# A Kriging Model Based Finite Element Model Updating Method for Damage Detection
## 4 Experimental Study  
In this section, the experimental data [42] of a three-story building structure made of aluminum tested at Los Alamos National Laboratory (LANL), shown in Figure 6, is used to verify the effectiveness and the practicability of the proposed approach. The test-bed structure consists of 4 plates and 12 columns assembled by bolted joints and was installed on rails that allow it to slide in only one direction. Additionally, a center column suspended from the top floor and an adjustable bumper mounted on the 2 nd floor was used to simulate the nonlinear behavior of damage. An electrodynamic shaker was used to provide a band-limited random lateral excitation (20-150 Hz) to the base floor along the center line of the structure and a force transducer was connected to the tip of the singer to measure the input force. At the same time, four accelerometers were attached at the center line of each floor on the opposite side from the excitation to measure the dynamic response of the system. The analog sensor signals were discretized into 8192 data points sampled at 3.125 m intervals corresponding to a sampling frequency of 320 Hz. Force- and acceleration-time histories for 17 different structural state conditions were collected. However, only 9 of them are utilized as the others are concerned with the nonlinearity which is not within the scope of this article. The 9 different states can be divided into 3 categories: the baseline structure, the increase of the mass and the reduction of the stiffness. The change of mass was realized by adding a 1.2 kg concentrated mass to the aluminum plates, while the reduction of stiffness was introduced by replacing the corresponding column with another one with half the cross section thickness in the direction of shaking. Each state was tested separately and the detailed state conditions [43] are shown in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Kriging** & **GA (Genetic Algorithm)** \\ \hline
1 & 448 & 119,600 \\
2 & 380 & 92,600 \\
3 & 421 & 161,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The required number of running the finite element (FE) program of the Kriging method, and the GA method (which is repeated 3 times). Figure 6: Overview of the three-story structure (Source: Courtesy of [42]). As recommended in [42], the structure is modeled as four lumped masses at the floors shown in Figure 7. All the mass and stiffness values are selected as the updating parameters except \(k_{0}\) since the friction between the rails and the structure is negligible. Besides, the damping matrix is introduced based on the proportional damping assumption which can also be affected by the mass factors. At first, the Baseline condition is used to update the theoretical model to get the original undamaged model. The FRF of the first floor after the updating procedure compared with the experimental data is illustrated in detail in Figure 8. It can be clearly seen that the model after updating shows good agreement with the experimental data. However, there is considerable noise in the imaginary part of the experimental FRF especially near the anti-resonance point and modeling error in the theoretical FRF around the same point. What is more, there is also a big difference between the real parts of two FRFs near 22 Hz, which should not be selected as the frequency points of the objective function, since such a big difference cannot be eliminated by model updating. Then the two methods are applied on the undamaged model obtained using the experimental data of the other states. The intervals of the mass and stiffness parameter ratios are set to \([0.98,\ 1.5]\) and \([0.4,\ 1.0]\), respectively. This setting is based on the pre-knowledge of the structure's change and prevents all parameters from becoming larger at the same time or smaller at the same time, which would make little change in the value of the objective function.