## 4 MP Modeling by Experimental and Numerical Methods
The main goal of the present review on this topic is not necessarily the conclusions of these papers, but the ability of the numerical modeling through the selected constitutive model (CM) to reproduce the behavior resulted from the experimental tests, whether it was MPs only or a complex system with MPs, soil, and superstructure, but to also give the reader an idea of the factors (Table 2) that have already been investigated and calibrated with the most used MC criterion. [PERSON] et al. [13] performed a 3D analysis to study the seismic performance of inclined MPs by using a non-associated MC criterion; the study was not explicitly calibrated, but it showed an agreement with the normal expected behavior when using inclined piles. Further, it showed the importance of accounting for soil plasticity in the analysis, as the response between elastic vs. elastic-plastic analysis showed high differences. According to the authors, this is due to the allowance of energy dissipation by plastic deformation. In fact, considering plasticity in the analysis can reduce the resulting response in many cases, which is also an expected behavior. Neither the soil property nor the interface elements are clear in the study [13]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Factor** & **Importancy** & **Comments** \\ \hline \multirow{4}{*}{Grouting} & Grouting can modify the surrounding soil, which has a major influence on the kinematic and inertial interactions & Influence of gravitational and low pressure grouting (type A and B) have been studied with experimental and numerical methods [6; 12; 15]; More research on high pressure grouting needs to be carried out, some experimental data already exists; \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \cline{1-1}  & & \\ \hline \hline \end{tabular}
\begin{tabular}{c c} \hline \hline \end{tabular}
\end{table} 
Table 5: Factors influencing the seismic behavior—synthesis from literature. [PERSON] et al. [35] also performed a numerical study to show the effect of considering soil plasticity in the analysis, which can influence the seismic soil-MP-structure interaction, but again no explicit validations/calibrations were presented. The failure criterion used was non-associated MC; the analysis was performed on both harmonic and actual earthquake records and the results showed again that the behavior is strongly influenced by soil plasticity. MPs were modeled as a three-dimensional elastic beam element, damping was considered with Rayleigh method, and a viscous dashpot was used to couple the main grid to the free-field grid. This is a common practice in numerical modeling of piles and it was used here with MPs [35]. With harmonic excitation it was found that the elastic response is 100% higher than the elastic-plastic one. The authors of the paper argued that this is due to the dissipation of energy by plastic deformation and the effect on the stiffness (plasticity leads to stiffness reduction). Additionally, the inertial forces in MPs, when considering elastic-plastic behavior, showed a reduction in their values in comparison to the elastic one, which is due, according to the authors, to the energy transmission reduction related to the plasticity. With seismic excitation, the dynamic amplification factor was reduced when considering elastic-plastic analysis; furthermore, the dominant frequency was modified. These phenomena are similar to those found when studying other types of piles and show the importance of plasticity in the analysis [35]. [PERSON] et al. [11] performed a parametric study and a sensitivity analysis on the seismic performance of soil-MPs-superstructure. The authors considered in the study a nonassociated MC criterion and it was validated by a single-degree of freedom shaking table test. The soil considered in the study was sandy with zero cohesion. They used a contact element to model the interface between MPs and soil.