# Damage Detection in Beam Bridges Using Quasi-static Displacement Influence Lines
## 2 Damage Detection Using the Quasi-static Displacement Influence Lines of Bridges
### Damage Detection Using the Null Space Method Based on Quasi-static Displacement Influence Lines
#### 2.2.2 Metric for Evaluating the Damage Feature  
With the generated Hankel matrix of the damage feature, a new metric for evaluating the damage feature is proposed in this section. Under the reference state (initial conditions) of a bridge, we generate the Hankel matrix \(\mathbf{H}_{r}\) (described in Equation (6)) by taking the displacement ILs of one quasi-static load test. The following equation is obtained:  
\[\mathbf{H}_{r}\cdot\mathbf{N}_{r}=0 \tag{8}\]  
where \(\mathbf{N}_{r}\in R^{q\times 1}\) is one arbitrary column of the right null space matrix of \(\mathbf{H}_{r}\). Under healthy bridge conditions, after repeating the load test K times, it is easy to generate the Hankel matrix \(\mathbf{H}_{c}\) by using one arbitrary test, for example, the cth test. Using the reference vector \(\mathbf{N}_{r}\) defined in Equation (8), the residual vector is obtained by the following equation:  
\[\boldsymbol{\alpha}_{c}=\mathbf{H}_{c}\cdot\mathbf{N}_{r} \tag{9}\]  
where \(\boldsymbol{\alpha}_{c}\in R^{p\times 1},c\in(1,2,\cdots,K)\) is the residual vector, which is approximately equivalent to the Gaussian distribution. For the practical application of the proposed method, a quasi-static load test will be carried out regularly (e.g., every year) at the field site. In this way, an increasing number of reference samples will be accumulated under a healthy state; consequently, the novel detection technique [31; 32; 33; 34], which is described below, can be used to detect the damage of an actual bridge. The sample mean of the residual of the damage feature for a bridge in a healthy state is denoted by the following equation:  
\[\overline{\boldsymbol{\alpha}}=\frac{1}{K}\underset{c=1}{\overset{K}{\sum}} \boldsymbol{\alpha}_{c} \tag{10}\]  
Accordingly, the sample covariance is defined by the following equation:  
\[\boldsymbol{\theta}=\frac{1}{K-1}\underset{c=1}{\overset{K}{\sum}}( \boldsymbol{\alpha}_{c}-\overline{\boldsymbol{\alpha}})(\boldsymbol{\alpha}_{ c}-\overline{\boldsymbol{\alpha}})^{\mathrm{T}} \tag{11}\]  
Given the sample mean and covariance for a bridge in a healthy state, a new metric of the damage feature is obtained using K samples obtained under the healthy state. \[\gamma_{c}=\sqrt{(\boldsymbol{\alpha}_{c}-\overline{\boldsymbol{\alpha}}) \boldsymbol{\theta}^{-1}(\boldsymbol{\alpha}_{c}-\overline{\boldsymbol{\alpha }})^{\mathrm{T}}} \tag{12}\]  
With the above equation, the K metric values are obtained for the healthy state, after which a threshold \(\lambda\) needs to be determined on the basis of an acceptable level of false positives or type I errors (e.g., 5%). \[\lambda=[\gamma_{1},\gamma_{2},\cdots,\gamma_{c},\cdots,\gamma_{K}]_{0.95} \tag{13}\]  
Here, \([\cdot]_{0.95}\) represents the operator within the 95% confidence interval. To investigate the state of damage, the damage metric is calculated with the same procedure, that is,  
\[\gamma_{d}=\sqrt{(\boldsymbol{\alpha}_{d}-\overline{\boldsymbol{\alpha}}) \boldsymbol{\theta}^{-1}(\boldsymbol{\alpha}_{d}-\overline{\boldsymbol{\alpha }})^{\mathrm{T}}} \tag{14}\]  
where \(\boldsymbol{\alpha}_{d}(d\in(1,2,\cdots))\) is the residual of the damage feature defined in Equation (9) using the data of the dth load test to investigate the damage and \(\overline{\boldsymbol{\alpha}}\) and \(\boldsymbol{\theta}\) are the sample mean and covariance,respectively, for a bridge in a healthy state. In this study, we test whether the residual of the damage feature \(\gamma_{d}\) satisfies the following conditions:  
\[\left\{ \begin{array}{ll}Z_{d}=1&\text{under}\quad\gamma_{d}\geq\lambda\\ Z_{d}=0&\text{under}\quad\gamma_{d}<\lambda\end{array} \right. \tag{15}\]  
where \(Z_{d}\) takes a value of either one or zero to represent the acceptance or rejection of the null hypothesis, that is, indicating either the presence or the absence of damage. As shown in Equations (9) and (13), each metric \(\gamma_{c}\) is calculated by using all the measured quasi-static displacement ILs from one load test.