## 1 Introduction  
Recent years have witnessed the rapid development of bridge construction, and bridges have become vital components involved in transporting infrastructure. Over time, in-service bridges will inevitably be damaged due to material aging, harsh natural environments, increasing traffic demands, as well as other extreme events, resulting in damage to the beam structure [1]. The American Society of Civil Engineers (ASCE) Infrastructure Report Card rates the bridge infrastructure in the United States as a C, with 7.5% of bridges being deemed structurally deficient [2]. Therefore, it is necessary to implement safety assessments effectively throughout the entire life cycle of bridges to ensure their integrity and reduce accident losses. Non-destructive damage evaluation (NDE) aims to identify damages in structures as early as possible to enable the taking on of appropriate measures to prevent further damage and avoid catastrophic failure. NDE provides a scientifically feasible solution for detecting the presence of damage in civil engineering structures [3; 4; 5]. Damage in the structure will lead to changes in some material parameters, such as cross-sectional area, moment of inertia, elastic modulus, and stiffness, thus affecting changes in structural static and dynamic properties, such as displacement, strain, mode shape, and natural frequency [6; 7]. These damages can be identified by non-destructive testing of the structural response. According to different load cases, these tests can be divided into dynamic tests [8; 9; 10] and static tests [11; 12; 13; 14]. The advantage of static-based NDE methods may include simplicity and effectiveness as they can provide high-precision results for structural damage identification. The NDE method based on the static response has attracted the attention of many scholars. [PERSON] et al. [15; 16] have been committed to structural parameter identification and damage assessment based on the static response, obtaining formulas for estimating structural parametersfrom static strain and conducting NDE on structures. Moreover, [PERSON] et al. [17, 18] identified the damage for the truss structures using static response and proposed a stiffness separation method to simplify the damage identification of large-scale space truss structures. [PERSON] et al. [19] proposed a damage identification method based on the correlation of the probability distribution of the quasi-static response data. This proposed method is validated by monitoring the strain and tension of bridge structures, and the results show that the method has good accuracy and robustness in identifying bridge structural damage. [PERSON] et al. [20] also proposed a method to identify structural damage based on the influence line of a sensor and an empirical Bayesian threshold estimator. This method utilizes the quasi-static displacement influence line to obtain displacement readings and infer load effects on the bridge. The accuracy of the method was verified through numerical simulation and field tests on bridges. [PERSON] et al. [21] investigated a new method for locating and quantifying damage in Euler-Bernoulli beams using the principle of static deflection changes. The effectiveness and reliability of this method have been verified through numerical simulations and experiments on multiple types of beams. [PERSON] et al. [22] proposed a two-step non-model-based damage localization and quantification method to study the identification of beams. By using the static deflection changes caused by damage, the location of the damage is determined, and the crack depth is evaluated based on the characteristic expression of the rotational spring model. Numerical examples were used to verify that the proposed method can accurately locate and quantify the existing damage in the beam. [PERSON] et al. [23] utilized static shear energy to detect damage in beam-like structures.