# An Integrated Machine Learning Algorithm for Separating the Long-Term Deflection Data of Prestressed Concrete Bridges

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Multiscale Characteristics of Long-Term Deflection Data

_{v}

_{,}which is an irreversible index that can be used to evaluate the condition of a bridge. D

_{v}can be expressed as: align symbols or replace with text versions

_{v}), which will increase each year, the frequency can be considered 0. The amplitude of the live load deflection and temperature deflection is significant in comparison with the yearly increment caused by irreversible structural deflection D

_{v}.

## 3. An Integrated Machine Learning Algorithm

_{i}) are extracted. The U

_{i}extracted by the PCA algorithm are uncorrelated, but they are not statistically independent. In the last step, a further procedure called FastICA needs to be performed to derive the independent deflection components.

#### 3.1. EEMD

- (1)
- Add a white-noise series to the targeted signal.$${x}_{m}(t)=x(t)+{n}_{m}(t)$$
_{e}), which starts from 1. - (2)
- Decompose the signal ${x}_{m}(t)$ into a series of IMFs, ${c}_{i,m},(i=1,2\cdots N-1)$, and a residual, ${r}_{m}$, using EMD. We can obtain:$${x}_{1}(t)={\displaystyle \sum _{i=1}^{N-1}{c}_{i,1}(t)}+{r}_{1}(t)$$
- (3)
- Repeat Step (1) and Step (2) for m
_{e}trials.$${x}_{{m}_{e}}(t)={\displaystyle \sum _{i=1}^{N-1}{c}_{i,{m}_{e}}(t)}+{r}_{{m}_{e}}(t)$$ - (4)
- The final IMFs are obtained by overall averaging the IMFs produced in each trial.$$\overline{{c}_{i}}(t)=\frac{1}{N}{\displaystyle \sum _{m=1}^{{m}_{e}}{c}_{i,m}(t)}$$$$\overline{r}(t)=\frac{1}{N}{\displaystyle \sum _{m=1}^{{m}_{e}}{r}_{m}(t)}$$

#### 3.2. PCA

_{L×R}is a factorization of the form ${Y}_{L\times R}=\left[{U}_{L\times L}{\sum}_{L\times R}\text{}{V}_{R\times R}^{T}\right]$, where U and V are orthogonal matrices (with orthogonal columns) and $\sum $ is a matrix of r singular values ${\lambda}_{r}={\lambda}_{r\times r}$, where ${\lambda}_{1}\ge {\lambda}_{2}\ge \cdots \ge {\lambda}_{r}\ge 0$. The contribution of each vector is ranked based on the magnitude of its corresponding singular value. The result of the PCA is usually presented by the component score ${\beta}_{i}$ for each IMF:

#### 3.3. FastICA

_{i}extracted by the PCA algorithm are uncorrelated, but they are not statistically independent. A further procedure, i.e., ICA, is needed to derive the independent deflection components. ICA [42] is a blind source separation technique that extracts statistically independent components from a set of recorded signals.

## 4. Numerical Simulation

#### 4.1. Characteristic of the Individual Deflection Component of Different Effects

#### 4.1.1. Live Load Effect

_{1}is a two-axle vehicle model with a distance of 4.4 m and a total weight of 110 kN, accounting for 16.34% of the total traffic volume. Model M

_{2}is a three-axle vehicle model with a wheelbase of 3.4 and 4.92 m and a total vehicle weight of 167 kN, accounting for 0.03% of the total traffic volume. Therefore, Model M

_{1}is used to calculate the deflection of the live loads in Midas/Civil software.

_{1}passes through the bridge at a speed of 60 km/h. The maximum deflection is −5.3 mm. The result of fast Fourier transformation (FFT) is shown in Figure 6b. It is evident that the main frequencies of live loads deflection are greater than 200 Hz, which belongs to the higher-frequency domain. Therefore, the live load deflection effect can be separated by a Butterworth filter.

#### 4.1.2. Temperature Effect

_{1}(+1.48 mm) with a global temperature increase of 1 °C but develops d

_{2}(−2.92 mm) with an increase in the gradient temperature of 1 °C. It is assumed that there is a linear relationship between the temperature and the deflection of the bridge structure considering the sinusoidal variation. According to Equation (13), the deflection time history of temperature effect deflection can be obtained at a sampling frequency of 1 time/hour, as shown in Figure 7a. The sampling frequency is regarded as 1 Hz. For FFT, the frequency of the daily temperature and annual temperature effects are 0.04 and 0 Hz, respectively.

