# Time-Series and Frequency-Spectrum Correlation Analysis of Bridge Performance Based on a Real-Time Strain Monitoring System

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fu-Sui Bridge and SHM System

## 3. Data Measurements and Preprocessing

#### 3.1. Temperature and Strain Preprocessing

#### 3.2. Traffic Strain Counter

## 4. Bridge Performance Assessment

#### 4.1. Performance Due to Affected Loads

_{1}, b

_{2}and b

_{3}are unknown parameters; therefore, the number of unknowns u = k + 1, k is the number of input quantities; i = 1, …, n; n is the number of observations. The unknown parameters U (Equation (2)), U

^{T}= [a, b

_{1}, b

_{2}, b

_{3}]

^{T}, can be estimated and tested for statistical significance using the least squares method. The estimation process is examined after removing the insignificant parameters.

_{1}, w

_{2}, …, w

_{n})), w

_{i}is the chosen weight function as follows:

_{1}, v

_{2}, …, v

_{n}). It is assumed that the response error is a normal distribution, and the extreme values are rare. However, extreme values do occur and are called outliers. The main disadvantage of the least square (LS) fitting is its sensitivity to outliers. Outliers have a large influence on the fit because squaring the residuals magnifies the effects of these extreme observation points. Bi-square weighted robust predictors are used in the regression analysis to minimize the influence of outliers. Bi-square weights minimize a weighted sum of squares, where the weight given to each observation point depends on how far the point is from the fitted line. Points near the line get full weight; while points further from the line get reduced weight. Robust fitting with bi-square weights uses an iteratively re-weighted LS algorithm, as shown in Equation (3).

_{f,95%}confidence limit of the t distribution, where f is the order of freedom. As a result of the test, insignificant parameters are excluded from the function, and this procedure is continued till all parameters became significant (Martin, 2007). The second criterion is Akaike’s final prediction error (FPE), which is defined as:

^{T}of the robust regression models and the covariance matrices of these parameters were predicted using the LS method (Table 1). However, due to the sensitivity of the LS method to incompatible measurements, the existence of outliers was checked with bi-square weighted robust predictors. As a result of the investigation, w

_{i}bi-square weights were found to be close, or near to “1”, which showed that there were no outliers in the observations for the smoothed strains. In addition, the model evaluation criteria, t, FPE and R-square, values were calculated. The statistical significance of the model coefficients, presented in Table 1, were tested by comparing them to the confidence boundary of the t-distribution related to the degree of freedom at a confidence level of 95%, t

_{f, 95%}. Test results revealed that the coefficient pertaining to the traffic (V) in Model “1”, representing the strain of the bridge in Section 5, was statistically insignificant since t

_{v}< 1.96. Hence, the effect of traffic on the bridge strain was ignored in the static strain time domain analysis.

#### 4.2. Temperature Correlation Performance Analysis

#### 4.3. Frequency Correlation Analysis

## 5. Conclusions

- The static strain can be estimated using smoothed strain measurements, while the dynamic strain behavior can be extracted by filtering the strain measurements. Based on this conclusion, the traffic volume can be estimated, and the study reveals that the traffic volume on Fu-Sui Bridge increased during one year by 55%.
- The multi-input single-output robust regression identification model of strain measurements reveals that the influent portion of traffic loads in the static strain is lower than the air temperature and temperature changes, and it can be neglected in the case of studying the performance of the bridge based on the strain monitoring system.
- The time-series correlation analysis of strain and temperature revealed that the winter temperature time has more effect on the upper and lower strain behavior than summer temperature, while the summer time strain behavior is less reliable than winter time, and the behavior of the bridge during winter time is more stable than summer time. Furthermore, the temperature changes of the bridge section affect the lower plate girder more than the upper plate during summer time. This means that the direct air temperature effect is higher than indirect temperature effects. The linear fitting between strain and temperature changes shows that the bridge performance during winter time is more stable than summer time.
- The correlation of frequency spectrum analysis of strain residuals shows that the increased traffic volume on the bridge increases the bridge stability in vibration modes with more controlled bridge vibration. In addition, the air temperature and temperature changes of the bridge sections do not affect the frequency modes and power spectrum density of strain signals. The correlation of power spectrum density reveals that the dynamic performance of the bridge in summer and winter times is safe.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Measured and smoothed data of: (

