# Experimental Studies of Scale Effect on the Shear Strength of Coarse-Grained Soil
## 2 Materials and Methods
### Research Method of Scale Effect  
In order to quantitatively explore the influence of the maximum particle size \(d_{\rm max}\) and gradation structure on the mechanical characteristic of CGS, the gradation equation of Equation (1) was selected to quantitatively express the CGS. Subsequently, the shear strength relationship of the maximum particle size \(d_{\rm max}\), gradation structure \(n\), and gradation area \(e\) were established. Through the analysis and summary of the test data, showing not direct relationship of \(e\) and \(n\) to \(c\) and \(\varphi\), a quantitative relationship of \(e\) and \(n\) with the grading area \(S\) is presented. The grading area \(S\) is surrounded by the grading equation curve, coordinate horizontal axis, maximum particle size \(d=d_{\rm max}\) line and \(d=d_{\rm max0}\) line (as given in Figure 5). Therefore, the calculation equation is as follows: \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & \(d_{\rm max}\)/mm & \(n\) & \(e\) & \(S\) \\ \hline A7-4 & 40 & 0.6 & 0.6 & 0.673 \\ A8-4 & 40 & 0.6 & 0.3 & 0.581 \\ A9-4 & 40 & 0.6 & \(-\)0.2 & 0.482 \\ A10-4 & 40 & 0.4 & 0.6 & 0.749 \\ A11-4 & 40 & 0.4 & 0.3 & 0.672 \\ A12-4 & 40 & 0.4 & \(-\)1.0 & 0.486 \\ D1-4 & 40 & 1.0 & 0.3 & 0.441 \\ D1-2 & 20 & 1.0 & 0.3 & 0.441 \\ D1-1 & 10 & 1.0 & 0.3 & 0.441 \\ D2-4 & 40 & 0.8 & 0.6 & 0.603 \\ D2-2 & 20 & 0.8 & 0.6 & 0.603 \\ D2-1 & 10 & 0.8 & 0.6 & 0.603 \\ D3-4 & 40 & 0.6 & \(-\)1.0 & 0.390 \\ D3-2 & 20 & 0.6 & \(-\)1.0 & 0.390 \\ D3-1 & 10 & 0.6 & \(-\)1.0 & 0.390 \\ D4-4 & 40 & 0.4 & \(-\)0.2 & 0.581 \\ D4-2 & 20 & 0.4 & \(-\)0.2 & 0.581 \\ D4-1 & 10 & 0.4 & \(-\)0.2 & 0.581 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No. ** & \(\rho_{\rm min}\)/**(g\(\cdot\)cm\({}^{3}\))** & \(\rho_{\rm min}\)/**(g\(\cdot\)cm\({}^{3}\))** & \(D_{\rm r}\) & \(\rho_{\rm 0}\)/**(g\(\cdot\)cm\({}^{3}\))** \\ \hline A1-4 & 1.905 & 2.163 & 0.8 & 2.106 \\ A2-4 & 1.815 & 1.994 & 0.8 & 1.955 \\ A3-4 & 1.750 & 1.959 & 0.8 & 1.913 \\ A4-4 & 1.878 & 2.122 & 0.8 & 2.068 \\ A5-4 & 1.838 & 2.049 & 0.8 & 2.003 \\ A6-4 & 1.78\[S=\frac{\ln(1-we)-\ln(1-e)}{ne\ln 10} \tag{6}\]  
where \(w\) is the percentage passing when \(d_{\max}=d_{\max 0}\), which can be expressed as:  
\[w=\frac{1}{(1-e)(d_{\max}/d_{\max 0})^{n}+e} \tag{7}\]  
According to Equations (6) and (7), a quantitative relationship between parameters \(e\) and \(n\) and the grading curve area is established. That is, the grading area value is affected by parameters \(e\) and \(n\). According to the single variable methods [2], it should ensure that the grading curve area \(S\) is the same value before and after scaling so as to reveal how the maximum particle size affects the shear strength of CGS. Based on the Equations (6) and (7), and the literature [25], the values of grading parameters \(e\) and \(n\) is maintained after the scaling of the similar grading method with a consistent \(d_{\max}/d_{\max 0}\), while the grading curve area \(S\) is the same before and after scaling. The literature [2] also suggested the boundary particle size of sand and gravel as taken to be considered to be about 5 mm. Therefore, \(d_{\max}\) and \(d_{\max 0}\) are taken as 40 and 5 mm, respectively, to solve the grading curve area \(S\). At this time, the grading curve area \(S\) is only related to the grading parameters \(e\) and \(n\), and the gradation structure is determined by both \(e\) and \(n\). As a result, the influence of the gradation structure on mechanical characteristics can be expressed by the grading area \(S\) as an alternative characteristic parameter. Figure 4: Gradation curves of test CGS. (**a**) Samples with same gradation structures and the different maximum particle sizes. (**b**) Samples with the different maximum particle sizes and same gradation structures.