# Damage Detection in Beam Bridges Using Quasi-static Displacement Influence Lines
## 2 Damage Detection Using the Quasi-static Displacement Influence Lines of Bridges
### Generation of the Quasi-static Displacement Influence Lines of Actual Bridges  
The principles of quasi-static load testing are described as follows. As shown in Figure 1, a quasi-static load is represented by a loading vehicle that drives over the bridge along one lane at a constant slow velocity; this procedure requires that the bridge be cordoned off from all traffic. In theory, the excitation frequency band of the loading vehicle should be lower than the first natural frequency of the tested bridge; that is, the bridge in question should be as close to static as possible. To achieve this type of excitation load, the loading vehicle is usually forced to drive over the bridge at a constant slow velocity. Because the natural frequency varies by bridge, the velocity of the loading vehicle may change according to the bridge. Of course, this loading procedure should be repeated several times along the transverse direction of the bridge or along one lane to obtain reliable test data while keeping the vehicle velocity constant for each case. For medium- and small-span bridges, the above procedure can be completed quickly than traditional load experiments on bridges; thus, this technique constitutes an effective way to avoid a long-term interruption of the traffic flow.  
Suppose that the bridge is simply supported and composed of several beams in the transverse direction, as shown in Figure 1. Three sensors are placed on cross-section A, cross-section B and cross-section C on the side beam beneath the bridge (Beam #1) to measure the vertical displacement. Although the loading vehicle has an additional axis along the longitudinal direction of the vehicle, the vehicle can be considered equivalent to a concentrated vertical load by using the translation theorem of forces theoretically. Therefore, under a quasi-static loading situation, the displacement time histories of the three cross-sections along which the abovementioned sensors are placed are similar to three static displacement ILs, that is, the quasi-static displacement ILs defined in this study. For an actual bridge, considering the transverse distribution of the load, there should be more than one displacement IL for each cross-section of the bridge superstructure. For example, cross-section B is composed of several beams, as shown in Figure 1 and thus, all ILs form the influence surface of cross-section B under quasi-static loading.  
Generally, the loading vehicle should be sufficiently heavy to ensure that the measured displacement induced by the loading vehicle is sensitive to the potential damage of the bridge. However, for actual bridges, especially existing bridges characterized by bad conditions, loading vehicles that are excessively heavy may damage the bridge. Therefore, the weight of the loading vehicle is determined on a case-by-case basis in consideration of the real technical conditions of the bridge. It is recommended that, for an actual bridge [25], the reference weight of the loading vehicle should first be equal to the weight of one loading vehicle determined by using the design of a transitional load test; then, this reference weight can be adjusted to the final load by evaluating the real structural conditions of the bridge.  
Figure 1: Schematic of the quasi-static load test of a bridge.  
Based on the above discussion, a method is proposed and described in detail in the next section to show how to detect bridge damage using quasi-static displacement ILs. In this study, we focus on a method of detecting damage in medium- and small-span beam bridges using measured quasi-static displacement ILs.