Shock and Vibration

Volume 2017 (2017), Article ID 6423039, 9 pages

https://doi.org/10.1155/2017/6423039

## Damage Detection in Railway Truss Bridges Employing Data Sensitivity under Bayesian Framework: A Numerical Investigation

Department of Civil Engineering, IIT Kanpur, Kanpur 208016, India

Correspondence should be addressed to Samit Ray-Chaudhuri; ni.ca.ktii@crtimas

Received 9 October 2016; Revised 23 February 2017; Accepted 12 March 2017; Published 6 April 2017

Academic Editor: Evgeny Petrov

Copyright © 2017 Kanta Prajapat and Samit Ray-Chaudhuri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In general, for a structure it is quite difficult to get information about all of its modes through its dynamic response under ambient or external excitation. Therefore, it is vital to exhaustively use the available information in the acquired modal data to detect any damage in the structures. Further, in a Bayesian algorithm, it can be quite beneficial if a damage localization algorithm is first used to localize damage in the structure. In this way, the number of unknown parameters in the Bayesian algorithm can be reduced significantly and thus, the efficiency of Bayesian algorithm can be enhanced. This study exploits a mode shape and its derivative based approach to localize damage in truss type structures. For damage quantification purpose, a parameter sensitivity based prediction error variance approach in Bayesian model updating is employed, which allows extracting maximum information available in the modal data. This work employs the sensitivity based Bayesian algorithm to determine the posterior confidence in truss type railway bridges. Results of the study show that the proposed approach can efficiently detect and quantify damage in railway truss bridges.

#### 1. Introduction

The collapse or damage, especially in large structures such as bridges, can lead to huge losses in terms of human lives and economic burden. The repair after a collapse is a tedious and time taking task, which renders the bridge inoperable (downtime) for a considerable period of time adding to the indirect loss. Thus, all the major bridges are needed to be monitored in regular basis and needed to be repaired or rehabilitated whenever required. As specified in a recent report on the condition of Indian railway bridges by Indian Institute of Science (IISc), Bangalore (condition assessment of railway bridges, 2010, http://civil.iisc.ernet.in/crbridge.pdf), the damage has been induced in many of these bridges because of increase in average traffic and associated traffic loads. In fact, the design axle load has been increased from 18 t to 22 t in recent years. These factors may lead to minor damage such as cracks, which can be worrisome to railway authority.

In the last few decades, Bayesian model updating has rapidly arisen as a reliable and effective approach to solve system identification problems probabilistically and, in turn, to detect damage in a system. The efficiency of Bayesian model updating depends on various issues such as the efficiency of simulation algorithm, data used for updating, prior distributions, and likelihood function. Many of these issues have been successfully resolved in recent years [1–13]. Any kind of damage in a structure causes change in its structural parameters, for example, stiffness and mass. Subsequently, the dynamic properties such as natural frequencies and mode shapes also get changed. The change in these dynamic properties due to a change in structural parameters has been utilized for years to identify damage in structures. Sohn and Law [14] employed incomplete and noisy modal data to localize and quantify the amount of damage under Bayesian inference. A damage localization technique in structures under Bayesian inference using modal data is presented by Huhtala and Bossuyt [15] for a steel cantilever beam. Damage detection in plate type structures is studied by Kurata et al. [16]. Simoen et al. [17] did damage assessment of a slice of 7-story RC building using Bayesian uncertainty quantification technique.

The efficiency of any Bayesian quantification algorithm can be improved significantly by reducing the number of unknowns in the problem. This can be achieved by using a damage localization algorithm to localize damage in the structure before quantifying the damage in the structure. The Bayesian algorithm basically works on error minimization between the response of a mathematical model and the response of the actual structure to update the parameters of the mathematical model. This is achieved in repeated cycles or repeated runs of Markov chain. In one run of Markov chain, each unknown parameter is picked up one by one and its new proposed value is being compared to accept or reject with the previous value for a better fit of the actual structural response. While doing this, all other unknown parameters are being kept on the best achieved values from the previous or current run, which are not the actual value of these parameters. Therefore, when an inverse problem is solved under Bayesian inference with many unknown parameters and limited Bayesian evidence, the convergence or getting the stationary distribution of Markov chain becomes difficult. It costs both the computational efficiency and the accuracy of the solution. This is particularly true when the employed data in Bayesian evidence are contaminated with noise, which is a general situation to find in any practical application. The present noise in the data may lead to the convergence of the chain to a false solution. By employing the information about the undamaged members using the damage localization algorithm, the Bayesian algorithm is made to search the optimum value of unknown parameters in right direction. This causes the algorithm to escape any false solution of the inverse problem. Hence, the accuracy and computational efficiency of the Bayesian algorithm can be improved by limiting the number of unknown parameters with the localization algorithm.

