# Model-Free Method for Damage Localization of Grid Structure
## 3 Damage Identification of Grid Structure
### Damage Localization of Grid Structure Based on Displacement Difference  
Measurement of static displacements can be performed by many different methods. Most often used are traditional techniques, using precise leveling, dial gauge, or inductive gauge. Recently, optical systems for displacement measurement have been successfully applied in actual engineering, and good effects were achieved, such as photogrammetry, laser techniques, and visual techniques. More details about displacement measurement can be found in References [21; 22; 23; 24; 25]. Figure 2: Force of element 3 (Fr is resultant force). According to the measured displacement difference before and after the damage of each node, the rate of length change of each element can be calculated. The calculation formula is derived as follows:  
The displacement difference can be decomposed as the displacement difference \(\Delta u_{x}\), \(\Delta u_{y}\), and \(\Delta u_{z}\) in the \(x\), \(y\), and \(z\) direction, respectively. The calculation formula of the length of each element in the damaged structure is expressed as:  
\[L_{d}=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z _{1}-z_{2}\right)^{2}} \tag{16}\]  
\[x_{1}=x_{0}+\Delta u_{x}\text{, }y_{1}=y_{0}+\Delta u_{y}\text{, }z_{1}=x_{0}+\Delta u_{z} \tag{17}\]  
where \(x_{0}\), \(y_{0}\), and \(z_{0}\) refer to original coordinates of the previous node of the element in the \(x\), \(y\), and \(z\) directions, respectively, \(x_{1}\), \(y_{1}\), and \(z_{1}\) refer to original coordinates plus the coordinates of \(\Delta u_{x}\), \(\Delta u_{y}\), and \(\Delta u_{z}\), respectively, and \(x_{2}\), \(y_{2}\), and \(z_{2}\) refer to original coordinates of the post node of the element plus coordinates of the displacement difference \(\Delta u_{x}\), \(\Delta u_{y}\), and \(\Delta u_{z}\) of the corresponding node. Using the results obtained in Equation (16), the change in length of the element before and after damage is expressed as:  
\[\Delta L=L_{d}-L \tag{18}\]  
where \(L\) refers to the original length of the structural element. According to Equation (18), the rate of length change of the element is expressed as:  
\[\varepsilon=\frac{\Delta L}{L} \tag{19}\]  
The determination of the damage position can be made based on the computed change rate of element length. In summary, the damage localization of grid structure based on the displacement difference has the following steps:  
1. Obtain the displacement difference \(\Delta u\) through experiment. It is preferred to use a laser range finder to obtain the displacement difference in the static test. If a static test cannot be performed, we can use the dynamic flexibility method to indirectly obtain the displacement difference (multiply the virtual force vector by the dynamic flexibility difference matrix before and after damage). Similarly, it is not necessary to establish a finite element model when the dynamic flexibility method is used. The approximate flexibility difference matrix can be obtained by the following formula: \[\Delta F=F_{d}-F\] (20) \[F=\sum_{j=1}^{m}\frac{1}{\lambda_{j}}\varphi_{j}\varphi_{j}^{T}\] (21) where \(F\) and \(F_{d}\) are the flexibility matrices before and after damage, respectively, \(m\) is the measured number of modes, and \(\lambda_{j}\) and \(\varphi_{j}\) are the \(j\)th order eigenvalues and eigenvectors (mode shapes), respectively. The flexibility matrix difference obtained by the dynamic flexibility method was considered in the following equation to obtain the change in displacement: \[\Delta u=\Delta F\cdot l_{v}\] (22) where \(l_{v}\) is the virtual force vector. Theoretically, the virtual vector can be selected arbitrarily according to need. For example, \(l_{v}\) = [0 -- 1 -- 0]\({}^{T}\). The specific steps for the selection of the virtual force vector can be obtained from literature [1];
2. The length of element after damage was calculated by Equation (16), and the change in length of the element before and after damage was calculated by Equation (18).