## 2 Crack Damage Simulation Algorithm  
Based on the statistical analysis of the number, length, width, and height of the cracks in the continuous rigid-frame bridge, this study simulates the crack damage by reducing the stiffness in the areas with high concentrations of cracks and establishes a crack model accordingly. This can improve the accuracy of the results and can also establish a theoretical foundation for the subsequent influence of the coupling vibration of the vehicle bridge on the dynamic response of the bridge structure. According to [PERSON]'s [22], the investigation of 74 continuous rigid-frame bridges in Shaanxi Province, the section cracking of continuous rigid-frame bridges is serious in the L/4, L/3, and midspan closure sections. Among them, the longitudinal cracks in the bottom plate of L/4 and the assembly section, such as the midspan closure section, are particularly serious and have become the most serious disaster [23] area for crack development. Therefore, L/4, L/3, and the midspan in the following text refer to the corresponding position of the second span (middle span). The cracks are divided into four categories, and each category is shown in Figures 1-4. Figure 1: Distribution ratio of cracks in the first type of bridge. Figure 2: Distribution ratio of cracks in the second type of bridge. The first type of bridge cracks is mainly distributed on the bottom plate, running longitudinally along the bridge span direction. The second type of bridge cracks is mainly distributed on the web of the beam body, and the longitudinal, vertical, and oblique cracks extend from the end of the beam to the middle of the span in an increasing trend. The third type of bridge cracks is mainly distributed in the web, roof, and flange plate. These cracks include longitudinal, vertical, and oblique cracks. Of these, the top plate and flange plate mainly have longitudinal cracks, while the web plate mostly has inclined cracks. The fourth type of bridge cracks is mainly distributed in the top plate, bottom plate, web and flange plate. The top plate, bottom plate and flange plate have longitudinal cracks, while the web has longitudinal cracks and oblique cracks. The beam cracks mainly appear at the midspan, and the length of the cracks along the longitudinal position (near the end, one-fourth, one-third, and the midspan) typically measures around 1.5 m; in addition, a small number of cracks are longer than 2 m and concentrated in the area from one-third to the midspan. From the perspective of crack width, cracks larger than 0.2 mm are mainly concentrated in the area from one-fourth span to approximately middle span position. Therefore, this research focused on the areas around these three positions. According to the results of the crack field investigation, a cracked beam section was selected as the research object (as shown in Figure 5), where \(h_{cri}\) represents the degree of concrete withdrawal from the work in the tension area; \(\delta_{cri}\) is the degree of cracking, i.e., the crack width; \(l_{cri}\) is the crack spacing, which represents the number of cracks in the curved section beam; and \(l_{cr}\) represents the range of crack distribution and is related to the position of the external load form. Figure 4: Distribution ratio of cracks in the fourth type of bridge. Figure 3: Distribution ratio of cracks in the third type of bridge. [PERSON] [24] pointed out that the statistical parameters of the cracks in the section are as follows: the average height of the cracks is \(\overline{h_{cr}}\), the average statistical spacing of the cracks is \(\overline{l_{cr}}\), the total width of the cracks is \(\Delta_{cr}\), etc. \[\left\{ \begin{array}{l}\overline{h}_{cr}=\dfrac{\sum\limits_{i=1}^{n}h_{cri}}{ n}\\ \overline{l}_{cr}=\dfrac{\sum\limits_{i=1}^{n}l_{cri}}{n-1}=\dfrac{l_{cr}}{n-1} \\ \Delta_{cr}=\sum\limits_{i=1}^{n}\delta_{cri}\end{array} \right.