# Evaluating Pile-Supported Embankment Considering the Soil Anisotropy Effect
## Appendix A  
The yield function expression of the Mohr-Coulomb yield criterion in ABAQUS is expressed by the stress invariant (Equation (A1)) [62]. \[F=R_{mc}q-p\tan\varphi-c=0\] (A1)where  
\[R_{mc}=\frac{1}{\sqrt{3}\cos\varphi}\sin(\theta+\frac{\pi}{3})+\frac{1}{3}\cos( \theta+\frac{\pi}{3})\tan\varphi \tag{10}\]  
\(R_{mc}\) defines the yield curve shape of the yield function on the partial plane;  
\(p=-\frac{1}{3}\text{trace}(\sigma)\) is the average stress;  
\(\cos(3\theta)=\left(\frac{r}{q}\right)^{3}\) defines the angle of polarization, namely, the Lode angle. When defined by the cosine function, the Lode angle ranges from 0 to 60\({}^{\circ}\);  
\(r=\left(\frac{9}{5}S\cdot S:S\right)^{\frac{1}{3}}\) defines the third deviatoric stress invariant;  
\(q=\sqrt{\frac{3}{2}(S:S)}\) defines the deviatoric stress;  
\(S\) defines the deviatoric stress, \(S=\sigma+pI\);  
\(I\) defines the identity matrix. The yield function and plastic potential function are derived using the implicit integration algorithm. We denote the first derivative of the yield function with respect to stress as vector \(a\) (Equation (11)). \[a=\frac{\partial F}{\partial\sigma}=C_{1}\frac{\partial p}{\partial\sigma}+C _{2}\frac{\partial q}{\partial\sigma}+C_{3}\frac{\partial\theta}{\partial\sigma} \tag{11}\]  
where  
\[C_{1}=\frac{\partial F}{\partial p},\ C_{2}=\frac{\partial F}{\partial q},\ C_{3}=\frac{\partial F}{\partial\theta}\]  
Specifically expressed as  
\[C_{1}=-\tan\varphi \tag{12}\]  
\[C_{2}=\frac{1}{\sqrt{3}\cos\varphi}\sin(\theta+\frac{\pi}{3})+\frac{1}{3}\cos (\theta+\frac{\pi}{3})\tan\varphi \tag{13}\]  
\[C_{3}=\left[\frac{1}{\sqrt{3}\cos\varphi}\cos(\theta+\frac{\pi}{3})-\frac{1}{3 }\sin(\theta+\frac{\pi}{3})\tan\varphi\right]\cdot q \tag{14}\]  
The derivative of the average principal stress \(p\) to stress is  
\[\frac{\partial p}{\partial\sigma}=-\frac{1}{3}[1,1,1,0,0,0]^{T} \tag{15}\]  
The derivative of the generalized shear stress \(q\) with respect to stress is as follows:  
\[\frac{\partial q}{\partial\sigma}=\frac{3}{2q}\big{[}\sigma_{x}-\sigma_{m}, \ \sigma_{y}-\sigma_{m},\ \sigma_{z}-\sigma_{m},\ 2\tau_{xy},\ 2\tau_{xz},\ 2\tau_{yz}\big{]}^{T} \tag{16}\]  
The derivative of Lode angle \(\theta\) to stress is expressed as  
\[\frac{\partial\theta}{\partial\sigma}=\frac{\partial\theta}{\partial q}\frac {\partial q}{\partial\sigma}+\frac{\partial\theta}{\partial\tau^{3}}\frac{ \partial\tau^{3}}{\partial\sigma} \tag{17}\]  
where the derivative of the third stress invariant to the stress multiplied by \(27/2\) is the derivative of \(r^{3}\) to the stress (Equation (17)). \[\frac{\partial r^{3}}{\partial\sigma}=\frac{27}{2}\left\{ \begin{array}{l}( \sigma_{y}-\sigma_{m})(\sigma_{z}-\sigma_{m})-\tau_{yz}^{2}\\ (\sigma_{z}-\sigma_{m})(\sigma_{x}-\sigma_{m})-\tau_{zz}^{2}\\ (\sigma_{x}-\sigma_{m})(\sigma_{y}-\sigma_{m})-\tau_{xy}^{2}\\ 2\left[\tau_{xz}\tau_{yz}-(\sigma_{z}-\sigma_{m})\tau_{xy}\right]\\ 2\left[\tau_{yz}\tau_{xy}-(\sigma_{y}-\sigma_{m})\tau_{xz}\right]\\ 2\left[\tau_{xy}\tau_{xz}-(\sigma_{x}-\sigma_{m})\tau_{yz}\right]\end{array}  \right.+\frac{3q^{2}}{2}\left\{ \begin{array}{l}1\\ 1\\ 0\\ 0\\ 0\end{array} \right\} \tag{18}\]The derivatives of Lode angle \(\theta\) with respect to mean principal stress \(p\) and generalized shear stress \(q\) are  
\[\frac{\partial\theta}{\partial q}=\frac{1}{q\tan 3\theta} \tag{11}\]  
\[\frac{\partial\theta}{\partial r^{3}}=\frac{1}{3q^{3}\sin 3\theta} \tag{12}\]  
The expression of vector \(a\) is  
\[a=\frac{\partial F}{\partial\sigma}=\left(\frac{1}{3}\tan\varphi+\frac{C_{3}}{ 2q\sin 3\theta}\right) \begin{bmatrix}1\\ 1\\ 0\\ 0\\ 0\end{bmatrix} +\left(\frac{3C_{2}}{2q}+\frac{3C_{3}}{2q^{2}\tan 3\theta}\right) \begin{bmatrix}\sigma_{x}-\sigma_{m}\\ \sigma_{y}-\sigma_{m}\\ \sigma_{z}-\sigma_{m}\\ 2\tau_{xy}\\ 2\tau_{xz}\\ 2\tau_{yz}\end{bmatrix} +\frac{9C_{3}}{2q^{3}\sin 3\theta}