# Investigation of Earthquake-Induced Pipe Damage in Liquefiable Soils
## 3 PM4 Sand Model and Verification Analyses  
The Plaxis PM4 Sand (Version 3.3) plasticity model has been used for the numerical analysis of soil liquefaction. The model follows the basic framework of the strain-controlled, critical state-based, bounding surface plasticity model initially developed by [PERSON] and [PERSON] [46] and extended by [PERSON] and [PERSON] [47]. In the model, the difference between the current relative density \(\mathrm{D_{R}}\) and the critical state relative density \(\mathrm{D_{R,\,cs}}\) is expressed as the relative state parameter index \(\xi_{\mathrm{R}}\). It has been described in the guides (starting from Boulanger [48], with the latest version being Version 3.3 in 2023) and in  
Figure 2: (**a**) Iskenderun city center (Atatürk Boulevard in the Cay Neighborhood) and Sarseki District/Google Earth Image; (**b**) geomorphological section view of the Iskenderun region [45] (revised 2023). associated publications [49]. The model has been encoded as a dynamic link library (DLL) for use with the dynamic module of PLAXIS 2D. The PM4 Sand model is a constitutive model developed to account for excessive pore water pressure, liquefaction, and post-liquefaction behavior observed during earthquakes. This model is based on the critical state concept and is formulated using the bounding surface plasticity theory. The constitutive model is stress ratio-controlled, critical state compatible, and operates based on the bounding surface plasticity theory. In this model, the behavior of the soil is represented by dilation and critical state surfaces and bounded by the bounding surface plasticity to simulate the response of soil under seismic loading and confinement pressure within a specific density range. The formulation of the model can simulate the volumetric response of soil to repeated loading and directly simulate excessive pore water pressures [49]. This constitutive model includes the following: revising the fabric formation/destruction to depend on plastic shear rather than plastic volumetric strains; adding fabric history and cumulative fabric formation terms; modifying the plastic modulus relationship and making it dependent on fabric; modifying the dilatancy relationships to include dependence on fabric and fabric history and to provide more distinct control of volumetric contraction versus expansion behavior; providing a constraint on the dilatancy during volumetric expansion so that it is consistent with [PERSON]'s (1986) dilatancy relationship [50]; modifying the elastic modulus relationship to include dependence on stress ratio and fabric history; modifying the logic for tracking previous initial back-stress ratios (i.e., loading history effect); recasting the critical state framework to be in terms of a relative state parameter index; simplifying the formulation by restraining it to plane strain without Lode angle dependency for the bounding and dilatancy surfaces; incorporating a methodology for improved modeling of post-liquefaction reconsolidation strains; and providing default values for all but three primary input parameters [49]. The initial relative density value has been estimated based on the SPT N value using the correlation suggested by [PERSON] and [PERSON] [51]:  
\[D_{R}=\sqrt{\frac{\left(N_{1}\right)_{60}}{C_{d}}} \tag{1}\]  
The fundamental variable controlling the small-strain shear modulus is \(G_{\text{max}}\). \(G_{\text{max}}\) is selected to match the estimated or measured shear wave velocities (\(G_{max}=\rho Vs^{2}\)). The value is calculated based on the modified correlation between SPT (N\({}_{1}\))\({}_{60}\) according to Equation (2). \[G_{0}=167\,\sqrt{\left(N_{1}\right)_{60}+2.5} \tag{2}\]  
The third fundamental input parameter is the constant h\({}_{\text{po}}\) used during calibration for a specific relative density value. The contraction rate parameter h\({}_{\text{po}}\), allows for calibration to specific values of the cyclic resistance ratio (CRR). This parameter should be calibrated last, after assigning the values of the other parameters.