# Effect of Mix Proportion Parameters on Behaviors of Basalt Fiber RPC Based on Box-Behnken Model
## 2 Experimental
### Box-Behnken Design (BBD)  
The Box-Behnken method (BBD) is a response surface design, which can evaluate the non-linear relation between indexes and factors. It does not need to carry out repeated tests. With the same number of experimental factors, it is more economical and efficient because it requires less times of experiment [27]. The number of experiments (\(N\)) required by BBD method is determined by Equation (1). \[N=2k(k-1)+C_{p} \tag{1}\]  
where \(k\) is the number of factors and \(C_{p}\) is the number of center points. Through the BBD method, a quantitative mathematical relationship (linear equation of quadratic polynomial) can be obtained to accurately reflect the relationship between process variables and response variables, Equation (2). \[y=\beta_{0}+\sum_{1}^{k}\beta_{i}x_{i}+\sum_{1}^{k}\beta_{ii}x_{i}^{2}+\sum \sum_{i<j}\beta_{ij}x_{i}x_{j}+\varepsilon \tag{2}\]  
where \(y\) is the response variable, \(x_{i}\), \(x_{j}\) are the process variables, where \(i\), \(j\) represent code values corresponding to different factors, \(k\) is the number of process variables, \(\beta_{0}\) is the intercept, \(\beta_{i}\) is the linear coefficient of \(i\)-th factor, \(\beta_{ii}\) is quadratic coefficient of \(i\)-th factor, \(\beta_{ij}\) is interaction coefficient of \(i\)-th and \(j\)-th factors and \(\varepsilon\) is a random error [28]. In this paper, Box-Behnken design was established by using Design-expert 8(r) software. Four factors were taken as process variables: Sand binder ratio (\(A\)), water binder ratio (\(B\)), basalt fiber content (\(C\)) and silica fume to cement ratio (\(D\)). The fluidity (\(R1\)), flexural strength (\(R2\)) and compressive strength (\(R3\)) of the mixture were set as response variables. The response surface model with 29 experimental points composed of 24 factorial and five center points was established. The model concept diagram is shown in Figure 1.  
Table 5 shows the levels of experimental factors, and the corresponding code values. The low and high levels of each factor were respectively encoded as \(-1\) and \(+1\) while the central points were encoded as 0. The mix proportions and corresponding response results of the 29 experimental groups in this paper are shown in Table 6, in which B25-B29 is the five central experimental groups used to evaluate the stability of the model. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Symbol**} & \multirow{2}{*}{**Factors**} & \multicolumn{3}{c}{**Actual Values**} & \multicolumn{3}{c}{**Code Values**} \\ \hline \(A\) & Quartz sand/binder ratio & 0.9 & 1 & 1.1 & \(-1\) & 0 & 1 \\ \hline \(B\) & Water/binder ratio & 0.18 & 0.2 & 0.22 & \(-1\) & 0 & 1 \\ \hline \(C\) & Basalt fiber & By weight (kg/m\({}^{3}\)) & 4 & 6 & 8 & \(-1\) & 0 & 1 \\ \(D\) & \(D\) & \(D\) & 0.15 & 0.225 & 0.3 & \(-1\) & 0 & 1 \\ \hline \(D\) & \multicolumn{3}{c}{Silica fume/cement ratio} & 0.2 & 0.3 & 0.4 & \(-1\) & 0 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Levels of experimental factors, given in actual and code values.