More and more attention has been paid to the research on bridge structure health monitoring and state analysis technology, especially the theory and technology of time series analysis based on bridge deflection information. In this paper, multi-fractal de-trend analysis (MF-DFA) is used to analysis the deflection information of bridges. The deformation characteristic parameters can be obtained by the multi-fractal de-trend fluctuation method, which can effectively identify the deformation degree of bridges. The analysis result shows that the multi-fractal de-trend fluctuation analysis method provides the theoretical basis and implementation approach for the health monitoring of the bridge.

#### Keywords

- deflection signal
- Multi-fractal
- De-trend analysis
- health monitoring of bridge

The bridge is an important component in the traffic network, its quality being crucial to people’s lives. As of 2016, one-quarter of China’s bridges have potential problems [1] according to statistics. These defective bridges need to be repaired regularly, which is an expensive task. If regular repairs are not undertaken, it will seriously affect the safety and stability of bridges. Bridge health inspection is an important measure to prevent and reduce accidents. Bridge deflection refers to the linear displacement of the centre of the bridge cross-section along the vertical direction of the axis when the bridge is stressed or the non-uniform temperature changes. In bridge health inspection, the bridge deflection information is often used to detect the deformation of the bridge.

The deflection signal will change due to the damage or mutation of the bridge [2]. Currently, the time-frequency analysis method is the main process of handling the non-stationary signal. For example, In [3] Wavelet transform is proposed to separate the transient and slow information of deflection signal. Hu et al. [4] verified the practicability of principal component analysis (PCA) in fault diagnosis, but the result also shows that PCA is not very sensitive to small faults. Due to the influence of nonlinear factors (impact load, damping, friction etc.), the deflection information of the bridge is nonlinear, non-stationary and non-periodic. Therefore, the structure state of the bridge cannot be fully reflected by the physical model of the bridge deflection data based on the traditional method [5].

Now, the method that is widely used to describe the object of study is fractal dimension [6]. As the bridge damage is affected by many factors, using the multi-fractal can depict the signal from overall irregularity and expresses the local behaviour finely [7, 8]. Therefore, this paper analyses the bridge deflection time-domain signal, and then uses the multi-fractal de-trend fluctuation analysis method to carry out the multi-fractal spectrum analysis of the bridge deflection signals, and summarises the connection between the parameters of the multi-fractal spectrum and the health state of the bridge, which provides the theoretical basis and the way of implementation for bridge health monitoring.

Bridge deflection signal is a non-stationary signal with small amplitude variation, and its inherent non-stationary characteristics are difficult to estimate. In this paper, the trend component of the bridge deflection signal is eliminated by the de-trended fluctuation analysis, and the deformation characteristics are described by the multi-fractal spectrum.

For deflection signal time series

Step 1: Determine the ‘profile’

Subtraction of the mean

Step 2: Divide the profile _{s}_{s}

Step 3: Calculate the local trend for each of the 2_{s}_{s}_{s}_{s}_{v}

Step 4: Average over all segments to obtain the _{q}_{q}_{q}_{q}

Step 5: Determine the scaling behaviour of the fluctuation functions by analysing log-log plots _{q}_{i}_{q}

Among solutions of

The generalised Hurst exponent _{0} and multi-fractal spectrum

The multi-fractal singularity spectrum obtained by the MF-DFA method is a set of parameters that can accurately describe the multi-fractal dynamic behaviour characteristics of deflection signals.

The singularity exponent _{0} corresponding to the maximum value of the spectral function _{0} represents the more irregular deflection signal. ∆_{max}) − _{min}). Multi-fractal spectrum width ∆_{max} − _{min}. When ∆_{0} is smaller, and the deflection signal fluctuation is more intense, the multi-fractal features are stronger [11]. Therefore, in this paper, we have selected _{0}, ∆

The deflection signals of bridges in a normal state, fatigue deformation and sudden deformation were collected from the detection signals of Chongqing xxx Bridge. The time-domain diagram is shown in Figure 1.

It can be seen from Figure 1, that the frequency and amplitude of bridge deflection signals are very small, which makes it difficult to obtain deformation characteristics by traditional spectrum analysis methods. Therefore, it is difficult to diagnose and predict bridge health state directly by amplitude spectrum.

By the theory of multi-fractal, the characteristics of deflection signals can be effectively described. Using this characteristic, trend analysis of deflection signals can be carried out by DFA to eliminate the non-stationary trend influence of time series multi-fractal features, and then the multi-fractal spectrum can be calculated for feature extraction.

Using DFA for fault prediction, the main operation is a de-trending operation, and the use of different order polynomial fitting means the removal of different types of a trend in the signal. The scale range of choice within the scale index can reflect the intrinsic characteristics of the signal. Only an appropriate scale can obtain reliable statistics, with the time sequence changing fast and with the maximum recommended being

Due to the non-periodic and non-stationary characteristics of the bridge deflection signals, the scaling exponents

Figure 3 shows the variation relationship between the generalised Hurst index _{q}

From Figure 3, we observe that the dynamic mechanism of deflection signal generated by the bridge under different deformation is different, which leads to great differences in the generalised Hurst index under different states. Moreover, the generalised Hurst index _{q}

Figure 4 shows the variation rule of the multi-fractal spectrum wave function of the bridge deflection signals with the singularity index in different states.

From Figure 4, it can be seen that the multi-fractal spectrum of the deflection signals of bridges varies with the degree of deformation. For example, the width of the singularity index ∆_{0} is also regular. On selection of the multi-fractal spectrum parameter, we ascertain _{0}, ∆

Feature parameters of the bridge deflection signal.

Parameter | Mutation | Fatigue Signal | Normal_1 | Normal_2 |
---|---|---|---|---|

_{0} | 2.072 | 1.873 | 1.835 | 1.780 |

∆ | 0.0254 | −0.2887 | 0.1633 | 0.04698 |

∆ | 2.55 | 1.31 | 0.775 | 0.576 |

Table 1 shows that when the bridge is in a healthy state, the deflection of the multi-fractal spectrum parameter ∆_{0} gradually increase. ∆

In conclusion, the multi-fractal spectrum parameter _{0} and ∆_{0} and multi-fractal spectrum width ∆

For bridge deflection signal, it is difficult to obtain effective fault characteristic quantity by the traditional spectrum analysis method. Therefore, the MF-DFA method is used to analyse the multi-fractal spectrum of the deflection signals of bridges under different states. The conclusions are as follows:

Deflection signals have obvious multi-fractal characteristics, and the multi-fractal characteristics of bridge deformation state are stronger than those corresponding to the normal state.

The MF-DFA method can effectively identify the fault state of the bridge deflection signals. The multi-fractal spectrum parameter _{0} and ∆

#### Feature parameters of the bridge deflection signal.

Parameter | Mutation | Fatigue Signal | Normal_1 | Normal_2 |
---|---|---|---|---|

_{0} | 2.072 | 1.873 | 1.835 | 1.780 |

∆ | 0.0254 | −0.2887 | 0.1633 | 0.04698 |

∆ | 2.55 | 1.31 | 0.775 | 0.576 |

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