## 5 Judgment of Damage Degree of Continuous Beam Bridge
### Damage Index  
The MSAA coefficient is used as the coefficient peak value of the wavelet transform coefficient at the damage position. By processing the MSAA coefficient, the damage degree index can be obtained. Compared with the low frequency coefficient (d5), the high frequency coefficient (d1) has more concentrated energy, and the MSAA coefficient peak value changes more regularly. Therefore, the bridge detection method based on bridge health data selects the bior4.4-d1 coefficient to determine the degree of damage.  
The MSAA coefficient is the peak coefficient of the damage location obtained after the discrete wavelet transformation of the vehicle body acceleration response difference.  
Figure 14: Wavelet transform coefficient (d5) under different noise levels.  
Therefore, the MSAA coefficient is closely related to the vehicle body acceleration response difference. It can be seen from Equation (28) that the sudden change point of the vehicle body acceleration response difference at the damage location is mainly related to the healthy bridge deflection \(f_{z}(x)\), so the MSAA coefficient at the damage location is divided by the healthy bridge deflection \(f_{z}(x_{\theta_{d}})\) at the same location. The \(D_{1}\) coefficient of continuous beam bridge damage detection method based on bridge health data is:  
\[D_{1}=\frac{w_{m}}{f_{z}\big{(}x_{\theta_{d}}\big{)}} \tag{31}\]  
\[f_{z}\big{(}x_{\theta_{d}}\big{)}=\frac{F_{d}}{EI}\cdot\left[\begin{array}{c} -\frac{\big{(}l-\big{(}x_{\theta_{d}}+\Delta a\big{)}\big{)}x_{\theta_{d}}}{ \delta l}\\ +\left(\frac{(x_{\theta_{d}}+\Delta a)^{3}}{\delta l}+\frac{\big{(}x_{\theta_ {d}}+\Delta a\big{)}l}{3}-\frac{(x_{\theta_{d}}+\Delta a)^{2}}{2}\right)x_{ \theta_{d}}\end{array}\right]+\frac{F_{c}x_{\theta_{d}}}{3 EI}(x_{\theta_{d}}- l)^{2} \tag{32}\]  
where \(w_{m}\) is the MSAA coefficient at the damage location, \(f_{z}\big{(}x_{\theta_{d}}\big{)}\) is the healthy bridge deflection value at the damage location, and \(x_{\theta_{d}}\) is the distance from the damage location to the left end of the span.  
The MSAA coefficients of the same degree of damage vary greatly in different positions. The change trend in MSAA coefficient is similar to that of healthy bridge deflection. Under the same damage degree, the MASS coefficient of the mid span is the largest, and it gradually decreases from the middle of the span to the support, approximately in the shape of a parabola. Therefore, damage location adjustment factor \(D_{2}\) should be introduced, so that the damage degree index of each position of the same damage degree is kept within a certain range as in Formula (33)  
\[D_{2}=x_{d}{}^{0.5} \tag{33}\]  
where \(x_{d}\) is the distance between the damage location and the nearest support of the span, \(x_{e1}\) is the distance between the damage location and the left end of the span, \(x_{e2}\) is the distance between the damage location of the side span and the nearest bridge end, and \(l_{b}\) is the length of the side span.  
From Formulas (27) and (29), the damage degree index D of the continuous beam bridge damage detection method based on bridge health data is:  
\[D=D_{1}\cdot D_{2}=\frac{w_{m}}{f_{z}\big{(}x_{\theta_{d}}\big{)}}\cdot x_{d} {}^{0.5} \tag{34}\]  
The MSAA coefficient of the same damage position increases as the damage degree increases, and the change trend is in the form of a power function. Multiple damage degree indicators at a certain point of the bridge are used to fit the damage degree function \(S(x)\), and the damage degree function \(S_{d1}(x)\). Multiple damage degree indexes at 20 m from the damage position are selected for fitting, and the damage degree is 10%, 20%, 30%, 40%, 50%, and Matlab is selected for function fitting. Substituting the damage degree index \(D\) into the damage degree function \(S(x)\), the damage degree function \(S_{d1}(D)\) is expressed as:  
\[S_{d1}(D)=10.49\times D^{0.4519}-0.1707 \tag{35}\]  
where \(D\) is an index of the degree of damage.