**a**) strain of strain point S19; (

**b**) temperature of May 2012.

**Figure 4.**Outside and inside cross-section temperature: (

**a**) May and October 2012; (

**b**) January and March 2013.

**Figure 5.**Section 5: (

**a**) strain measurements; (

**b**) wavelet filter of S19; and (

**c**) traffic volume calculation (unit: number of vehicles).

**Figure 8.**Strain response variation for: (

**a**) May 2012; (

**b**) October 2012; (

**c**) January 2013; and (

**d**) March 2013.

**Figure 9.**Correlation scatter plots between S19 and S20 and temperature changes for the (

**a**,

**b**) S20 and (

**c**,

**d**) S19.

**Figure 10.**PSD of strain during: (

**a**) May 2012; (

**b**) October 2012; (

**c**) January 2013; and (

**d**) March 2013.

**Table 1.**Robust fit model of the strain S19 of Section 5 of the bridge. FPE, final prediction error.

Model | S = a + b_{1}(T) + b_{2}(DT) + b_{3}(V) | t_{(T,DT,V)} | FPE | R^{2} |
---|---|---|---|---|

1 | S = −93.726 + 12.306(T) + 11.630(DT) − 0.185(V) | 6.28, 5.76, −1.28 | 33.62 | 0.64 |

2 | S = −307.081 + 12.828(T) + 12.200(DT) | 6.62, 6.12 | 31.16 | 0.66 |

3 | S = −20.775 + 0.913(T) | 4.95 | 35.43 | 0.32 |

Parameter | S19 | S20 | S21 | S22 | To | |||||
---|---|---|---|---|---|---|---|---|---|---|

May | January | May | January | May | January | May | January | May | January | |

S19 | 1.00 | 1.00 | 0.57 | 0.95 | 0.47 | 0.92 | 0.78 | 0.89 | 0.46 | 0.95 |

S20 | 0.57 | 0.95 | 1.00 | 1.00 | 0.98 | 0.98 | 0.04 | 0.97 | 0.93 | 0.97 |

S21 | 0.47 | 0.92 | 0.98 | 0.98 | 1.00 | 1.00 | −0.06 | 0.98 | 0.96 | 0.97 |

S22 | 0.78 | 0.88 | 0.04 | 0.97 | −0.06 | 0.98 | 1.00 | 1.00 | −0.04 | 0.96 |

To | 0.46 | 0.95 | 0.93 | 0.97 | 0.96 | 0.97 | −0.04 | 0.95 | 1.00 | 1.00 |

**Table 3.**Coefficient of correlation among the calculated power spectrum density (PSD) of strain measurements.

Parameter | S19 | S20 | S21 | S22 | ||||
---|---|---|---|---|---|---|---|---|

May | January | May | January | May | January | May | January | |

S19 | 1.00 | 1.00 | 0.93 | 0.99 | 0.99 | 0.99 | 0.92 | 0.99 |

S20 | 0.93 | 0.99 | 1.00 | 1.00 | 0.93 | 0.99 | 0.91 | 0.99 |

S21 | 0.99 | 0.99 | 0.93 | 0.99 | 1.00 | 1.00 | 0.92 | 0.99 |

S22 | 0.92 | 0.99 | 0.91 | 0.99 | 0.92 | 0.99 | 1.00 | 1.00 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Kaloop, M.R.; Hu, J.W.; Elbeltagi, E.
Time-Series and Frequency-Spectrum Correlation Analysis of Bridge Performance Based on a Real-Time Strain Monitoring System. *ISPRS Int. J. Geo-Inf.* **2016**, *5*, 61.
https://doi.org/10.3390/ijgi5050061

**AMA Style**

Kaloop MR, Hu JW, Elbeltagi E.
Time-Series and Frequency-Spectrum Correlation Analysis of Bridge Performance Based on a Real-Time Strain Monitoring System. *ISPRS International Journal of Geo-Information*. 2016; 5(5):61.
https://doi.org/10.3390/ijgi5050061

**Chicago/Turabian Style**

Kaloop, Mosbeh R., Jong Wan Hu, and Emad Elbeltagi.
2016. "Time-Series and Frequency-Spectrum Correlation Analysis of Bridge Performance Based on a Real-Time Strain Monitoring System" *ISPRS International Journal of Geo-Information* 5, no. 5: 61.
https://doi.org/10.3390/ijgi5050061