# Mechanism and Application of Soilbags Filled with Excavated Soil in Soft Soil Subgrade Treatment
## 4 FEM Simulation  
### FEM Model  
The settlement of the soilbag-treated subgrade was observed to be closely related to the consolidation of the soft soil. However, the consolidation during construction and the settlement distribution in the subgrade were not recorded in the field tests. Thus, an FEM numerical model was developed to estimate the consolidation performance of the soilbag-treated subgrade in the rural roadway project. Figure 11 showcases the mesh and boundary conditions for the soilbag-treated subgrade model. The FEM model covered a span of 30 m horizontally and a depth of 6.67 m vertically. The filling area of the soilbag-treated subgrade was 10 m in width and 1.67 m in height. The horizontal range of the FEM model was three times the width of the filling area, and the vertical range of the FEM model was four times the height of the filling area. The influence of the boundary conditions was relatively small. Thus, the horizontal displacements were restrained at both the left and right boundaries, while permitting vertical movements. Conversely, both the horizontal and vertical displacements were anchored at the bottom boundary. The top layer was permeable in its entirety, whereas the left, right, and bottom confines were impermeable. The modified Cam-Clay (MCC) model, which is a widely accepted elastoplastic model for describing the stress-strain behavior of soft soil, was adopted [34, 35]. For long-term  
Figure 11: Mesh and boundary conditions of soilbag-treated subgrade FEM model. Figure 10: Settlement of soilbag-treated subgrade and the original design of subgrade: (**a**) settlement of typical measuring points: (**b**) settlement along the road after 18 months. operation analysis, a coupled consolidation analysis based on [PERSON]'s theory was used in this numerical simulation. [PERSON]'s covering equation for two- or three-dimensional consolidation theory is given by:  
\[\frac{\partial u}{\partial t}=C_{v}\
abla^{2}u+\frac{\partial\sigma_{m}}{ \partial t} \tag{3}\]  
where \(u\) is the excess pore pressure, \(\sigma\) is the mean normal stress equal to \(\frac{1}{3}\big{(}\sigma_{xx}+\sigma_{yy}+\sigma_{zz}\), \(\frac{\partial u}{\partial t}\) is the variation of pore water pressure with time, \(\
abla^{2}\) is the Laplace operator, and \(C_{v}\) is the coefficient of consolidation. In the FEM consolidation analysis, the deformation and pore water pressure changed with time at each point of the subgrade model. By solving the equilibrium equation and continuity equation using the finite differences method, [PERSON]'s governing equation in the FEM simulation was obtained as:  
\[ \begin{bmatrix}[k_{s}]&[k_{c}]\\ [k_{c}]^{\mathsf{T}}&-\theta\Delta t[k_{h}]\end{bmatrix} \bigg{\{} \begin{aligned} \{\Delta\delta^{e}\}\\ \{\Delta H^{e}\}\end{aligned} \bigg{\}}=\bigg{\{} \begin{aligned} \{\Delta R^{e}\}\\ \{\Delta Q^{e}\}\end{aligned} \bigg{\}} \tag{4}\]  
where \(k_{s}\) is the stiffness matrix, \(k_{c}\) is the coupling matrix, \(k_{h}\) is the seepage matrix, \(\Delta\delta^{e}\) is the deformation of the nodes, \(\Delta H^{e}\) is the pore water pressure of the nodes, \(\Delta R^{e}\) is the equivalent node force vector of the external load, and \(\Delta Q^{e}\) is the equivalent node flow vector of the external load. The model parameters were deduced from a fusion of onsite geological experiments, survey data, and analogous studies, as delineated in Table 2. The polypropylene bags were modeled separately, with their permeability coefficient determined to be 7.1 \(\times\) 10\({}^{-3}\) cm/s. The simulation was designed to closely mimic the staged construction observed in the field for soilbag-treated subgrades, capturing key phases, such as the placement of soilbags, intermittent construction periods, and subsequent consolidation. Specifically, the simulation's initial phase consisted of six steps correlating to the layered construction of the large- and common-sized soilbags, a lime soil cushion, and a cement-gravel stabilized cushion.