# Model-Free Method for Damage Localization of Grid Structure
## 1 Introduction  
In construction, buildings with large spans are becoming increasingly common, and lightweight, economical, beautiful grid structures have become the first choice for roofs. While these grid structures remain in service, damage will accumulate with the long-term effects of loads, material aging, fatigue, accidental loads, and environmental corrosion. If the damaged elements in the structure cannot be identified in time, serious accidents may result. In recent decades, many scholars have comprehensively studied [1; 2; 3] the damage identification of large structures. For example, [PERSON] et al. [4] studied the degree of element damage in the quadrangular pyramid structure by simulating the displacement-time curve of the unit node, indicating that the displacement relationship can reflect the damage to the grid structure. [PERSON] et al. [5] can accurately determine the damage positions of grid element units using hammer strokes to obtain the structural time domain signals. [PERSON] [6] combined the static virtual deformation method and the sequence quadratic programming algorithm for damage identification to quickly and accurately identify damage positions. [PERSON] et al. [7] identified damage by the cubic interpolation of bending deflection of the beam in the case of smaller stress redistributions, after assuming structural damages. [PERSON] et al. [8; 9] used a random damage index to quantify the damage of a random beam structure through the static test, while considering the model and measurement errors. [PERSON] et al. [10] recognized damage identification as the process of optimizing the stiffness to match the external force with the displacement and proposed a method for calculating thestatic displacement using flexibility, as well the cuckoo and optimization algorithm to identify damages. The displacement influence line [11] and strain influence line [12] are also used to identify the damage, and [PERSON] conjectured that the time to measure the structural displacement also affects the result [13]. Most of the above studies are based on the model correction technique, which is a common method for damage identification [14; 15; 16], using finite element models. Nevertheless, it is difficult to establish accurate finite element models for large structures. In order to avoid finite element modeling, [PERSON] et al. [17] used the Sherman-Morrison-Woodbury (SMW) formula to discuss the theoretical basis of structural damage localization in detail. Additionally, they took a beam structure and rigid frame structure as examples to explain the application of this theoretical basis to achieve damage localization. However, the grid structure has not been specifically discussed.  
In view of this, the principle in literature [17] was applied to the grid structure to deduce the physical connection between the position of damage and variation in displacement of the grid structure in detail. It was concluded that the basic principle of damage localization of the grid structure before and after the structural damage were small and only the rigid body displacement occurred, except for tensile and compressive deformations of damaged elements. Based on this, the rate of length change of each element was used as the index to identify the damage position for the space grid structure. The proposed method can realize damage recognition without a model, thus avoiding the complicated finite element modeling process, and can be implemented by using static or dynamic test data. Taking the space grid structure with 71 elements as an example, the static and dynamic test parameters were simulated respectively. The length change rate of the element was calculated according to the obtained displacement difference. At the same time, data noise was added to the simulation. The simulation results indicated that the element with the larger rate of length change is the most likely to appear damaged, and the damaged element can be accurately evaluated even if the data are noisy.