later, the amplitude-dependent decrease in the bridge frequency was amplified by crack propagation. These results are important evidences of a nonlinear and nonstationary system. Besides solving real problems, the proposed method is expected to significantly contribute to condition-based maintenance and structure-health monitoring of PC bridges. In particular, it enables early detection of damage and deterioration in addition to bridge performance evaluation over time.

[1]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[2]  Biswajit Basu,et al.  ASDAH: An automated structural change detection algorithm based on the Hilbert–Huang transform , 2014 .

[3]  Yuan Wang,et al.  Assessment and prediction of high speed railway bridge long-term deformation based on track geometry inspection big data , 2021 .

[4]  Mahir Ülker-Kaustell,et al.  Seasonal effects on the stiffness properties of a ballasted railway bridge , 2013 .

[5]  Richard L. Weaver,et al.  On crack identification and characterization in a beam by non-linear vibration analysis , 1995 .

[6]  Joerg F. Hipp,et al.  Time-Frequency Analysis , 2014, Encyclopedia of Computational Neuroscience.

[7]  K. Terada,et al.  Simulations of cohesive fracture behavior of reinforced concrete by a fracture-mechanics-based damage model , 2019, Engineering Fracture Mechanics.

[8]  Raimundo Delgado,et al.  Finite element model updating of a bowstring-arch railway bridge based on experimental modal parameters , 2012 .

[9]  T. Chondros,et al.  VIBRATION OF A BEAM WITH A BREATHING CRACK , 2001 .

[10]  Mahir Ülker-Kaustell,et al.  Time-frequency analysis of railway bridge response in forced vibration , 2016 .

[11]  R. Ghanem,et al.  A WAVELET-BASED APPROACH FOR THE IDENTIFICATION OF LINEAR TIME-VARYING DYNAMICAL SYSTEMS , 2000 .

[12]  Hajime Wakui,et al.  IMPACT FACTORS OF CONCRETE GIRDERS COPING WITH TRAIN SPEED-UP , 2005 .

[13]  S. Mukherjee,et al.  MODAL ANALYSIS OF A CRACKED BEAM , 1997 .

[14]  J. Beck,et al.  Updating Models and Their Uncertainties. I: Bayesian Statistical Framework , 1998 .

[15]  Arnaud Castel,et al.  Modeling the dynamic stiffness of cracked reinforced concrete beams under low-amplitude vibration loads , 2016 .

[16]  Hitoshi Shiohara,et al.  Tendon Model for Nonlinear Analysis of Prestressed Concrete Structures , 2001 .

[17]  Arturo González,et al.  A wavelet-based damage detection algorithm based on bridge acceleration response to a vehicle , 2012 .

[18]  Jer-Nan Juang,et al.  An eigensystem realization algorithm for modal parameter identification and model reduction. [control systems design for large space structures] , 1985 .

[19]  Yozo Fujino,et al.  Bridge monitoring in Japan: the needs and strategies , 2011 .

[20]  Siem Jan Koopman,et al.  A simple and efficient simulation smoother for state space time series analysis , 2002 .

[21]  Michael Feldman,et al.  Time-varying vibration decomposition and analysis based on the Hilbert transform , 2006 .

[22]  Huajiang Ouyang,et al.  Moving-load dynamic problems: A tutorial (with a brief overview) , 2011 .

[23]  Nao-Aki Noda,et al.  ANALYSES OF DYNAMIC RESPONSE OF VEHICLE AND TRACK COUPLING SYSTEM WITH RANDOM IRREGULARITY OF TRACK VERTICAL PROFILE , 2002 .

[24]  Abhishek K Gupta,et al.  Numerical Methods using MATLAB , 2014, Apress.

[25]  Raid Karoumi,et al.  Closed-form solution for the mode superposition analysis of the vibration in multi-span beam bridges caused by concentrated moving loads , 2013 .

[26]  Thanh Q. Nguyen,et al.  A data-driven approach based on wavelet analysis and deep learning for identification of multiple-cracked beam structures under moving load , 2020 .

[27]  Paul R. White,et al.  BISPECTRAL ANALYSIS OF THE BILINEAR OSCILLATOR WITH APPLICATION TO THE DETECTION OF FATIGUE CRACKS , 1998 .

[28]  Yang Xiang,et al.  Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load , 2008 .

[29]  Paul Sas,et al.  Parametric modal identification of time-varying structures and the validation approach of modal parameters , 2014 .

[30]  Y. Chu,et al.  Analysis of forced bilinear oscillators and the application to cracked beam dynamics , 1992 .

[31]  M. Mahmoud,et al.  DYNAMIC RESPONSE OF A BEAM WITH A CRACK SUBJECT TO A MOVING MASS , 2002 .

[32]  Mahir Ülker-Kaustell,et al.  Application of the continuous wavelet transform on the free vibrations of a steel–concrete composite railway bridge , 2011 .

[33]  Satish Nagarajaiah,et al.  Behavior Analysis and Early Warning of Girder Deflections of a Steel-Truss Arch Railway Bridge under the Effects of Temperature and Trains: Case Study , 2019, Journal of Bridge Engineering.

[34]  Hong Hao,et al.  Damage detection in bridge structures under moving loads with phase trajectory change of multi-type vibration measurements , 2017 .

[35]  G. Kitagawa.  Smoothness priors analysis of time series , 1996 .

[36]  Bojana V. Rosić,et al.  Model updating techniques for damage detection in concrete beam using optical fiber strain measurement device , 2016 .