#### 4.1.3. Effect of Concrete Shrinkage and Creep

_{0}denote the calculating age and loading age of the concrete, respectively. t

_{1}equals 1 day. The shrinkage strain can be expressed as:

_{0}are the theoretical thickness of components and 100 mm, respectively; and t

_{s}is the age of concrete at the beginning of shrinkage. As shown in Figure 8, the bridge develops a deflection of −20.14 mm caused by shrinkage and creep over three years. According to the deflection curve, the expression of fitting curve is obtained by an exponential function, shown as Equation (16):

#### 4.1.4. Effect of Prestress Loss

#### 4.2. Validation Result

#### 4.2.1. Total Deflection

_{n}is added to the noise-free signal to obtain the source signal. The signal vectors $\widehat{s}\left(i\right)$ and $x(i)$ are established as:

_{S}and P

_{N}represent the effective power of the noise-free signal and noise, respectively. $x(i)$ and $\widehat{s}(i)$ represent the signal vectors of the source signal data and the reference noise-free signal data, respectively.

#### 4.2.2. Procedure of Separation

_{e}) are defined as 0.4 and 100, respectively. As a result, we obtained 10 IMFs and a residue (as shown in Figure 11).

_{i}is the i-th source signal component, and z

_{i}is the i-th estimated output signal component, shown in Figure 13. The range of the correlation coefficient is [−1, 1]. The closer $\left|\psi \right|$ is to 1, the closer the correlation is. Usually, when $\left|\psi \right|$ is greater than 0.8, the two vectors are considered to have a high linear correlation.

## 5. Practical Verification

#### 5.1. Description of the Structural Health Monitoring (SHM) System

#### 5.2. Processing of the Monitored Deflection Data

#### 5.2.1. Live Loads Effect

#### 5.2.2. Temperature Effect

#### 5.2.3. Structural Deflection

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Time history and frequency spectrum of live load deflection. (

**a**) Time history data. (

**b**) Frequency spectrum.

**Figure 7.**Characteristics of the deflection of temperature effect D

_{T}. (

**a**) Three years’ time history of temperature effect deflection D

_{T}. (

**b**) Power spectrum of D

_{T}. The sampling frequency is regarded as 1 Hz.

**Figure 10.**Total deflection of different effects with 10% noise level (not including live load effect).

**Figure 13.**Individual deflection component after separation. (

**a**) Daily temperature deflection effect (D

_{T}

_{1}). (

**b**) Annual temperature deflection effect (D

_{T}

_{2}). (

**c**) Structural deflection D

_{V}(including D

_{S}, D

_{C}, and D

_{P}).

**Figure 17.**Separated daily temperature deflection effect. (

**a**) Time history data. (

**b**) Power spectrogram.

**Figure 18.**Separated annual temperature deflection effect. (

**a**) Time history data. (

**b**) Power spectrogram.

**Figure 19.**Separated structural deflection effect (including D

_{S}, D

_{C}, and D

_{P}). (

**a**) Time history data. (

**b**) Power spectrogram.

Effects | Frequency Domain | |||
---|---|---|---|---|

Low | Medium | High | ||

Live loads | D_{L} | |||

Daily temperature variation (D_{T}_{1}) | D_{T} | |||

Annual temperature variation (D_{T}_{2}) | ||||

creep | D_{v} | |||

shrinkage | ||||

material deterioration | ||||

noise | - |

**Table 2.**Correlation coefficients ($\psi $ ) between the source signals and estimated components under different noise levels.

Noise Level (SNR) | Daily Temperature Effect D_{T}_{1} | Annual Temperature Deflection Effect D_{T}_{2} | Structural Deflection D_{V} |
---|---|---|---|

5% | 0.921 | 0.937 | 0.916 |

10% | 0.822 | 0.837 | 0.804 |

15% | 0.691 | 0.728 | 0.703 |

20% | 0.413 | 0.506 | 0.398 |

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**MDPI and ACS Style**

Ye, X.; Chen, X.; Lei, Y.; Fan, J.; Mei, L.
An Integrated Machine Learning Algorithm for Separating the Long-Term Deflection Data of Prestressed Concrete Bridges. *Sensors* **2018**, *18*, 4070.
https://doi.org/10.3390/s18114070

**AMA Style**

Ye X, Chen X, Lei Y, Fan J, Mei L.
An Integrated Machine Learning Algorithm for Separating the Long-Term Deflection Data of Prestressed Concrete Bridges. *Sensors*. 2018; 18(11):4070.
https://doi.org/10.3390/s18114070

**Chicago/Turabian Style**

Ye, Xijun, Xueshuai Chen, Yaxiong Lei, Jiangchao Fan, and Liu Mei.
2018. "An Integrated Machine Learning Algorithm for Separating the Long-Term Deflection Data of Prestressed Concrete Bridges" *Sensors* 18, no. 11: 4070.
https://doi.org/10.3390/s18114070