## 4 Numerical Simulation  
### Numerical Model  
To comprehensively assess the bearing characteristics of GRPS embankment under static and dynamic loads, the nonlinear finite element software ABAQUS 2020 is employed to develop a numerical model as depicted in Figure 10. The finite element models for the RACFCTs consist of soil, pile, pile cap, embankment, gravel cushion, and geogrid. In the model, 8-node brick elements (C3D8) are utilized to simulate the soil, pile, pile cap, embankment, and gravel cushion, respectively. Since geogrids are mainly subjected to tensile forces with negligible flexural and compressive properties and their thickness dimensions are much smaller than their flat dimensions, geogrids are modeled by a three-dimensional 4-node membrane element (M3D4). Additionally, to simulate the actual condition of the GRPS embankment, the boundary of the bottom surface was supposed to be restricted in the X, Y, and Z directions. The side boundaries and symmetry boundaries were fixed in the X and Y directions, respectively. When calculating the response of the GRPS embankment under dynamic loading, the infinite element boundary is used to reduce the influence of reflected stress waves from the artificial boundary on the simulation results.  
Figure 10: Numerical model of the GRPS embankment.  
Figure 9: Tension of geogrid under dynamic load; (**a**) A-5. (**b**) A-4. (**c**) A-1. (**d**) B-1.  
The contact behavior of pile-soil, embankment-cushion, and cushion-soil contacts is simulated via interface elements, and the surface-to-surface contact algorithm for the interface is used. The geogrid is embedded in the gravel cushion using \"Embedded region\". General contact is set between the pile and the soil; the tangential behavior of the contact surface is hard contact; and normal behavior is defined as penalty contact. The friction coefficient is taken as the tangent of the friction angle within the soil. The material parameters of the pile, soil, embankment, cushion, and geogrid used in the analysis are given in Tables 2 and 3, all of which are based on tests. To simulate the loading of static loads, following the first step of in situ geostatic stress generation, a total of five levels of loads are applied step by step to the embankment. The finite element dynamic load is calculated using the same sinusoidal cyclic load as the test.  
In the pile-soil cooperative work of the GRPS embankment, the soil is more likely to be destroyed than the pile and the geogrid. Therefore, isotropic linear elastic material is used to calculate the pile and the geogrid, and it is assumed that the pile will not be damaged during the loading process. The interaction between pile and soil is simulated by using the contact surface, and the friction coefficient is assumed to be constant during the simulation. The embankment, cushion, and soil are modeled using a linearly elastic-perfectly plastic model with the Mohr-Coulomb failure criterion, assuming that all soils are homogeneous. The yield function of the model criterion can be expressed as:  
\[F=R_{mc}q-p\tan\varphi-c=0 \tag{1}\]  
\[R_{mc}=\frac{1}{\sqrt{3}\cos\varphi}\sin\Bigl{(}\theta+\frac{\pi}{3}\Bigr{)}+ \frac{1}{3}\cos\Bigl{(}\theta+\frac{\pi}{3}\Bigr{)}\tan\varphi \tag{2}\]  
where \(q\) is the Mises equivalent stress; \(p\) is the equivalent pressure stress; \(\varphi\) is the friction angle; \(c\) is the cohesion of the material; and \(\theta\) is the deviatoric polar angle.