# Performance Assessment of a Renovated Precast Concrete Bridge Using Static and Dynamic Tests
## 5 Comparisons of Calculation and Test Results to Verify and Validate the FEM Model
### Natural Frequencies and Mode Shapes  
An operational modal analysis (OMA) was performed on the bridge to obtain data for the validation of the FEM model [16]. Moreover, the aim of OMA was also to confirm that the global frequencies do not have a significant influence on the natural frequency of the free cables. The data obtained from the NI 9234 modules (analog to digital converter (ADC)) were raw, without any physical unit. Because of that, the measured accelerations (sampled at 2 kHz) were pre-processed in the first step. The raw data were calibrated using the developed code and the data calibration list of each sensor. After this procedure, the time series had the appropriate unit of acceleration (m/s\({}^{2}\)). Then, the digital low pass filter with an eighth-order Chebyshev type I filter with a 15 Hz cut-off frequency was applied in both the forward and reverse direction in order to remove all phase distortions. Subsequently, records were down sampled to 50 samples/s. After that, each signal was high pass filtered with a fourth-order Butterworth filter with a 0.2 Hz cut-off frequency, again in both the forward and reverse direction, similarly to [24]. Finally, the pre-processed data were analyzed in the ModalVIEW software. The possibilities of the ModalVIEW software (stabilization charts) were used for extracting modal parameters such as natural frequency, damping ratio, and corresponding mode shape. Stabilization charts were calculated using the stochastic subspace identification (SSI) method, which is described in [25] in detail. These cut-off frequencies were set because the natural frequencies calculated by the initial FEM model fell within this interval. In this case, the stability criteria from the stabilization charts used were chosen similarly to [26], specifically: 1% for frequency stability, 5% for damping stability, and 2% for mode shape stability. System identification (SI) of the bridge was done in accordance with the mentioned criteria, and the obtained experimental results were compared with the FEM calculations (Figures 9-12). Figure 8: The strains in the middle of span no. 2 caused by two heavy trucks (**a**) measurement by strain gauges; (**b**) results from FEM model. Figure 10: The 5 th identified mode shape (**a**) measurement, (**b**) FEM model. Table 6 shows the comparison of natural frequencies of the FEM model and the measurement. Since the FEM model is only 2D (coordinate system \(X\) axis vs. \(Z\) axis), natural frequencies and their corresponding mode shapes are only shown in Z directions. There is a good conformity between the measured and the calculated natural frequencies for the corresponding mode shapes. Additionally, Cross-MAC (Modal Assurance Criterion) values, similar to [27] are presented. The formula is as follows:  
\[Cross-MAC(i,j)=\frac{\left(\[PERSON]^{A^{T}_{i}}\[PERSON]^{B}{}_{j}\right)^{2}}{\left( \[PERSON]^{A^{T}_{i}}\[PERSON]^{A^{T}_{j}}\right)\left(\[PERSON]^{B^{T}_{i}}\[PERSON]^{B}{}_{j} \right)} \tag{3}\]  
where \(\[PERSON]^{A}{}_{i}\) is the \(i\)th mode shape vector obtained by the FEM model, and \(\[PERSON]^{B}{}_{j}\) is the \(j\)th mode shape vector of the measured structure as computed by the SSI method. If the value of Cross-MAC is nearing 1, the match between two mode shapes is complete. On the other hand, if the value of Cross-MAC is 0, then the mode shapes are entirely different. The gathered differences of the Cross-MAC values are smaller than 10%, which means that the mode shapes were identified correctly. Such a good conformity between the real response and the response from the prepared FEM model can be later used for SHM purposes, e.g., for the simulation of damages or a check of a suitable method of damage detection. The result of the SSI method (the third natural frequency) was compared to the natural frequency obtained by the IBIS-S radar. As seen in Figure 13, the response of the structure after the truck left the bridge (at about 13 s) can be described as free damped vibration.