## 2 Materials and Methods
### Theoretical Equation  
Deriving the undrained shear strength ratio \(S_{u}/\sigma_{vi}^{\prime}\) from the soil intrinsic relationship is a common method for solving soil strength. Based on the Cambridge model, [PERSON] et al. [33] proposed the undrained shear strength derivation equation under the condition of triaxial \(K_{0}\) (\(K_{0}\) is the coefficient of earth pressure at rest) consolidation. \[\left(\frac{S_{u}}{\sigma_{vi}^{\prime}}\right)_{\text{CCA}}=OCR^{\overline{ \Lambda}}\cdot\left(\frac{S_{u}}{\sigma_{v0}^{\prime}}\right)_{\text{NCA}} \tag{2}\]  
where OCA is the overconsolidated state of anisotropic clay, NCA is the normally consolidated state of anisotropic clay, the overconsolidation ratio is \(OCR=\sigma_{v0}^{\prime}/\sigma_{vi}^{\prime}\), \(\overline{\Lambda}=1-\overline{\kappa}/\lambda\), \(\overline{\kappa}\) is the slope of the swelling lines described by the effective overburden pressure, and \(\lambda\) is the slope of the anisotropical normally consolidated soil compression curve:  
\[\overline{\kappa}=\frac{(e_{i}-e_{0})}{\left(ln\sigma_{v0}^{\prime}-ln\sigma_{ vi}^{\prime}\right)} \tag{3}\]  
\[\kappa=\frac{(e_{i}-e_{0})}{\left(ln\rho_{0}^{\prime}-ln\rho_{i}^{\prime} \right)} \tag{4}\]  
where \(\kappa\) is the slope of the swelling lines described by the effective average stress, \(p_{0}^{\prime}\) is the effective average vertical pressure in the anisotropical normally consolidated state, and \(p_{i}^{\prime}\) is the effective average overburden pressure. \(e_{i}\) is the void ratio, and \(e_{0}\) is the initial void ratio. \[\frac{p_{0}^{\prime}}{p_{i}^{\prime}}=OCR^{(\overline{\kappa}/\kappa)} \tag{5}\]  
Substitute \(p_{0}^{\prime}=\frac{1}{3}(1+2K_{0 nc})\cdot\sigma_{v0}^{\prime}\) and \(p_{i}^{\prime}=\frac{1}{3}(1+2K_{0})\cdot\sigma_{vi}^{\prime}\) into Equation (5), to obtain the following:  
\[\overline{\kappa}=\kappa\cdot\frac{ln\left(\frac{1+2K_{0 nc}}{1+2K_{0}}\right) +ln(OCR)}{ln(OCR)} \tag{6}\]  
where \(K_{0}\) is the coefficient of earth pressure at rest in the soil in the overconsolidated state, \(K_{0 nc}\) is the coefficient of earth pressure at rest in the soil in the normally consolidated state, and the undrained shear strength ratio of the normally consolidated soil is  
\[\left(\frac{S_{u}}{\sigma_{v0}^{\prime}}\right)_{\text{NCA}}=\frac{1+2K_{0}}{ 3}\frac{M}{2}exp\left(-\Lambda\pm\frac{\Lambda}{M}\eta_{0}\right) \tag{7}\]  
where \(\Lambda=1-\kappa/\lambda\), \(\eta_{0}=3\cdot(1-K_{0})/(1+2\cdot K_{0})\), \(M\) is the critical state stress ratio, \(M\)= \(q/p\), \(q\) is generalized shear stress, \(q=(\sigma_{1}-\sigma_{3})\), \(p\) is the average principal stress, and \(p=(\sigma_{1}+\sigma_{2}+\sigma_{3})/3\). When \(\eta_{0}=0\) and \(K_{0}=1\), the ratio of undrained shear strength of the isotropical normally consolidated state and isotropically overconsolidated state clay is obtained, respectively. \[\left(\frac{S_{u}}{\sigma_{v0}^{\prime}}\right)_{\text{NCI}}=\frac{M}{2^{\Lambda +1}} \tag{8}\]  
\[\left(\frac{S_{u}}{\sigma_{vi}^{\prime}}\right)_{\text{OCI}}=OCR^{\Lambda}\cdot \left(\frac{S_{u}}{\sigma_{v0}^{\prime}}\right)_{\text{NCI}}=OCR^{\Lambda}\frac {M}{2^{\Lambda+1}} \tag{9}\]  
where OCI is the overconsolidated state of isotropic clay, and NCI is the normally consolidated state of isotropic clay. Based on the experimental results of isotropic consolidation and swelling tests, [PERSON] [34] proposed an empirical equation \(M=1.75\cdot\Lambda\). From Equations (8) and (9), the equations for calculating the undrained shear strength of the isotropical normally consolidated and isotropically overconsolidated clays can be obtained.