It is evident from literature that comparison of mode shape and its derivatives for undamaged and damaged structures effectively localizes damage in a structure. Modes shapes curvatures idea for damage localization was first proposed by Pandey et al. [18] in which mode shape curvature was said to be sensitive parameter for damage localization. Sampaio et al. [19] extended the idea of Pandey et al. [18] by using the curvature-based method for frequency response function instead of mode shape and demonstrated the potential of this approach by considering real time data. Later, Wahab [20] utilized mode shape curvature along with natural frequencies and mode shapes for finite element model updating. Ray Chaudhuri [21] investigated the behaviour of eigenproperties, namely, frequencies and mode shapes, in presence of stiffness degradation in simulated structures such as shear building and steel moment-resisting frame (SMRF). A mathematical formulation with the use of perturbation approach was developed in this study to inspect the change in frequencies as well as mode shapes for a given location of stiffness degradation. It may be noted that higher modes are usually difficult to get for a typical civil engineering structure as these modes do not get excited under ambient conditions due to higher input energy requirements. Thus, the fundamental mode is more frequently used for damage detection. Zhu et al. [22] demonstrated the efficiency of the change in slope of the first mode shape as a damage sensitive feature by performing a numerical study on an eight-story shear building and conducting experiments on a three-story building model. Dilena et al. [23] demonstrated that mode shape curvature can be a useful term for damage location on a reinforced concrete single span bridge. Bai et al. [24] applied mode shape curvature-based technique on a two-dimensional plate grid structure to locate the damage in the structure. Recently, Roy and Ray-Chaudhuri [25] proposed a mathematical basis to establish a correlation between structural damage and change in the structure’s fundamental mode shape and its derivatives using a perturbation approach. In this work, this approach is used to localize the damage in the truss type structure.

When modal data is used as evidence in a Bayesian damage detection algorithm, many studies have suggested different ways to take variance for prediction error model of frequency and mode shape data types [26–30]. Most of these studies consider only two variances: one for frequencies of all modes and the other for mode shape components of all modes. Only a few studies consider separate variances for data of different modes. However, depending on various conditions, all frequencies and mode shape components of all modes may require separate variances for the prediction error models for an efficient information extraction from these data points. This study employs a sensitivity based approach recently developed by the authors [30] to derive the variances for prediction error models of different data points to efficiently extract the information from these data points.

This work is structured in three parts: the first part deals with damage localization, whereas the second and third parts emphasize damage quantification and posterior resolution of unknown parameters, respectively, in a railway truss bridge. For localization purpose, change in the fundamental mode shape based algorithm is used. Quantification and posterior resolution are achieved employing the sensitivity based algorithm under Bayesian inference. The modal data from the first two modes are used for updating the stiffness parameters of the truss bridge model. Markov Chain Monte Carlo simulation technique is employed using Metropolis-Hasting algorithm to simulate the samples from the posterior distribution. The mean of the posterior distribution is taken as the parameter estimation of distribution to represent the unknown stiffness parameters. Results of the study show the efficiency of used approaches for damage detection in truss type bridges.

#### 2. Mode Shape Based Damage Localization

From literature it is evident that the damage in a structure can be localized by comparing its damaged and undamaged mode shapes and their derivatives. Recently, Roy and Ray-Chaudhuri [25] proposed a mathematical basis to establish a correlation between structural damage and change in the structure’s fundamental mode shape and its derivatives using a perturbation approach.

For a shear beam (Figure 1), they have illustrated that the difference between the damaged and undamaged fundamental mode shapes can be expressed ashere is a constant as given in Roy and Ray-Chaudhuri [25] and is as follows:A jump of can be observed in the function between the locations and (1). This causes a steep slope around (location of damage). It may also be observed that if the beam is discretized into a large number of elements, that is, , this slope approaches a Dirac delta-type function. Further, this implies that the second derivative of , that is, , changes its sign at . Therefore, by looking at the difference of damaged and undamaged mode shape and the first and second derivatives of this difference, the location of the damage in a structure can be determined.