# Dilatant Nature of Sand Shear Strength
## 3 Shear Strength of Sand for TXC, BXC, and TXE
### Plane Strain Conditions  
For the plane strain conditions (BXC), the stress ratio at failure can be calculated using [PERSON]'s theory (Equation (1)), with \(\phi^{\prime}_{f}=\phi^{\prime}_{cv}\).  
By comparing [PERSON]'s Equations (10) and (11), we can derive the following equation:  
\[\phi^{\prime}_{b}-\phi^{\prime}_{cv}=-16.67(\delta\varepsilon_{v}/\delta \varepsilon_{a})_{b} \tag{27}\]  
The stress ratio calculated using the FSC depends on the intermediate principal stress (parameter \(b_{b}\); angle \(\theta_{b}\)). The intermediate principal stress (\(b_{b}\) values) depends on the initial porosity, the contact of the belt plate [51], and the height-to-width ratio of the specimen [4].  
Figure 5 shows some experimental values of \(b_{b}\) as a function of \(\left(\phi^{\prime}_{b}-\phi^{\prime}_{cv}\right)\). The \(b_{b}\) values range from 0.18 (\(\theta_{b}\)= 20.3\({}^{\circ}\)) to 0.35 (\(\theta_{b}\)= 9.8\({}^{\circ}\)), with mean values of \(b_{b}\)= 0.265 (\(\theta_{b}\). = 15.2\({}^{\circ}\)). Similar \(b_{b}\) values were observed by [PERSON] and [PERSON] [51], [PERSON] et al. [54], and [PERSON] et al. [55]. At the post-failure shearing stage, the \(b_{b}\) value does not change [8,50].  
Figure 5: The \(b_{b}\) values for some sands at BXC.  
For BXC, the friction angle can be calculated from Equation (16) using Equations (13) and (15). \(\delta\varepsilon_{2}=0\); therefore, at failure:  
\[\theta_{\varepsilon}=-sin^{-1}(D_{b}/3) \tag{28}\]  
and  
\[D_{b}=\frac{3(\delta\varepsilon_{v}/\delta\varepsilon_{a})_{b}}{2\sqrt{3+( \delta\varepsilon_{v}/\delta\varepsilon_{a})_{b}^{2}-3(\delta\varepsilon_{v}/ \delta\varepsilon_{a})_{b}}} \tag{29}\]  
For the BXC failure states, the experimental stress-dilatancy relationship is accurately described by the FSC equation (Equation (13)), with \(\alpha_{f}=0\) and \(\beta_{f}=1.4\)[24]. \(A_{b}\) (Table 1) depends on the Lode angles for the stresses (\(\theta_{b}\) or \(b_{b}\)) and the strain increments (\(\theta_{cb}\)). The effect of \(b_{b}\) on the friction angle, (\(\phi^{\prime}_{b}\)), for \(\phi^{\phi}=\phi^{\prime}_{cv}=30^{\circ}\) and \(34^{\circ}\), which was calculated using the FSC, is shown in Figure 6.  
For the observed ranges of \(b_{b}\), the maximum impact is less than \(1^{\circ}\). The difference increases with a decrease in the strain increment ratio (\((\delta\varepsilon_{v}/\delta\varepsilon_{a})_{b}\)), and it is almost independent of the critical state angle. Therefore, further analyses will be performed only for \(b_{b}=0.265\) (\(\theta_{b}=15.2^{\circ}\)).  
Figure 7 shows the relationship between \(\phi^{\prime}_{b}\) and \((\delta\varepsilon_{v}/\delta\varepsilon_{a})_{b}\), which is calculated using the Rowe, Bolton, and FSC relationships.  
The values of the \(\phi^{\prime}_{b}\) obtained from the Rowe and FSC relationships are very similar. The maximum difference is smaller than \(0.5^{\circ}\). A good convergence of the \(\phi^{\prime}_{b}\) values, which were calculated using the Bolton and FSC relationships, is obtained for \((\delta\varepsilon_{v}/\delta\varepsilon_{a})_{b}>-1\). For \((\delta\varepsilon_{v}/\delta\varepsilon_{a})_{b}<-1\), the \(\phi^{\prime}_{b}\) values calculated using [PERSON]'s equation (Equation (27)) are slightly higher than those calculated using the FSC.  
\((\delta\varepsilon_{v}/\delta\varepsilon_{a})_{b}=(\delta\varepsilon_{v}/ \delta\varepsilon_{a})_{c}\)[7,8]; therefore, the relationship between \(\phi^{\prime}_{b}\) and \(\phi^{\prime}_{c}\) can be calculated using the FSC. The relationships between the \(\phi^{\prime}_{b}\) and the \(\phi^{\prime}_{c}\), which are calculated using the FSC, and which are obtained in some experiments [4,8,9,21,56,57,58,59,21,59, 56,60, 57, 58], are shown in Figure 8.  
It can be seen that the FSC correctly expresses the \(\phi^{\prime}_{b}\) and \(\phi^{\prime}_{c}\) relationship.  
Figure 6: Influence of \(b_{b}\) on the sand strength for BXC.