## 5 Calculation of Fiber-Sprayed Concrete Stacked Beams with Flexural Stiffness Correction
### Theoretical Verification of the Modified Theory of Laminated Beams  
Taking the thickness ratio \(\lambda_{h}=0\) as the base group, the interfacial influence factor under different thickness ratios is calculated using Table 7, as shown in Table 8.  
The tests show that the effective flexural stiffness of the thickness ratio \(\lambda_{h}=1\) to the thickness ratio \(\lambda_{h}=5\) is lower than the thickness ratio \(\lambda_{h}=0\), indicating that the layered construction reduces the overall effective stiffness. With the downward shift of the interface position, the effective flexural stiffness of fiber shotcrete beams \(EI_{eff}\) gradually decreases, the interface influence factor \(f_{\lambda_{h}}\) decreases, the thickness ratio \(\lambda_{h}=5\) is the lowest, and the interface influence factor \(f_{\lambda_{h}}\) is only 0.44. Excessive downward shifting of the interface will lead to interface slip, weakening the synergistic nature of the whole, aggravating the loss of stiffness, and ultimately decreasing the effective flexural stiffness.  
In order to analyze the relationship between the thickness ratio of fiber shotcrete beams \(\lambda_{h}\) and the interface influence factor \(f_{\lambda_{h}}\), we performed a linear fitting of the experimental  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Working Condition** & \(\lambda_{h}\) & **Effective Flexural Stiffness, \(EI_{eff}\)/\(\mathrm{N}\cdot\mathrm{mm}^{4}\)** & \(f_{\lambda_{h}}\) \\ \hline A1 & 0 & 222.01 \(\times\) 10\({}^{9}\) & 1 \\ A2 & 1 & 180.73 \(\times\) 10\({}^{9}\) & 0.81 \\ A3 & 2 & 156.00 \(\times\) 10\({}^{9}\) & 0.70 \\ A4 & 3 & 125.36 \(\times\) 10\({}^{9}\) & 0.56 \\ A5 & 4 & 107.62 \(\times\) 10\({}^{9}\) & 0.46 \\ A6 & 5 & 98.76 \(\times\) 10\({}^{9}\) & 0.44 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Calculation of interfacial influence factor.  
data using Origin 2024 version software, and the fitting results are shown in Figure 14. According to the fitting results, the following linear equation is obtained.  
\[f_{\lambda_{h}}=0.9975-0.19982\lambda_{h}+0.01768\lambda_{h}^{2} \tag{11}\]  
In the formula, \(f_{\lambda_{h}}\) denotes the interfacial influence factor, and \(\lambda_{h}\) denotes the thickness ratio.  
The fitted curve well describes the relationship between the thickness ratio and the interfacial influence factor. The fitting results show that there is a significant linear relationship between the trend of the interfacial influence factor and the thickness ratio at different thickness ratios.  
Combining Equations (7), (10), and (11), the deflection-to-thickness ratio of the concrete span is calculated as follows:  
\[\delta=\frac{23 FL^{3}}{648\big{(}0.9975-0.19982\lambda_{h}+0.01768\lambda_{h} ^{2}\big{)}EI_{0}}+\delta_{crack} \tag{12}\]  
In the formula, \(F\) is the elastic stage load, \(F=F_{cr}\), N, and \(\delta_{crack}\) is the axial deflection at mid-span during the crack development stage, mm.  
The mid-span deflections of fiber shotcrete beams with different thickness ratios are calculated, as shown in Figure 15a, and the linear fitting of the calculated results of the modified method to the experimental results is shown in Figure 15b, which shows that the error between the calculated results of the modified method and the experimental results is relatively small, and it proves that the formula can describe the relationship between the thickness ratios and the structural deflections very well.  
Figure 14: Relationship between the thickness ratio and the interfacial influence factor.