## 1 Introduction
Moreover, the contact force measured by sensors in the direct method is not equivalent to train axle loads, which involve sophisticated wheel-rail interaction mechanisms. These two existing gaps finally limit the practical engineering application for evaluating structural responses under train axle loads for urban rail transit bridges. An indirect method used in urban rail transit estimates train axle loads by analyzing the dynamic structural responses of rails or existing bridges. This approach allows the monitoring of axle loads from multiple trains using a single instrumented section. Initially, researchers utilized the structural response of the rail to infer the axle load of moving trains. [PERSON] [13] introduced an indirect method based on transfer functions derived from dynamic responses measured at specific longitudinal locations, such as the acceleration response of the rail and the corresponding vertical wheel-rail contact forces. In another research [14], the direct problem of identifying the weight of a moving train was tackled first. Subsequently, the inverse problem of determining the train load from a measured strain time history at the rail foot was addressed as a minimization problem. The objective function of this approach was based on minimizing the difference between experimental and measured strains. In addition to analyzing the structural response of rails, scholars have increasingly utilized static structural responses, including strains and displacements of railway bridges, to reversely identify train axle loads. [PERSON] et al. [15] employed strain measurements from a long-term monitoring system with strain gauges on a railway bridge to accurately determine train axle loads, axle distances, and train speeds, providing a robust statistical characterization of traffic conditions. With strain monitoring data of a continuous steel truss railway bridge, an identification method of moving train load parameters was proposed based on the basic theories of strain influence line [16]. [PERSON] et al. also [17] developed mechanics-based algorithms using strain data from specific locations on bridge girders to estimate wheel loads and train speeds. Furthermore, [PERSON] et al. [18; 19] introduced a system for identifying multi-axle train loads based on time history displacements from measurement stations on a continuous steel truss bridge using a displacement imaging monitoring system. These studies predominantly employed traditional electrical displacement and strain sensors in field applications to monitor bridge structural responses for estimating train axle loads in urban rail transit. Additionally, emerging advanced technologies such as piezoelectric sensors [20], fiber-optic sensors [21; 22], and even self-sensing structural materials [23] have also been applied to evaluate train axle loads. To the authors' knowledge, all the above research has primarily focused on individual railway trains in experiments or operations, often overlooking the dynamic amplification effects related to train operating velocities. Another key point to emphasize is that none of the above studies has conducted long-term monitoring of structural responses, resulting in limited measurement data size, making it impossible to study the inherent variability of train axle loads. In practical engineering, train axle loads must exhibit a degree of randomness and adhere to specific statistical laws. However, there remains a lack of relevant information on the statistical distribution pattern of train axle loads in urban rail transit, particularly considering the dynamic amplification effect. This gap is much more crucial than static train axle loads for the effective management and maintenance of urban rail transit bridges. The study implemented a strain field measurement scheme on an in-service simply supported composite beam bridge in urban rail transit. This involved numerical modeling and validation of the bridge's behavior, followed by numerical calculation of strain influence lines at measurement points.