# Evaluating Pile-Supported Embankment Considering the Soil Anisotropy Effect
## 5 Soil Arching Considering Friction Angle Anisotropy
### Friction Angle Anisotropy  
Based on the Mohr-Coulomb yield criterion, in the finite element calculation, the value of the friction angle at each integral point is regarded as a function of the shear stress direction, and it is developed into a yield criterion that can consider the anisotropy of the friction angle. For the calculation of the friction angle, the calculation formula proposed by [PERSON] [58; 59] is derived from the fabric tensor \(A\) proposed by [PERSON] and [PERSON] [60] from the microscopic point of view. For orthotropic materials, \(A\) can be expressed by Equation (24):  
\[A_{ij}=a_{1}e_{i}^{(1)}e_{j}^{(1)}+a_{2}e_{i}^{(2)}e_{j}^{(2)}+a_{3}e_{i}^{(3) }e_{j}^{(3)} \tag{24}\]  
where \(a_{1}\), \(a_{2}\) and \(a_{3}\) are eigenvalues of tensors. By introducing the unit vector \(n\), the scalar value \(\alpha\) of tensor A in different directions in the orthogonal coordinate systems \(e^{(1)}\), \(e^{(2)}\), and \(e^{(3)}\) can be obtained. As shown in Formula (25), any material parameter value in the \(n\) direction can be considered according to Formula (25).  
\[\alpha=An\cdot n=A_{ij}n_{i}n_{i} \tag{25}\]  
The normalized partial tensor of vector \(A\) is  
\[\Omega_{ij}=\frac{A_{ij}-\frac{1}{3}\delta_{ij}A_{kk}}{\frac{1}{3}A_{kk}} \tag{26}\]  
For isotropic materials:  
\[\Omega=\left[\begin{array}{ccc}-\Omega_{o}/2&0&0\\ 0&\Omega_{o}&0\\ 0&0&-\Omega_{o}/2\end{array}\right] \tag{27}\]  
where \(\Omega_{o}=\frac{2(a_{2}-a_{1})}{a_{2}+2a_{1}}\). By combining Formulas (25) and (26), and assuming that the anisotropy coefficient of the difference between vertical and horizontal strength is \(A_{r}\) (anisotropy ratio), the calculation formula for strength parameters in the n direction can be obtained, as shown in Formula (28):  
\[\alpha=\alpha_{0}\bigg{\{}1-\frac{A_{r}-1}{A_{r}+2}\Big{[}1-3(n_{v})^{2}\Big{]} \bigg{\}} \tag{28}\]  
where \(A_{r}=\frac{a_{o}}{a_{3}}=\frac{1+\Omega_{o}}{1-\Omega_{o}/2},\alpha_{0}= \frac{1}{3}A_{ii}\), \(n_{v}\) denotes the vertical component of the unit vector \(n\). Therefore, the anisotropic internal friction angle can be calculated using Equation (29):  
\[\varphi^{\prime}=\arctan\left[\tan\varphi_{0}\bigg{\{}1-\frac{A_{r}-1}{A_{r}+ 2}\Big{[}1-3(n_{v})^{2}\Big{]}\bigg{\}}\right] \tag{29}\]  
where \(A_{r}=\frac{\tan\varphi_{o}}{\tan\varphi_{h}}\), tan \(\varphi_{0}=\frac{\tan\varphi_{o}+2\tan\varphi_{h}}{3}\), \(\varphi_{0}\) represents the average value of the friction angle in each direction; \(\varphi_{o}\) and \(\varphi_{h}\) are the friction angles when the shear stress direction is vertical and horizontal, respectively.  
In the calculation of the model, the strength of the soil is the largest when the vertical shear is considered, and the strength is the smallest when the horizontal shear is considered. Therefore, the \(n_{v}\) value is determined by the angle between the shear stress and the vertical direction at different stress states, and \(\varphi^{\prime}\) is brought into the Mohr-Coulomb yield function. The Mohr-Coulomb yield criterion considering the anisotropy of the internal friction angle can be obtained:  
\[F=R_{mc}q-p\tan\varphi^{\prime}-c=0 \tag{30}\]  
When calculating and analyzing, it is necessary to input the internal friction angle anisotropy coefficient (\(A_{r}\)) value in the user-defined material parameter window. For specific projects, it should be accurately obtained through multiple sets of tests.