## 1 Introduction
[PERSON] and [PERSON] [29] used static response parameters and a grouping optimization algorithm to identify structural damage parameters. [PERSON] et al. [25] used a combination of static displacement sensitivity and frequency sensitivity to identify structural damage. Based on static error function, [PERSON] and [PERSON] [30] used a genetic algorithm to solve the damage parameters for each element of damage identification. [PERSON] et al. [31] proposed an optimization algorithm for solving nonlinear static equations by taking the minimum difference of load vectors between damaged and undamaged structures as the objective function. [PERSON] et al. [32] first used the grey system theory to determine the location of structural damage and then used an optimization algorithm to solve the degree of structural damage. Using static test data, [PERSON] et al. [33] proposed a coordinated evolutionary optimization algorithm for structural damage identification in two stages. Using a moving load, [PERSON] et al. [34] presented a static algorithm for determining the damage locations in beam structures based on the pure bending theory. In recognizing the fact that bridge girder damage causes force redistribution in stay cables, [PERSON] et al. [35] used the measured force changes of cables to identify bridge girder damage. The damage identification was formulated as an optimization problem in which the cable force error between testing values and analytical values was minimized. Using the virtual distortion method (VDM), [PERSON] et al. [36] studied damage detection for cable structures in bridge engineering using both static VDM and dynamic VDM. It was found that static VDM had higher accuracy than dynamic VDM in damage identification. Recently, [PERSON] et al. [37] presented a static residual force vector method for structural damage assessment. Based on the static displacement data and the stiffness matrix of the structural finite element model, the static residual force vector was defined, and the damage degrees of freedom were judged according to the nonzero elements in the vector. It was found from the numerical results that the presented algorithm was feasible for damage diagnosis. However, the above static damage detection methods all require knowledge of the exact static load data in advance. Thus, the loading position and level of the static load both need to be accurately measured in practice for structural damage assessment. This operation increases the cost and time of damage evaluation. To overcome this drawback, an improved residual force algorithm, which is independent of static load vector, was proposed in this paper. The proposed method mainly makes use of the particularity that only a few coefficients in residual force vector and static load vector are nonzero. By combining two different static loading modes, a new damage indicator vector was defined in the method for damage localization and quantification. Compared with existing static residual force methods, the outstanding advantage of the improved algorithm is that the specific value and loading position of the static load are not required in the damage identification process. This special advantage causes this method to not require special static loading, but instead uses any load vehicle. This advantage can make the operation process of structural damage identification based on static tests easier and faster. This is of great significance to reducing the cost of bridge damage detection and quickly restoring traffic on the bridge. The improved method was also applied for the first time to cable damage identification in cable-stayed bridges. The framework of this paper is as follows: In Section 2, the static residual force algorithm is briefly reviewed, and then the proposed improved approach is presented. In Section 3, a single tower cable-stayed bridge structure is used to verify the feasibility of the proposed method in cable damage identification. Finally, the conclusions of this study are summarized in Section 4.