# Damage Detection in Beam Bridges Using Quasi-static Displacement Influence Lines
## 1 Introduction  
In contrast to dynamic methods [1; 2; 3], static methods [4; 5] exhibit obvious advantages in the detection of damage in civil structures: (1) static methods easily ensure a high accuracy of the measured static deformation of structures; (2) they effectively avoid the difficulties associated with identifying damage in structures; and (3) they do not need to consider various effects, such as the mass of the structure. Therefore, static methods have been increasingly employed to detect damage in structures using various static features, for example, the static displacement [6; 7; 8], static strain [9] and static flexibility of structures [10; 11]. Compared with the static deformation of other civil structures, the static deformation of bridges is relatively easy to measure during the operational period. The static load test is commonly implemented to measure the static deformation of operational bridges to, for example, ascertain the displacement and strain along a cross-section of the superstructure. Similar to dynamic methods, static methods boost increasingly accurate damage detection results with increasingly large numbers of sensors. The static influence line (IL) technique can be used to effectively overcome limitations on the number of sensors because, in theory, only one sensor is needed to obtain the complete IL. Taking advantage of the above property, some researchers have focused on identifying damage in structures using ILs [12; 13; 14] and have identified the location and extent of structural damage using displacement IL curvatures [15; 14], the stress IL curvatures [16], the second derivative of displacement IL [17; 18] and so forth. Therefore, the IL technique has been deemed a promising damage index [19; 20]. However, the theoretical static ILs cannot be obtained for operational bridges because only quasi-static moving loads, that is, loads at low uniform velocities, are implemented under actual loading conditions. With quasi-static loading, the displacement or strain IL of the bridge superstructure is approximated to a static IL and the abovementioned IL is referred to as a quasi-static IL. A two-stage damage detection method utilizing quasi-static displacement ILs was proposed to rapidly locate and quantify damage and this method does not necessitate time-consuming finite element model updating [21]. Furthermore, an intuitive damage index based on quasi-static ILs was presented to locate damage in a bridge and the effects of local damage on different components of the displacement IL were investigated in detail [22]. A damage locating method using stress ILs under a quasi-static moving load was proposed and applied to an actual suspension bridge [16]. In addition, both the quasi-static displacement IL and the strain IL techniques were utilized to locate the damage in a bridge without the baseline finite element model; accordingly, the feasibility, effectiveness and limitations of this approach were investigated both numerically and experimentally [23]. Similar to other static-based methods, the accuracy and robustness of quasi-static IL-based damage detection techniques depend mainly on the consistency of the loading conditions before and after bridge damage occurs [24], that is, the quasi-static loading conditions before and after damage should be identical. However, the above loading condition is difficult to implement during actual load testing of a bridge. To address this issue, a method for identifying damage using the strain ILs based on the Brillouin optical time domain analysis (BOTDA) technique is proposed by the authors of this study [25]. Quasi-static strain ILs are applied to generate a damage location index that is independent of the difference between the weight of the loading vehicle before and after damage.