# 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
#### 2.2.3 Discussion of the Quasi-static Loading Conditions Before and After Damage  
As mentioned in previous research [12; 13; 14], applying the same loading conditions before and after damage is fundamental to successfully detecting the state of damage of a bridge by using the difference in the displacement ILs between the healthy and damaged states. However, during the practical implementation of a quasi-static load test on a bridge, it is impossible to ensure that the weight of the loading vehicle remains constant for every test. Therefore, the effect of an inconsistent loading vehicle weight during quasi-static load tests before and after bridge damage is worth discussing.  
The displacement IL of a bridge superstructure describes the vertical deformation of a certain cross-section due to the unit load acting at different positions along the longitudinal direction of the bridge. The unit load is replaced by a certain load for practical applications. The vertical displacement of the bridge superstructure directly reflects the stiffness performance of the bridge and thus, the displacement ILs are deemed effective features with respect to the damage of the bridge superstructure. A simply supported beam bridge is assumed to be acted upon by a quasi-static moving load \(F_{R}\) characterized by a longitudinal position \(x_{t}\) and the displacement of the cross-section A of the main beam \(\omega_{\text{A}}(x_{t})\) is calculated by the following equation with the virtual work principle:  
\[\omega_{\text{A}}(x_{t})=\text{F}_{R}\int_{0}^{l}\frac{1}{EI(x)}M_{A}(x)M_{R}(x )dx \tag{16}\]  
where \(EI\) represents the stiffness of the main beam of this bridge, \(x\) is the longitudinal position of a certain cross-section and I is the length of the main beam of the bridge. Similarly, under the action of a virtual unit force \(F_{I}\) acting upon cross-section A, the bending moment of a certain cross-section is \(M_{A}(x)\). Under the action of the unit load, the bending moment of a certain cross-section is \(M_{R}(x)\).  
Assuming that the quasi-static load after damage changes from \(F_{R}\) to \(\eta F_{R}\), the displacement IL through cross-section A after damage is obtained by:  
\[\omega_{A}{}^{\prime}(x_{t})=\eta F_{R}\int_{0}^{l}\frac{1}{EI(x)}M_{A}(x)M_{R }(x)dx=\eta\omega_{\text{A}}(x_{t}) \tag{17}\]  
where \(\omega_{\text{A}}{}^{\prime}(x_{t})\) is the displacement IL through cross-section A after bridge damage occurs and \(\eta\) is the coefficient describing the change in the quasi-static load. Using the above two equations,the correlation coefficient between two displacement ILs defined in Equation (3) is calculated by the following equation:  
\[\beta^{\prime}=\sum_{a=1}^{m}\eta^{2}\Big{(}\big{(}X_{aj}-\mu_{j}\big{)}\big{(}X_ {ak}-\mu_{k}\big{)}\Big{)}/\eta^{2}\Bigg{(}\sum_{a=1}^{m}\big{(}X_{aj}-\mu_{j} \big{)}^{2}\cdot\sum_{a=1}^{m}(X_{ak}-\mu_{k})^{2}\Bigg{)}^{\frac{1}{2}},j\
eq k \tag{18}\]  
Similar to the method presented in Reference [25], the effect of the difference in the weight of the loading vehicle before and after damage is eliminated when the damage feature proposed in this study is applied to detect the damage state of a bridge. Therefore, the proposed method based on quasi-static displacement ILs has no relation to the inconsistency in the loading conditions before and after damage (the detailed results of the numerical example are shown in Section 3.2). Of course, this method is limited insomuch as the damaged bridge approximates a linear elastic state, that is, the damage is not sufficiently severe to force the mechanical performance of the bridge into a nonlinear state. However, for most medium- and small-span bridges, this limitation is easily satisfied in theory.