# Model-Free Method for Damage Localization of Grid Structure
## 4 Cases  
The space grid structure in Figure 3 had a total of 71 elements. The main parameters were as follows: \(E\) = 200 GPa, \(\rho\) = 7.8 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\); \(A\) = 3.14 \(\times\) 10\({}^{-4}\) m\({}^{2}\); \(L\) = 3 m. The three translational degrees of freedom were constrained at nodes 1, 7, 8, and 14, and a concentrated force of 20 kN in the negative \(Y\)-axis direction was applied at nodes 4 and 11, as shown in Figure 3a. In the example, the element damage was simulated by the reduction in elastic modulus. Consider two cases of single damage and multiple damages: (1) simulate 20% stiffness damage for element 5 between nodes 5 and 6; and (2) simulate 20% stiffness damage for element 35 between nodes 1 and 15, and 20% stiffness damage for element 40 between nodes 7 and 19, respectively. First, we simulated the static test. According to the obtained static displacement data, under a concentrated force, the rates of length change of 71 elements before and after the damages occurred were calculated by Equation (15). The results obtained from the calculation without considering the noise are shown in Figures 4 and 5. Figure 3: (**a**). X-Y axis elevation structure and force; (**b**). Standard view of the space grid structure of element. It can be seen from Figures 4 and 5 that when the data used were noise-free, the rate of change of length of the damaged element was abnormally increased, and the length change rates of the remaining elements were all small, indicating that the length change rate of the element was feasible as the basis for damage localization. In order to test the noise resistance of the proposed method, data noise of 5% was added to the static displacement data to simulate the real measurement condition, and the results of the calculation using the noise-containing data are shown in Figures 6 and 7. The formula for adding data noise is as follows:  
\[\Delta u^{*}_{(j)}=\Delta u_{(j)}\times(1+nl\times unifrnd[-1,1]) \tag{23}\]  
where \(\Delta u^{*}\) denotes the \(j\)th component in the displacement change \(\Delta u\), \(\Delta u^{*}_{(j)}\) denotes the corresponding component contaminated with noise, unifrnd [-1, 1] denotes a random number in the range of \(-1\) and \(1\), and \(nl\) denotes the noise level. As stated before, \(nl=5\%\) is used in this example. Figure 4: Rate of change of element length obtained from the static data (element 5 had damage with no noise). Figure 5: Rate of change of element length obtained from the static data (elements 35 and 40 were damaged with no noise). As can be seen from Figures 6 and 7, even if there was a noise interference of 5%, the damage of element 5 can be clearly judged from Figure 6, and the damage of elements 35 and 40 can be clearly judged from Figure 7. This indicates that the proposed method had good anti-noise ability. As mentioned above, actual static loading may be difficult to perform on the actual engineering structure to obtain the displacement parameters. In such a case, a dynamic test can be used to obtain the modal parameters and the virtual displacement parameters. That is, a method that uses the dynamic flexibility combined with virtual force was adopted to obtain the displacement data indirectly. The results of the rate of length change of the element calculated by this method are shown in Figures 8-11. Figure 6: Length change rate of element obtained from the static data (element 5 was damaged with an added 5% noise). Figure 7: Length change rate of element obtained from the static data (elements 35 and 40 were damaged with an added 5% noise). Figure 8: Length change rate of element obtained from the dynamic data (element 5 was damaged with no noise). Figure 10: Length change rate of element obtained from the dynamic data (element 5 was damaged adding 5% noise). Figure 9: Length change rate of element obtained from the dynamic data (elements 35 and 40 were damaged with no noise).