## 1 Introduction  
Concrete bridges have been widely used and developed in recent years. Regular bridge inspections are essential to ensure safe operation throughout their service life and prevent major disasters. These inspections help maintain the bridges in a safe working condition, ensuring their normal operation [1; 2; 3; 4]. However, concrete structures are inherently compact, making it impossible to assess the development of internal damage based solely on their external appearance. Non-destructive testing methods are the preferred choice for detecting internal damage while avoiding any damage to the beams. Common non-destructive testing methods for detecting internal damage include acoustic emission, infrared thermography, impact-echo, ultrasound testing, microwave testing, and ground-penetrating radar [5; 6]. Acoustic emission technology is a passive testing method, in contrast to other methods [7; 8; 9]. It involves analyzing elastic wave signals emitted from the material or internal structure of the bridge using sensors, enabling dynamic and non-intrusive detection and monitoring of structural damage. Acoustic emission technology, unlike active testing methods, does not require energy input as it relies on the reception of elastic wave signals. Numerous experimental studies have shown that the ultimate failure of concrete structures results from the gradual development of internal microcracks. Scholars from various countries have extensively researched the evolution process of concrete cracks and their damage using different methods and theories [10; 11]. In their study, [PERSON] et al. [10] examined the mechanical behavior of concrete and rock samples under constant-rate-to-failure loading conditions. They utilized acoustic emission (AE) and electromagnetic emission (EME) techniques for analysis. The result shows that the presence of AE signals has been always observed during the damage process. Magnetic signals are usually observed only when there is a sharp drop in stress or an eventual collapse. [PERSON] et al. [12] combined AE characteristic parameters with the K-nearest neighbor (k-NN) algorithm to propose a concrete structure crack evaluation index (CAI). This method extracts important AE features from recorded AE signals, such as rise time, decay time, peak amplitude, AE energy, and AE count. Based on AEB characteristics, the normal state, microcrack state and macro crack state of concrete structures are effectively distinguished. [PERSON] et al. [12] developed a fractal multi-scale approach based on fracture mechanics and acoustic emission technology to predict the evolution of structural damage and failure time. Under load conditions, cracks rapidly develop and undergo multiple bifurcations, significantly increasing the irregularity of the fracture surface. Previous research has investigated crack bifurcation phenomena using fractal theory. It examines the crack bifurcation mechanism from the perspective of energy release and presents theoretical formulas describing the variation of fracture toughness with crack bifurcation angle [13]. Currently, acoustic emission (AE) technology is being applied to investigate concrete structures under various loading conditions [14], including bending, shear, and torsion. Scholars from both domestic and international contexts have successfully integrated AE technology with signal processing, mathematical modeling, and damage mechanics, achieving promising results [15; 16; 17; 18; 19]. For instance, the combination of Gaussian mixture models and AE technology effectively addresses the randomness of experimental data. The polling scheduling method effectively combines damage scalars from damage mechanics with AE technology to assess the evolution of concrete damage [20]. [PERSON] et al. [21; 22] conducted a comprehensive review and summary of research on sound propagation prediction methods for plate structures from 1967 to the present. They also delved into the propagation and diffusion of waves in infinite hyperbolic laminated composite shells. [PERSON] et al.