## 4 Modeling Wind Turbine Foundations  
The FE models established herein simulate the soil-structure interaction, including the steel WT tower, concrete foundation, and soil. Three wind turbine models with rated powers of 5 MW, 8 MW, and 10 MW are modeled, with details presented in [3]. It is assumed that the wind turbine foundation is constructed in a clay layer with the same properties as the San Francisco Bay mud for which the UMAT parameters are calibrated in Section 3. The clay layer is 100 m thick and is underlain by rigid bedrock. A circular shallow footing with a diameter of 20 m supports the wind turbine, with its base at a depth of 5 m from ground surface. The thickness of the footing increases linearly from 0.7 m at the edge to 3.2 m, where it is attached to a pedestal with 10 m diameter. WT loads are calculated from aerodynamic analyses using the FAST software [57] for two cases: operational conditions and shut-down due to the occurrence of an earthquake (EQ). The aerodynamic thrust forces on the rotors of the WTs are plotted in Figure 4. Three components of the Val des Bois EQ at Bancroft station (23 June 2010) and Saguenay EQ at Newcomb station (25 November 1988) are matched to the seismic spectrum for bedrock at Alexandria, Ontario, with a 2% probability of exceedance in 50 years, based on [58]. Time-history plots and the response spectra of the matched records are shown in Figures 5-7. Figure 8 shows the dynamic  
Figure 3: Comparison of settlement–rotation curves of the experiment with FE analysis; (**a**) with hard contact, (**b**) cohesive contact. Figure 9 displays the FE model of the WT tower, foundation, and soil. Table 4 shows the tower dimensions and top masses in the FE models. A close-up view of half of the model near the foundation is shown in Figure 10. It is shown in Figures 9 and 10 that the sizes of the soil elements are smaller near the foundation for better accuracy, and they become larger as they get further away, for computational efficiency. Sizes of the elements are determined by several trial analyses with different element sizes, so that the further refinement of elements does not significantly affect the response quantities of interest. The part of the soil above the foundation slab is meshed by quadratic wedge elements, to avoid bad-shaped cube elements that would have small corner angles and would cause convergence issues. The concrete foundation is modeled by eight-node linear elements with reduced integration (C3D8R) and the steel tower is modeled with four-node shell elements with reduced integration (S4R), with the inertial effects of the nacelle and rotor modeled as a concentrated mass located at the center of gravity of the rotor-nacelle assembly. The soil domain is discretized with linear hexahedral elements (C3D8), quadratic hexahedral elements (C3D20), and quadratic wedge elements (C3D15). The quadratic elements are used in the regions in the vicinity of the foundation, to prevent the distortion of elements as a result of contact between the foundation and soil. The material properties for concrete  
Figure 8: Aerodynamic loads and EQ components applied in the analyses. Figure 7: Seismic spectrum of Alexandria, Ontario, and response spectra of the matched EQ records. and steel are summarized in Table 5. The soil is assumed to be San Francisco Bay mud, and its behavior is simulated with the UMAT model described in Section 2, utilizing the parameters given in Table 3. Interaction between the foundation and soil is modeled with the cohesive contact, as explained in Section 3. Side boundaries of the soil are modeled with the free-field type of boundaries similar to the work of [59]. For analyses under operational loads, the bottom boundary of the soil (i.e., the bedrock) is fixed. On the other hand, for analyses of shut-down with seismic loads, the acceleration time histories shown in Figures 5 and 6 are applied to the bottom boundary of the soil. For each EQ record, the three components (east-west, north-south, and vertical) are applied simultaneously.