[37]  Kiyoyuki Kaito,et al.  Bayesian time–frequency analysis of the vehicle–bridge dynamic interaction effect on simple-supported resonant railway bridges , 2020 .

[38]  Maciej Niedzwiecki,et al.  Identification of Time-Varying Processes , 2000 .

[39]  Helmut J. Pradlwarter,et al.  Reliability assessment of railway bridges subjected to high-speed trains considering the effects of seasonal temperature changes , 2016 .

[40]  Paolo Bonato,et al.  USE OF CROSS-TIME–FREQUENCY ESTIMATORS FOR STRUCTURAL IDENTIFICATION IN NON-STATIONARY CONDITIONS AND UNDER UNKNOWN EXCITATION , 2000 .

[41]  C. Williams,et al.  Review of full-scale dynamic testing of bridge structures , 1995 .

[42]  Bijan Samali,et al.  Time-varying characteristics of bridges under the passage of vehicles using synchroextracting transform , 2020, Mechanical Systems and Signal Processing.

[43]  Peter Avitabile,et al.  Photogrammetry and optical methods in structural dynamics – A review , 2017 .

[44]  Spilios D. Fassois,et al.  Non-stationary random vibration modelling and analysis via functional series time-dependent ARMA (FS-TARMA) models – A critical survey , 2014 .

[45]  Andrea Collina,et al.  Drive-by methodology to identify resonant bridges using track irregularity measured by high-speed trains , 2021 .

[46]  Jiahao Lin,et al.  Symplectic random vibration analysis of a vehicle moving on an infinitely long periodic track , 2010 .

[47]  Yeoshua Frostig,et al.  Free vibrations of cracked prestressed concrete beams , 2004 .

[48]  Filipe Magalhães,et al.  Vibration based structural health monitoring of an arch bridge: From automated OMA to damage detection , 2012 .

[49]  Christopher M. Bishop,et al.  Pattern Recognition and Machine Learning (Information Science and Statistics) , 2006 .

[50]  Shuncong Zhong,et al.  Crack detection in simply supported beams without baseline modal parameters by stationary wavelet transform , 2007 .

[51]  H. P. Lee,et al.  Dynamic response of a cracked beam subject to a moving load , 1994 .

[52]  O. S. Salawu.  Detection of structural damage through changes in frequency: a review , 1997 .

[53]  Tesheng Hsiao.  Time-varying system identification via maximum a posteriori estimation and its application to driver steering models , 2008, 2008 American Control Conference.

[54]  Ting-Hua Yi,et al.  Modal Identification of High-Speed Railway Bridges through Free-Vibration Detection , 2020 .

[55]  Ting-Hua Yi,et al.  Continuous tracking of bridge modal parameters based on subspace correlations , 2020, Structural Control and Health Monitoring.

[56]  Weichen Xue,et al.  Prediction of Long-Term Deflections for High-Speed Railway Prestressed Concrete Beams , 2016 .

[57]  Heung-Fai Lam,et al.  Multicrack Detection on Semirigidly Connected Beams Utilizing Dynamic Data , 2008 .

[58]  M. Olsson.  Finite element, modal co-ordinate analysis of structures subjected to moving loads , 1985 .

[59]  Paul R. White,et al.  THE ANALYSIS OF NON-STATIONARY SIGNALS USING TIME-FREQUENCY METHODS , 1996 .

[60]  A Matsuura.  DYNAMIC BEHAVIOR OF BRIDGE GIRDER FOR HIGH SPEED RAILWAY BRIDGE , 1979 .

[61]  Spilios D. Fassois,et al.  MSSP special issue on the identification of time varying structures and systems , 2014 .

[62]  Ton-Lo Wang,et al.  Impact in highway prestressed concrete bridges , 1992 .

[63]  Ahsan Kareem,et al.  Non-stationary frequency domain system identification using time–frequency representations , 2016 .

[64]  Y. C. Chu,et al.  Vibrations of beams with a fatigue crack , 1992 .

[65]  N. Roveri,et al.  Damage detection in structures under traveling loads by Hilbert–Huang transform , 2012 .

[66]  Sheng-En Fang,et al.  Probabilistic damage identification incorporating approximate Bayesian computation with stochastic response surface , 2019 .

[67]  Gang Wang,et al.  Bayesian approach to breathing crack detection in beam structures , 2017 .

[68]  Fumiaki Uehan.  Development of the U-Doppler Non-contact Vibration Measuring System for Diagnosis of Railway Structures , 2008 .

[69]  Raid Karoumi,et al.  Train–bridge interaction – a review and discussion of key model parameters , 2014 .

[70]  Boris A. Zárate,et al.  Bayesian model updating and prognosis of fatigue crack growth , 2012 .

[71]  A. Rama Mohan Rao,et al.  A novel vibration based breathing crack localization technique using a single sensor measurement , 2019 .

[72]  Jin-Song Pei,et al.  Solving Dynamical Systems Involving Piecewise Restoring Force Using State Event Location , 2012 .

[73]  Wieslaw J. Staszewski,et al.  Wavelet-based Frequency Response Function for time-variant systems—An exploratory study , 2014 .

[74]  Saeed Ziaei-Rad,et al.  Vibration analysis of beams with open and breathing cracks subjected to moving masses , 2009 .

[75]  Ting-Hua Yi,et al.  Clustering Number Determination for Sparse Component Analysis during Output-Only Modal Identification , 2019, Journal of Engineering Mechanics.

[76]  M. D. Martínez-Rodrigo,et al.  Influence of the vehicle model on the prediction of the maximum bending response of simply-supported bridges under high-speed railway traffic , 2014 .