Shock and Vibration

Volume 2015, Article ID 741618, 9 pages

http://dx.doi.org/10.1155/2015/741618

## Effect of Temperature Variation on Vibration Monitoring of Prestressed Concrete Girders

^{1}Department of Ocean Engineering, Pukyong National University, 599-1 Daeyeon-3-dong, Nam-gu, Busan 608-737, Republic of Korea^{2}Structural Engineering Division, Korea Institute of Construction Technology, Gyeonggi-do 411-472, Republic of Korea^{3}Department of Ocean Engineering, Pukyong National University, Busan 608-737, Republic of Korea^{4}BT Consultant Co., Gyeonggi-do 411-472, Republic of Korea

Received 5 September 2014; Accepted 10 November 2014

Academic Editor: Ting-Hua Yi

Copyright © 2015 Thanh-Canh Huynh et al. 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

The effect of temperature variation on vibration monitoring of prestressed concrete (PSC) girders is experimentally analyzed. Firstly, vibration features such as autoregressive (AR) coefficient, correlation coefficient of power spectral density (CC of PSD), natural frequency, and mode shape are selected to estimate the effect of temperature variation on vibration characteristics of PSC girders. Secondly, vibration experiments on a lab-scale PSC girder are performed under the condition of temperature variation. Finally, the vibration features with respect to the temperature variation are analyzed to estimate the effect of temperature in vibration characteristics of the PSC girder.

#### 1. Introduction

For the past decades, the interest in structural health monitoring of prestressed concrete (PSC) structures has been increased. In a variety of civil engineering structures, PSC girders are main components to resist against external loadings [1]. For a PSC girder bridge, flexural stiffness in girder and prestress force in tendon are two essential parameters that should be secured for its serviceability and safety against external loadings. Unless the PSC girder bridges are instrumented at the time of construction, the occurrence of damage cannot be directly monitored and other alternative methods should be sought.

Since as early as 1970s, many researchers have focused on the possibility of using vibration characteristics of a structure as an indication of its structural damage [2–6]. The most appealing feature associated with using vibration properties is that they are relatively simple to measure and to utilize for a prompt diagnosis. Recently, research efforts have been made to monitor the change in modal properties of the PSC structures in relation to the change in prestress forces [7], to investigate the dynamic behaviors of prestressed composite girder bridges [8], and to identify the change in prestress forces by measuring dynamic responses of prestressed beams [9–11].

However, the practicality of the so-called “bridge diagnosis via vibration monitoring” is limited for real PSC girder bridges since even significant damage sources may not be revealed as remarkable changes in vibration features due to temperature-induced uncertainty [12–17]. In a real-life situation of a PSC girder bridge, it is reasonable to say that temperature differences are about 10~20°C during day and night and about 20~50°C during a year. To account for the significant condition, therefore, temperature-induced changes in vibration properties should be accounted for the PSC girder bridge.

This study presents the effect of temperature on vibration responses of the prestressed concrete girder and also estimates the relative sensitivities of the selected vibration features. The following approaches are implemented to achieve the objective. Firstly, vibration features such as autoregressive coefficient, power spectral density, natural frequency, and mode shape are selected for estimating the effect of temperature variation on PSC girders. Secondly, experiments on a lab-scale PSC girder are described. The experiments are performed under the condition of temperature variation. Finally, vibration characteristics of the PSC girder are analyzed with respect to the temperature variation. Temperature-induced effects on vibration monitoring of the PSC girder are estimated for the selected vibration features.

#### 2. Vibration Features for Estimation of Temperature Effect

Vibration features selected to estimate the effect of temperature variation on the PSC girder include autoregressive model (AR model), correlation coefficient of power spectral density (CC of PSD), natural frequency, and mode shape. Once acceleration responses are measured at distributed location, the vibration features are extracted for the structure. For each type of vibration features, the effect of temperature is estimated by regression analysis.

##### 2.1. Autoregressive Model

AR model is one of the time series analysis methods. It forecasts future responses from past time history responses. In this study, the AR model is defined in terms of acceleration responses as follows [18]:where is the time history of acceleration at time step , is the th AR coefficient, is the order of AR model, and is the residual error. The change in AR coefficients represents the change in structural parameters.

