## 1 Introduction  
Bridge structures may be damaged or even destroyed in earthquake disasters. Therefore, many large bridges have been equipped with long-term structural health monitoring (SHM) systems for structural safety. Since the seismic excitations are unpredictable, it is still a great challenge to use only the monitoring data to evaluate the state of the structure, detect and localize the abnormal characteristics of structures, and evaluate and extend the service life of the structure [1; 2; 3; 4]. Data-driven damage detection has become one of the most popular methods in SHM. It does not need external excitations and only relies on the monitoring data of responses [5]. Since the transmissibility function (TF) reflects the response transfer characteristics of the structure and does not need the information of the external excitation, TF-based methods have attracted more attention in the field of SHM. Because of its advantages, TF is mainly applied to three categories of SHM system: modal analysis [6; 7; 8], damage detection [9; 10], and model updating [11; 12]. In addition, in recent years, TF has been widely used in vibro-acoustics for transfer path analysis [13; 14], force identification or reconstruction [15; 16], and characterization of the dynamic behavior of vibration isolation systems [17]. This paper mainly studies the application of TF in damage identification. Many scholars used TF to construct various damage indicators for damage identification. [PERSON] et al. [18; 19] proposed a response vector assurance criterion (RVAC) and a relative damage quantification indicator (DRQ) with transmissibility. [PERSON] et al. [20; 21; 22; 23] proposed a variety of damage indicators based on TF and combined the concepts of coherence function (ATC/TMAC), similarity measure (CI/MAC), and correlation analysis (CDI)to determine the occurrence of damage. [PERSON] et al. [24] proposed a new damage indicator based on the TF and studied the damage localization of the offshore platform structure support. [PERSON] et al. [25] also established an indicator to compare the advantages and disadvantages of the angular velocity TF and the acceleration TF on damage localization. [PERSON] et al. [26] studied the mechanism of damage localization using TF to establish the corresponding damage indicator. [PERSON] et al. [27; 28] proposed a structural damage identification strategy based on the weighted transmissibility assurance criterion. In addition, some literature combined TF and a distance measure to construct damage indicators, which is an outlier analysis, such as Cosh distance, Euclidean distance, Mahalanobis squared distance (MSD), Itakura distance, City Block distance, Chebyshev distance, and Minkowski distance [29; 30; 31; 32; 33; 34; 35]. Due to its good applicability and sensitivity to damage, MSD has been widely used in structural damage identification [36]. Some scholars studied MSD by calculating the TF of all measuring points at a certain frequency point as the test vector and detecting damage by the MSD vector varying with the frequency point [29; 30; 31]. [PERSON] et al. [32] and [PERSON] et al. [33] formed the TF matrix by considering the responses of all measuring points, used the principal component analysis to reduce the dimension, and detected the damage based on MSD. [PERSON] et al. [34] only used the acceleration response time history of two measuring points to obtain the wavelet TF and judged whether the observation value of the unknown state was an abnormal value according to MSD. [PERSON] et al. [35] grouped the TFs before and after damage and constructed the TF matrix. Then, the principal components were extracted by principal component compression. Moreover, the MSD between the principal components before and after structural damage was calculated, which was used as the sample of statistical analysis. Finally, the statistics were constructed, and the damage identification was studied by \(t\)-test.