# Study on Impact-Echo Response of Concrete Column near the Edge
## 3 Impact Response of Concrete Column
### Cross-Sectional Mode  
The above experimental results show that the measurement accuracy is biased when the impact-echo is within the influence range of boundary conditions. This is because the transient response excited by the impact-echo is composed of multiple resonance frequencies caused by the cross-sectional vibration mode. The echo response from the concrete column is mainly controlled by the wave propagating in its cross-section. Various reflections and interferences produce cross-sectional vibration modes with different frequencies. The sensor accepts the natural frequency excited by the ball in different modes. During the test, the experimenter has different frequencies received by the sensor because of the sensor's different placement and the impact force of the hammer on the measured object. At this point, the excited wave is also affected by the boundary, resulting in a deviation between the measurement accuracy and the amplitude. The corresponding vibration mode is excited when the specified frequency is one of the structure's natural frequencies.  
It has been shown that the response of a rod subjected to a short-term transverse impact consists of frequencies corresponding to the cross-sectional vibration pattern of the rod [30; 31]. In the case of concrete columns, it is shown that the boundary response also consists of multiple natural frequencies associated with the vibration pattern. The eigenvalue analysis of the two-dimensional finite element model of the concrete column were used to obtain the vibration mode and the corresponding natural frequency. The plane strain finite element model can be used to solve the cross-section mode shape and the corresponding eigenvalues. In the case of two-dimensional cross-section concrete columns, the signal response is also composed of multiple natural frequencies related to the vibration mode. The numerical model uses linear elastic and isotropic material properties to simulate the working conditions of the cross-section of the concrete column. When there are steel bars in the model, the measured dominant frequency is the same as that of the dense concrete with the same thickness. The reflection frequency generated by the interface between concrete and steel bars cannot be identified. This phenomenon is mainly due to the limited interface of the steel bars and the curved shape, resulting in weak echo reflection intensity. Therefore, in the finite element simulation, the influence of steel bars is small and can be ignored, except for large-diameter steel bars and exposed bars. The density, elastic modulus, and Poisson's ratio were 2360 kg/m\({}^{3}\), 30,000 MPa, and 0.2, respectively. The longitudinal wave velocity was 4000 m/s. The following is the analysis results of the cross-sectional eigenvalues of concrete columns.  
Figure 8 shows the shapes and eigenvalues of the two main cross-sectional modes of concrete columns. The shape coefficient for a square cross-section was usually 0.87, and n was 2. The predominant frequency calculated by the Formula (1) was 2.5 kHz. Consistent with the simulation results and compatible with the modal two eigenvalues. It can be seen that when the acquisition area is located at the center of the column, the predominant frequency obtained by the analysis is the eigenvalue corresponding to Mode 2 in the cross-section vibration mode. At this point, wave propagation is not affected by the boundary.  
Figure 8: Sectional vibration mode and eigenvalue. **(a)** Mode 1 (\(f\) = 2324 kHz); **(b)** Mode 2 (\(f\) = 2540.2 kHz).  
When the acquisition area is located in the influence range of the edge, the finite element simulation result is 2.391 kHz, which conforms to the eigenvalue corresponding to mode one. According to the above research, the predominant frequency in the middle of the column, the second eigenvalue of the mode, is taken as the primary frequency \(f_{0}\). The eigenvalue of the first mode is 0.9 \(f_{0}\).