## 2 Experimental Program
### Impact Test Setup
Figure 7 shows the measured acceleration signal from the west accelerometer of Beam D1. The figure presents the important characteristics of the signals that were discussed. The initial peak shows the maximum acceleration measured as the contact force develops and is resisted by the beam. Thereafter, the retardation phase initiates, where the beam decelerates due to the development of the strain energy in the beam and prevailing energy dissipation mechanisms. The retardation phase is succeeded by what resembles the damped free vibration of the beam with internal forces in the linear range. Figure 6: (**a**) Measured acceleration at the west accelerometer for A1–D1; (**b**) acceleration after signal processing; (**c**) velocity due to measured signal; (**d**) velocity after signal processing; (**e**) displacement due to measured signal; and (**f**) displacement after signal processing. The energy density of measured acceleration signals in the time-frequency domain, represented by the time-frequency scale-space (see [24]), were determined in [25] using the continuous wavelet transform (CWT); see [26]. From this procedure, frequency ranges with high energy content could be determined, corresponding to resonance frequencies of the system. A similar procedure was adopted in the current work, where a Morlet wavelet was used in combination with the MATLAB Wavelet Toolbox to determine the wavelet coefficients; see [27]. The procedure was applied on the free vibration phase of the signals post-retardation, such that the fundamental mode of the beams and other resonance frequencies with high relative energy content could be determined after sustaining damage during retardation. This is presented in Figure 7, where CWT was applied on the linear free vibration phase spanning 11-20 ms. Here, the response frequencies with high energy content are coloured black, and white indicates relatively low energy content. The frequency ranges with high energy content may be compared with the analytical solution for natural frequencies for the same beam configuration. The resonance frequency \(f_{analytical,n}\) for mode \(n\) of a simply supported slender beam may be determined as  
\[f_{analytical,n}=n^{2}\frac{\pi}{2}\sqrt{\frac{EI}{\rho_{b}Al^{4}}} \tag{1}\]  
where \(E\) is the Young's modulus of the beam, and \(\rho_{b}\) is the density of the beam; see, e.g., ref. [28]. Table 5 presents the first two analytical natural frequencies of the beam determined assuming uncracked concrete sections \(f_{uncracked}\) and cracked concrete sections \(f_{cracked}\), where the steel and concrete in compression are in the linear range. Comparing the resonance frequencies between the beams post-retardation is interesting since the mass generally is equal between the beams. A relatively low resonance frequency thus indicates a lower element stiffness from sustaining more damage. CWT was applied on all beams after the first impact test to study their degree of damage before the residual static and impact tests were conducted. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode \(n\)** & \(f_{uncracked}\) [Hz] & \(f_{cracked}\) [Hz] \\ \hline
1 & 514.37 & 317.57 \\
2 & 2057.48 & 1270.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Calculations of the first two natural frequencies for cracked and uncracked concrete beams. Figure 7: The initial peak; retardation; and linear free vibration phase of Beam D1 and time–frequency scale-space of the linear free vibration phase determined using CWT.