## 4 Range Analysis  
By conducting nine sets of orthogonal tests with various proportions, the physical and mechanical parameters were measured for material specimens with different proportions, including density \(\rho\) (g/cm\({}^{3}\)), compressive strength \(\sigma_{C}\), tensile strength \(\sigma_{t}\), cohesion \(C\), elastic modulus \(E\), and friction angle \(\varphi\). Table 3 shows the test results, and the density of the similar material is 2.23-2.65 g/cm\({}^{3}\). The similar material has a higher bulk density, which is consistent with the actual cohesionless soil bulk density on site. Therefore, it can simplify the conversion of physical parameters between the model and the prototype. On the other hand, the tensile-compression ratio of the similar material is 1/8.5-1/11.7, indicating that the similar material can better simulate the tensile properties of the prototype. Furthermore, the compressive strength ranges from 0.37 to 5.37 MPa, the elastic modulus ranges from 42.0 to 279.0 MPa, the cohesion ranges from 42.7 to 57.3 kPa, and the friction angle ranges from 28.37\({}^{\circ}\) to 37.04\({}^{\circ}\). The above experiment results demonstrate that these mechanical parameters have a large variation range which can meet the requirements of the various mechanical parameters of the similar material in different testing conditions. This paper adopts range analysis to study the results. Generally, \(X_{ij}\) is defined as the value of the \(i\)-th level of factor \(j\) (\(i=1\), \(2\),..., \(n\), \(j=\) A, B, C, D). Experiments under \(X_{ij}\) are conducted in order to obtain the result index \(Y_{ij}\) of the \(i\)-th level of factor \(j\), where \(Y_{ij}\) are random variables that follow the normal distribution. Conducting \(P_{1}\) tests under \(X_{ij}\) can produce \(P_{l}\) test results (\(Y_{ijk}=1\), \(2\),..., \(P_{l}\)). In Formula (7), \(K_{ij}\) is the statistical parameter of the \(j\)-th factor at the \(i\)-th level, while \(Y_{ijk}\) is the index value of the \(k\)-th test result of the \(j\)-th factor at the \(i\)-th level. \[K_{ij}=\sum_{k=1}^{P_{l}}Y_{ijk} \tag{7}\]  
The range refers to the difference between the extreme values of the data, which reflects the degree of dispersion of the data. The larger the range, the more sensitive the factor [36]. Figure 4: Direct shear test. (**a**) Direct shear specimens. (**b**) Direct shearing apparatus. (**c**) Specimens’ failure state. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Test** & \(\rho\) **(g/cm\({}^{3}\))** & \(\sigma_{c}\) **(MPa)** & \(\sigma_{t}\) **(MPa)** & \(E\) **(MPa)** & \(C\) **(kPa)** & \(\varphi\) (\({}^{\circ}\))** \\ \hline
1 & 2.46 & 0.69 & 0.071 & 248.67 & 46.359 & 37.04 \\
2 & 2.47 & 3.47 & 0.363 & 237.33 & 50.904 & 30.73 \\
3 & 2.33 & 1.10 & 0.097 & 96.33 & 51.813 & 28.37 \\
4 & 2.23 & 2.24 & 0.211 & 256.00 & 57.267 & 31.12 \\
5 & 2.46 & 5.37 & 0.498 & 279.00 & 42.723 & 29.72 \\
6 & 2.53 & 0.34 & 0.029 & 50.67 & 46.359 & 29.80 \\
7 & 2.39 & 2.02 & 0.235 & 217.33 & 55.449 & 28.77 \\
8 & 2.46 & 0.62 & 0.058 & 157.33 & 46.359 & 30.96 \\
9 & 2.65 & 0.37 & 0.033 & 42.00 & 45.450 & 28.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of test results. The factors with the largest range would be the most significant. Generally, the five main parameters of density, compressive strength, elastic modulus, cohesion, and friction angle can effectively control the physical and mechanical characteristics of materials. Therefore, these five parameters are utilized to carry out range analysis and determine the order of effect of the factors. The range analysis results of influencing factors are shown in Table 4.  
In order to more intuitively investigate the effect of each factor on the five parameters, a diagram of the relationships among the levels of each factor and parameter was produced, and this is shown in Figure 5. The results show that the four factors have certain effects on the mechanical properties of similar materials.