## 4 Mechanical Properties of Paste Backfill Materials Made of Cement
### Calculations and Analysis of Experimental Results  
In this study, the data were analyzed using the variance analysis and intuitive analysis approaches. The best value of many components and their arrangement can be ascertained using the qualitative data analysis method known as intuitive analysis. Variance analysis is a quantitative method that may be used to ascertain each factor's confidence and amount of influence on performance outcomes. In order to determine the ideal mix, these two analytical techniques can be utilized to more thoroughly determine the influence law of different influential components and levels on the performance of cemented backfilling materials. To process data, some specific formulas were used to input the necessary variables and perform the calculations accordingly. The relationship between the unconfined compressive strength (UCS) and the factor of safety is not a direct correlation. However, we can establish a relationship between them through the concept of shear strength. The factor of safety is a measure of the ability of a system to withstand loads without failing. In geotechnical engineering, the factor of safety is often calculated as the ratio of the shear strength of the soil to the shear stress applied. The unconfined compressive strength (UCS) is a measure of the strength of a soil in unconfined compression. It is related to the shear strength of the soil, which is a critical parameter in determining the factor of safety. According to Equations (5)-(12), the parameter values were calculated [27; 28; 29]. The experimental results can be seen in Table 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & **Curing Time (d)** & **Paste Density (\(\mathbf{\rho_{p}\%}\))** & **Fly Ash Content (\(\mathbf{X_{1}\%}\))** & **Cement Content (\(\mathbf{X_{2}\%}\))** & **Waste Rock Content (\(\mathbf{X_{3}\%}\))** \\ \hline S6 & 3 & 74 & 30 & 10 & 60 \\ S7 & 3 & 76 & 30 & 12 & 58 \\ S8 & 3 & 78 & 30 & 6 & 64 \\ S9 & 7 & 72 & 35 & 10 & 55 \\ S10 & 7 & 74 & 35 & 12 & 53 \\ S11 & 7 & 76 & 35 & 6 & 57 \\ S12 & 7 & 78 & 35 & 8 & 57 \\ S13 & 28 & 72 & 40 & 12 & 48 \\ S14 & 28 & 74 & 40 & 6 & 54 \\ S15 & 28 & 76 & 40 & 8 & 52 \\ S16 & 28 & 78 & 40 & 10 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type of Paste Fill Specim** & **Density (kg\(\cdot\)m\({}^{-3}\))** & **Peak Load (kN)** & **UCS (MPa)** & **Strain** & **Shear Stress (MPa)** \\ \hline
6\% Cement D1 = S1 & 34,571.4 & 3.0223 & 0.62 & 0.023 & 0.31 \\
6\% Cement D3 = S8 & 34,333.3 & 4.7335 & 0.97 & 0.018 & 0.485 \\
6\% Cement D7 = S11 & 34,523.8 & 12.3163 & 2.51 & 0.014 & 1.255 \\
6\% Cement D28  
The shear strength of a soil or cement past backfill can be estimated from the UCS using the following equation:  
\[\tau_{max}\ =cos\varphi\ *UCS/2 \tag{5}\]  
where \(\tau\) is the shear strength, \(\varphi\) is the internal friction angle, and \(UCS\) is the unconfined compressive strength. Once the shear strength is known, the factor of safety can be calculated as:  
\[FOS=\tau_{max}\ /\tau \tag{6}\]  
where \(FOS\) is the factor of safety, \(\tau_{max}\) is the shear strength, and \(\tau\) is the applied shear. The axial strain was calculated as  
\[\varepsilon=\Delta L/L_{0} \tag{7}\]  
where \(\varepsilon\) is the axial strain, \(\Delta L\) is the change in measured axial length, and \(L_{0}\) is the initial length of the sample. The compressive stress was calculated as  
\[\sigma=\text{P}/\text{A}_{0} \tag{8}\]  
where \(\sigma\) is the compressive stress, P is the load and A\({}_{0}\), is the initial cross-section area of the specimen. Therefore, the unconfined compressive strength is calculated for the maximum load applied:  
\[\sigma_{\text{UCS}}=\text{P}_{max}\ \ /\ \text{A}_{0} \tag{9}\]  
The shear stress was calculated as  
\[\tau=\text{F}/\ A \tag{10}\]  
where \(\tau\) is the shear stress, F is the force applied, and \(A\) is the cross-sectional area of the material.