# Fragility Assessment of Traffic Embarkments Exposed to Earthquake-Induced Liquefaction
## 3 Model Description
### 2D Numerical Model  
The numerical simulations in this study were performed with FLAC 2D software package in conjunction with the advanced material model PM4 Sand, which was developed for simulating the progress of earthquake liquefaction-induced deformations. The model can be calibrated with elementary or more advanced laboratory and in-situ tests with a reasonable amount of engineering effort and has been successfully validated in the literature [27; 28; 29]. The advantage of the material model was that it supports the simultaneous calculation of pore water pressure build-up due to seismic excitation and the corresponding soil densification and its dissipation [27]. \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
**No. of Model Variations** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** & **15** & **16** & **17** \\ \hline Embankment height—H [m] & 4 & & & 2 & & & 4 & & & 6 & & 8 \\ \hline Crest width—B [m] & 6 & 12 & & & & & & 24 & & & & & \\ \hline Soil profile ID & S1 & S1 & S2 & S3 & S4 & S1 & S2 & S3 & S4 & S1 & S2 & S3 & S4 & S1 & S3 & S4 \\ \hline \end{tabular}
\end{table} 
Table 2: Geometry variations. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & \multirow{2}{*}{\(\rho_{dys}\) [kg/m\({}^{3}\)]} & \multirow{2}{*}{\(k\) [m/s]} & \multicolumn{6}{c}{**Mohr-Coulomb**} & \multicolumn{6}{c}{**PM4 Sand**} \\ \cline{3-10}  & & & **K [MPa]** & & **\(G\) [MPa]** & & \(c_{\mathbf{e}}\) [kPa]** & \(\varphi^{\prime}\) [°] & \(c^{\prime}\) [°] & **\(\mathbf{KPa}\)** & \(D_{r}\) [\(l\)]** & \(G_{0}\) [\(l\)]** & \(h_{p\varphi}\) [\(l\)]** \\ \hline Crust layer & 1784 & 8\(\cdot 10^{-8}\) & 64 & 30 & 80 & & & & & & \\ \hline Liquefiable layer—medium dense & 1486 & 1.6\(\cdot 10^{-5}\) & / & / & / & / & / & 0.60 & 760 & 0.55 \\ \hline Liquefiable layer—loose & 1486 & 1.6\(\cdot 10^{-5}\) & / & / & / & / & / & 0.35 & 476 & 0.5 \\ \hline Base layer & 1436 & 1\(\cdot 10^{-9}\) & 227 & 105 & 150 & / & / & / & / & / \\ \hline Embankment & 1800 & 1.18\(\cdot 10^{-5}\) & 83.3 & 38.5 & / & 35 & 5 & / & / & / \\ \hline \end{tabular}
\begin{tabular}{c} Notations: \(\rho_{dys}\)—dry density, \(k\)—soil permeability, \(K\)—bulk modulus, \(G\)—shear modulus, \(c_{\mathbf{c}}\)—undrained shear strength, \(\varphi^{\prime}\)—friction angle, \(c^{\prime}\)—chesion, \(D_{r}\)—relative density, \(G_{0}\)—shear modulus coefficient, \(h_{p\varphi}\)—contraction rate parameter.  
\end{table} 
Table 3: Material properties. The FLAC 2D software package was used for the numerical simulations following a successful validation of the modelling technique being demonstrated in the case of liquefaction-induced deformations of the Naruse River levee in Japan [30]. When comparing the results of the numerical simulation to field measurements from on and under the Naruse embankment, the magnitude of the final displacements and the failure mechanism itself, as well as the increase in excess pore water pressure through the time of dynamic loading in a liquefiable layer, were satisfactorily captured. Initially, the ground stratigraphy, with a groundwater level and hydrostatic pore water pressure distribution, soil parameters and boundary conditions for static calculation, were applied successively in the model. Vertical displacements at the sides were allowed, while movements at the base were restricted in both the vertical and horizontal directions for this modelling phase. In addition, water drainage was assumed only through the model surface. In following phases, the embankment was constructed on the free field ground, the material model was modified in the liquefiable layer from the MC to the advanced PM4 Sand (the MC material model remained assigned to the other, non-liquefiable layers), and the boundary conditions were transformed, consistent with the dynamic analysis requirements. A compliant base was adopted at the bottom of the model, and free field boundary conditions were adopted at the sides.