Mahalanobis squared distance (MSD) is utilized to calculate the change of AR coefficients due to structural change. Considerwhere is the MSD value, is the AR coefficient vector (outlier), is the mean vector of sets of AR coefficient, and is the covariance matrix of sets of AR coefficient. The variation of MSD indicates the variation of AR coefficients due to structural change. The number of set, , is determined by partial autocorrelation function of the first acceleration sample in intact state. It is noted that the number of set should be larger than the order of AR model.

##### 2.2. Correlation Coefficient of Power Spectral Density

Assume there are two acceleration signals, and , measured before and after structural change, respectively. The corresponding power spectral densities, and , are calculated from Welch’s procedure as [18]where and are the frequency response transformed from correspondent acceleration signal; is the number of divided segments; and is the data length of divided segment.

The correlation coefficient (CC) of PSDs represents the linear identity between the two PSDs obtained before and after structural change, as follows: where is the expectation operator and and are the standard deviations of PSDs of acceleration signals measured before and after structural change, respectively. If any structural change occurs in target structure, its acceleration responses would be affected and, consequently, the decrement of CC can be a warning sign for the presence of the special cause in the structure.

For estimating CC of PSDs due to temperature variation, the excitation force should be consistently maintained during experiments. In the lab-scale experiment, the impact testing can be easily maintained to be consistent. However, the excitation magnitude can be different for each testing in the full-scale civil engineering structures. For such cases, the vibration responses should be normalized according to the excitation magnitude. In addition, the ambient vibration data seems to be difficult to apply as the input excitation varies over time, which results in different CC of PSDs although the temperature remains the same.

##### 2.3. Modal Parameters

Frequency domain decomposition (FDD) method [19, 20] is used to extract modal parameters such as natural frequency and mode shape. The singular values of the PSD function matrix are used to estimate the natural frequencies instead of the PSD functions themselves as follows [20]:where is the diagonal matrix consisting of the singular values, ’s , and and are unitary matrices. Since is symmetric, becomes equal to . In this FDD method, the natural frequencies can be determined from the peak frequencies of the singular value and the mode shape from anyone of the column vectors of at the corresponding peak frequencies. In this study, the first singular value is used to estimate the modal parameters.

Mode shapes measured before and after structural change are examined by modal assurance criterion (MAC). The MAC provides a measure of the least squares deviation or scatter of the points from the linear correlation where and are the modal vectors which are extracted from acceleration signals measured before and after structural change, respectively. The MAC value provides quantifying of the comparison between the two sets of mode shape data. In practice, a value of MAC close to the unity is expected if the two sets of mode shapes have no differences caused by structural change; otherwise, the MAC value would decrease. This method can be utilized to obtain the information on what vibration modes are more sensitive to the structural change.

##### 2.4. Regression Analysis

Regression analysis gives the information on the relationship between a response variable and one or more independent variables. From regression analysis, the response variable is expressed as a function of the predictor variables. The relationship obtained from regression analysis can be used to predict values of the response variable and identify variables that most affect the response. The value of each predictor variable can be accessed through statistical tests on the estimated coefficients of the predictor variables.

An example of a regression model is the simple linear regression model which is a linear relationship between response variable (RV) and the predictor variable of the form in which and are regression coefficients which are unknown modal parameters. In this study, the response variables are MSD of AR coefficient (2), CC of PSDs (3), and natural frequency and mode shape calculated by (6). Also, the predictor variable is temperature measured from the experiment. The error term has to be equal to zero on average. In statistics, simple linear regression is the least squares estimator of a linear regression model with a single predictor variable. Simple linear regression fits a straight line through the set of points in such a way that makes the sum of squared residuals of the model (i.e., vertical distances between the points of the data set and the fitted line) as small as possible.

#### 3. Experiments on PSC Girder

As illustrated in Figure 1, dynamic tests were performed on a lab-scaled PSC girder. While room temperatures were handled to vary in the range of about 5~23°C, a set of prestress cases were simulated to the PSC girder from which vibration responses were measured to determine vibration features and modal parameters. In this study, vibration features of interest include autoregressive coefficient and power spectral density. Also, modal parameters of interest are natural frequencies and mode shape. Note that damping properties of the PSC girder are not dealt with due to the difficulties in measurement and interpretation.