## 5 Calculation of Fiber-Sprayed Concrete Stacked Beams with Flexural Stiffness Correction
### Comparative Analysis of the Flexural Stiffness Calculation Theory and the Test in This Paper  
In four-point bending tests, fiber shotcrete beams usually exhibit typical flexural failure. For unstratified fiber shotcrete beams, friction does not have a direct effect on their performance due to the absence of interfacial interaction between different material layers. At this point, the calculation of the flexural load and stiffness of the beam can be directly based on the geometric properties and material properties of the beam, and the interlayer factor can be ignored. In the four-point bending test, the maximum bending moment usually occurs at the mid-span position with the following expression:  
\[M=\frac{FL}{6} \tag{7}\]  
In the formula, M is the bending moment at the mid-span position of the beam, N \(\cdot\) mm; F is the load, N; and L is the span of the lower support, mm.  
The deflection equation of a beam is the central equation that describes the deformation of a beam under a bending load, which is determined by the elastic modulus of the material and the geometric properties of the beam. The basic form of the bending equation is given below as follows:  
\[\frac{d^{2}y(x)}{dx^{2}}=\frac{M(x)}{EI} \tag{8}\]  
In the formula, \(y(x)\) is the deflection function of the beam, denoting the deflection of the beam at the location \(x\), mm; \(M(x)\) is the bending moment at location \(x\), N \(\cdot\) mm; and \(EI\), is the beam flexural stiffness, N \(\cdot\) mm\({}^{4}\).  
Combining Equations (7) and (8), the effective flexural stiffness of fiber shotcrete beams in the four-point test is calculated as follows:  
\[EI_{eff}=\frac{23F_{eff}L^{3}}{648\delta_{eff}} \tag{9}\]  
In the formula, \(F\) is the elastic phase load, \(F_{eff}=F_{cr}\), N; \(L\) is the span of the lower support, mm; and \(\delta_{eff}\) is the axial deflection at mid-span of the elastic phase, \(\delta_{eff}=\delta_{cr}\), mm.  
Calculate the flexural stiffness of fiber shotcrete beams with different thickness ratios, as shown in Table 7.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Working Condition** & \(\lambda_{h}\) & \multicolumn{3}{c}{**EI/N \(\cdot\)mm\({}^{4}\)**} \\  & & **Assumption (1)** & **Assumption (2)** & **Assumption (3)** \\ \hline A1 & 0 & 241.67 \(\times\) 10\({}^{9}\) & 241.67 \(\times\) 10\({}^{9}\) & 241.67 \(\times\) 10\({}^{9}\) \\ A2 & 1 & 241.67 \(\times\) 10\({}^{9}\) & 241.67 \(\times\) 10\({}^{9}\) & 60.42 \(\times\) 10\({}^{9}\) \\ A3 & 2 & 2 The results of plotting the test and three assumptions for calculating the flexural stiffness are shown in Figure 13:  
In Figure 13, the following can be seen:  
**Assumption 1:**_The flexural stiffness is the same for each thickness ratio;_  
**Assumption 2:**_The flexural stiffness is lowest when the thickness ratio is 0 or 1, and then it increases with the thickness ratio;_  
**Assumption 3:**_The flexural stiffness is maximum when the thickness ratio is 0, minimum when the thickness ratio is 1, and then gradually increases, but they are all lower than the flexural stiffness when the thickness ratio is zero._  
The trends of the results of the three hypothetical calculations are in conflict with the experimental results, and, therefore, the method of calculating the flexural stiffness of the stacked beams needs to be corrected.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Working Condition** & \(\lambda_{h}\) & \(F_{\textit{eff}}\)**/kN** & \(\delta_{\textit{eff}}\)**/mm** & **Effective Flexural Stiffness, \(EI_{\textit{eff}}\)**/N\(\cdot\)mm\({}^{4}\)** \\ \hline A1 & 0 & 20.85 & 0.09 & 222.01 \(\times\) 10\({}^{9}\) \\ A2 & 1 & 22.63 & 0.12 & 180.73 \(\times\) 10\({}^{9}\) \\ A3 & 2 & 22.79 & 0.14 & 156.00 \(\times\) 10\({}^{9}\) \\ A4 & 3 & 20.93 & 0.16 & 125.36 \(\times\) 10\({}^{9}\) \\ A5 & 4 & 19.09 & 0.17 & 107.62 \(\times\) 10\({}^{9}\) \\ A6 & 5 & 18.55 & 0.18 & 98.76 \(\times\) 10\({}^{9}\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Calculation of flexural stiffness.  
Figure 13: Calculation of flexural stiffness.