## 1 Introduction  
As the key node of road networks, bridge safety operation is very important to the normal operation of the transportation system. With the extension of the service period, the service function of the bridge will gradually deteriorate under the repeated action of the external environment and external load (such as floods, earthquakes, and vehicles, etc.) [1; 2; 3; 4; 5; 6; 7]. Therefore, in addition to optimizing the design of the bridge, it is of vital importance to the accurate judgment of the state of the bridge, which can not only provide a reasonable basis for the bridge maintenance decision, but also ensure its safe operation [8; 9; 10; 11; 12; 13; 14]. Hinged slab bridges are widely used in medium- and small-span bridges due to their simple structure and convenient construction, with one of the biggest defects being hinge joint damage, so it is of great significance to evaluate the status of hinged slab bridges under hinge joint damage [15; 16; 17; 18]. As the material characteristics and external load have randomness, most commonly used probability-based assessment methods mainly include the first-order second-moment method (FOSM), Monte Carlo method (MC), and response surface method (RSM) [19]. The first-order second-moment method is based on the first-order function of the random variable in the structural function, and combinesthe first-order moment and second-order moment probabilistic statistical parameters of the random variable to calculate the reliable index. For reliability analysis of a few dimensions and functions with clear expressions, the FOSM method is simple and efficient, which can usually obtain accurate results, [20]. [PERSON] et al. used the FOSM method to calculate the reliability index of cracking at the upper and lower edge of the section of prestressed concrete box girder [21]. [PERSON] et al. used the FOSM method to calculate the cable reliability of a suspension bridge under the effect of temperature [22]. The basic idea of the MC method is to carry out large-scale random sampling on random variables affecting the reliability of the structure, and substitute the sampling values into the structural function one by one to determine the failure times and calculate the failure probability. [PERSON] used the MC method to conduct a reliability assessment of the Zhaobashan cable-stayed bridge under the limit state considering the live load effect [23]. [PERSON] evaluated the reliability of bending and shear failure of piers by MC simulation [24]. The MC method can avoid complex mathematical formulas in structural reliability analysis, but it has the disadvantage of an intensive calculation workload [25; 26]. The clear functional expression cannot be obtained in the analysis of complex reliability, but the RSM can explicitly express the implicit functional function through a series of deterministic tests to fit an approximate limit state [27]. [PERSON] adopted the adaptive RSM based on the least-squares method to carry out seismic reliability analysis of multi-span piers [28]. [PERSON] conducted a reliability analysis by improved RSM on the limit state of typical self-anchored suspension bridges [29]. The above reliability calculation methods are all aimed at a single member or a single failure mode; however, the bridge structure is a complex structural system composed of multiple members, and the state of a single member cannot reflect the overall state of the bridge. Therefore, it is very important to evaluate the reliability of the bridge system. For a statically indeterminate bridge structural system, there are often multiple failure modes, but only a few major failure modes have great effect on the failure probability of the system. In addition, there is a certain correlation between members, and the load acting on the structure may also have the same random source, resulting in a correlation between the various failure modes.