# 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
#### 2.2.1 Hankel Matrix of the Damage Feature Established with Quasi-static Displacement ILs  
Unlike the Hankel matrix described in Reference [25], the Hankel matrix of the damage feature is reconstructed; thus, it is suitable for establishing the damage feature using a few quasi-static displacement ILs. We obtain the displacement IL of one quasi-static load test; then, the corresponding matrix \(\mathbf{Y}\) is defined by the following equation:  
\[\mathbf{Y}=\left[ \begin{array}{cccc}X_{11}&X_{12}&\cdots&X_{1n}\\ X_{21}&X_{22}&\cdots&X_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ X_{m1}&X_{m2}&\cdots&X_{mn}\end{array} \right]_{m\times n} \tag{1}\]  
where \(\mathbf{Y}\) is the set of all measured displacement ILs from one quasi-static load test; \(n\) is the number of sensors placed within the cross-sections of the bridge superstructure; m represents the total number of sampling points along each displacement IL; and \(X_{mn}\) is an element of \(\mathbf{Y}\), where each column of the matrix \(\mathbf{Y}\) represents only the quasi-static displacement IL of a certain sensor, that is, \(\boldsymbol{\omega}_{j}\in R^{m\times 1},j\in(1,2,\cdots,n)\)  
\[\boldsymbol{\omega}_{j}=\left\{X_{1j},X_{2j},\cdots,X_{aj},\cdots,X_{mj} \right\}^{\text{T}}{}_{m\times 1} \tag{2}\]  
where \(X_{aj}\) is the \(a\)th (\(a\in(1,2,\cdots,m)\)) element of the denoised quasi-static displacement IL of the \(j\)th sensor. According to Equation (1), it is easy to obtain the correlation coefficient \(\beta\) between two quasi-static displacement ILs through a certain cross-section of the bridge superstructure, that is,  
\[\beta=\sum_{a=1}^{m}\Bigl{(}\bigl{(}X_{aj}-\mu_{j}\bigr{)}\bigl{(}X_{ak}-\mu_{ k}\bigr{)}\Bigr{)}/\left(\sum_{a=1}^{m}\Bigl{(}X_{aj}-\mu_{j}\Bigr{)}^{2}\cdot \sum_{a=1}^{m}\bigl{(}X_{ak}-\mu_{k}\bigr{)}^{2}\right)^{\frac{1}{2}},j\
eq k \tag{3}\]  
where \(j\) and \(k\) are different measured points of displacement, that is, \(k\
eq j\), \(k\in(1,2,\cdots,n)\). \(\mu_{j}\) is the mean of the quasi-static displacement IL at the \(j\)th measured point of displacement and is calculated by the following equation. where \(\beta\) is the ordered set of \(\beta_{i}(i=1,2,\cdots,s)\), which is defined in Equation (3); \(sort(\cdot)\) is the operator used to order the elements from small to large; and \(s\) is the total number of correlation coefficients of two vectors of quasi-static displacement ILs. The correlation coefficient \(\beta_{i}\) defined in Equations (3) and (4) is completely different from the subspace between the quasi-static strain ILs at two different measured points, as defined in Reference [25]. Herein, the correlation coefficient represents the correlation between two quasi-static displacement ILs at different measured points under the same load test, while the subspace defined in Reference [25] is used to describe the change in the relationship between two quasi-static strain ILs before and after bridge damage occurs. In this study, vector \(\beta\) is defined as the damage feature because this vector includes all information regarding the displacement ILs of the bridge superstructure, as defined in Equation (5). The success of damage detection depends mainly on the change in feature \(\beta\) before and after damage. Thus, it is advisable to arrange all vectors of damage feature \(\beta\) according to certain regulations to maintain consistency in damage feature \(\beta\) results among different load tests. Considering the structural symmetry of beam bridges, if we arrange all damage features \(\beta\) in order of location, the same effects at each pair of symmetrical locations can be ignored. Therefore, the numerical order is adopted in this study, as shown in Equation (5). During subsequent damage detection of bridges, damage feature \(\beta\) is considered in the form of a Hankel matrix, which is defined as follows:  
\[\mathbf{H}=\left[