# Experimental Studies of Scale Effect on the Shear Strength of Coarse-Grained Soil
## 3 Results  
### Relationship between Shear Strength Index and Maximum Particle Size  
According to the Mohr-Coulomb theory [2; 16], the shear strength of CGS can be expressed as Equation (8):  
\[\tau=c+\sigma\tan\varphi \tag{8}\]  
where \(\sigma\) is the vertical pressure, \(\tau\) denotes shear strength, \(c\) is cohesion, and \(\varphi\) represents the internal friction angle. According to Equation (8), the shear strength of coarse-grained material is composed of cohesion \(c\) and internal friction angle \(\varphi\). In particular, for the coarse-grained soil composed of coarse and fine particles, the cohesion \(c\) includes not only the interaction between coarse particles, but also the particle cohesion between fine particles. A total of 24 groups of soil samples were tested; the vertical pressures of each group were 50, 100, 150 and 200 kPa, respectively. The relationship curve of shear strain (\(\tau\)) versus shear strain (\(\gamma\)) is drawn in Figure 6. The stress-strain curves of grade batching with \(S=0.581\) and \(d_{\text{max}}\) values of 40, 20, and 10 mm are listed. The peak strength of shear stress is taken as the shear strength and fitted with Equation (8). The intercept of the straight line on the ordinate is presented as the cohesion \(c\), and the inclination angle is presented as the internal friction angle \(\varphi\) (refer to Figure 7). According to the test results of D1-4 to D4-4, the maximum particle size \(d_{\text{max}}\) and shear strength indexes are sorted out by cohesion \(c\) and internal friction angle \(\varphi\). The variation relationship is shown as the discrete points in Figures 8 and 9. As can be observed, the maximum particle size has a great impact on the shear strength characteristics of CGS. Furthermore, regardless of the gradation structure, both cohesion \(c\) and internal friction angle \(\varphi\) increase with the improvement of the maximum particle size \(d_{\text{max}}\). The triaxial shear test results of rockfill materials for \(c\) and \(\varphi\) carried out by [PERSON] et al. [11] are consistent with those obtained in this test results; the change law is also similar to those of \(d_{\text{max}}\). The ratio of coarse and fine particle content in coarse-grained soil is one essential factor affecting its shear strength [16]. From Equations (6) and (7), the grading area \(S\) is proportional to the coarse content. The increase or decrease of gradation area \(S\) will cause the degree of filling of coarse and fine particles to change, and the values of cohesion and internal friction angle will also change with it. Figure 5: Schematic diagram of gradation curve area. Figure 6: Relationship of shear stress and strain. (**a**) \(S=0.581\), \(d_{\max}=40\). (**b**) \(S=0.581\), \(d_{\max}=20\). (**c**) \(S=0.581\), \(d_{\max}=10\). Figure 7: Relationship between internal axial pressure and shear strength of test CGS. Further analysis found that the relationship of the \(d_{\max}\), \(c\) and \(\varphi\) can be expressed as a logarithmic equation. Hence, the relation of can be expressed as:  
\[\left\{ \begin{array}{l}c=a_{1}\ln d_{\max}+b_{1}\\ \varphi=a_{2}\ln d_{\max}+b_{2}\end{array} \right. \tag{9}\]  
where \(a_{1}\), \(b_{1}\), \(a_{2}\), and \(b_{2}\) are parameters; \(b_{1}\) and \(b_{2}\) represent the cohesion and internal friction angle of the sample when the maximum particle size is 1 mm, and the units are kPa and (\({}^{\circ}\)), respectively; and \(a_{1}\) and \(a_{2}\) represent the change rate of cohesion and internal friction angle of the sample when the maximum particle size is 1 mm, and the units are kPa/ln(mm) and (\({}^{\circ}\))/ln(mm), respectively. The fitting curves according to Equation (9) are given in Figures 8 and 9. Furthermore, it is found that the fitting curves are in good agreement with the test data points, and the error between the fitting value and the corresponding test data points is basically less than 3% and 1%. The maximum error is less than 6.87% and 1.29%, and the determination coefficient is more than 0.937.