## 3 Verification by Comparison with Previous Experimental Results
### Investigation of the Accuracy and Applicable Range of the Deflection Estimation Formula  
The accuracy of the maximum deflection, \(\delta_{\rm u}\), can be calculated using Equation (20), and its application range is discussed below. The accuracy evaluation indicator (AEI) is defined as follows:  
\[\rm AEI=\delta_{\rm u}/\delta_{\rm u.exp}. \tag{20}\]  
Figure 11 shows the relationship between the AEI and plasticity ratio for the design \(R_{\rm p}\) (\(=\delta_{\rm u}/\delta_{\rm y}\)) in the same manner as that in Reference [14]. From the figure, many plots can be obtained when the plasticity ratio \(R_{\rm p}\leq 10\), and the AEI is densely distributed between 0.8 and 1.6. Conversely, when the plasticity ratio \(R_{\rm p}>10\), the AEI is widely distributed between 1.0 and 2.0.  
Therefore, the application range of the proposed formula can be determined as \(1<R_{\rm p}\leq 10\). When \(R_{\rm p}\leq 1\), the deflection of the RC beam is within the elastic range, and it does not exhibit a complete plastic collision. Therefore, \(R_{\rm p}<1\) was excluded from the above application range.  
To examine the estimation accuracy, Figure 12 shows the AEI when \(1<R_{\rm p}\leq 10\). As shown in the figure, the AEI was distributed between 0.81 and 1.56, with an average value of 1.15, and a coefficient of variation of 0.113.  
Figure 10: Investigation of the estimation accuracy based on the plasticity ratio \(R_{\rm p}\).  
4 Simplified Estimation Method for the Maximum Deflection of the RC Beam Subjected to an Impact Load  
Thus far, we have proposed a method for estimating the maximum deflection based on the assumption that the energy \(E_{\mathrm{t}}\) transmitted to the beam by a weight collision is equivalent to the absorption energy \(E_{\mathrm{a}}\), due to the bending deformation of the RC beam. Here, if the load-deflection curve of the RC beam can be simplified to a bilinear model, as shown in Figure 13, \(E_{\mathrm{a}}\) can be easily calculated, as expressed in Equation (21).  
\[E_{\mathrm{a}}=P_{\mathrm{y}}\ \delta_{\mathrm{ud}}-P_{\mathrm{y}}\ \delta_{ \mathrm{y}}/2 \tag{21}\]  
The deflection, when \(E_{\mathrm{a}}\) corresponds to \(E_{\mathrm{t}}\), is the maximum deflection for the design of \(\delta_{\mathrm{ud}}\). Therefore, \(\delta_{\mathrm{ud}}\) can be estimated as follows:  
\[E_{\mathrm{t}}=P_{\mathrm{y}}\ \delta_{\mathrm{ud}}-P_{\mathrm{y}}\ \delta_{ \mathrm{y}}/2 \tag{22}\]  
\[\delta_{\mathrm{ud}}=E_{\mathrm{t}}/P_{\mathrm{y}}+\delta_{\mathrm{y}}/2 \tag{23}\]  
Figure 11: Accuracy of the estimated maximum deflection for the design \(\delta_{\mathrm{ud}}\) (\(R_{\mathrm{p}}<40\)).  
Figure 12: Accuracy of the estimated maximum deflection for the design \(\delta_{\mathrm{ud}}\) (\(1<R_{\mathrm{p}}\leq 10\)).  
\[\delta_{\rm ud}=\frac{M_{\rm w}^{2}V^{2}}{2P\big{(}\frac{17}{35}\rho AL+M_{\rm w} \big{)}}+\frac{\delta_{\rm y}}{2} \tag{24}\]  
\[1<R_{\rm P}\leq 10 \tag{25}\]  
To confirm the difference between \(\delta_{\rm ud}\) and \(\delta_{\rm u}\), Figure 14 illustrates the relationship between these values. The figure shows that \(\delta_{\rm ud}\) is nearly equivalent to \(\delta_{\rm u}\). Therefore, the maximum deflection \(\delta_{\rm ud}\) can be calculated using the transmitted energy \(E_{\rm t}\), yield load \(P_{\rm y}\) of the RC beam, and yield deflection \(\delta_{\rm y}\). Such an evaluation is possible when the relationship between the load and deflection is nearly bilinear, as in the case of a single bar RC beam. However, if the load-deflection relationship is composed of a curved line or multi-line, such as a prestressed concrete (PC) beam and other composite structural members, further investigations are required.