## 4 Numerical Results
Figures 3 and 4 illustrate the time evolution of the acceleration and shear strain, respectively, at 3 different layers (1, 3, and 5), taking into account different methodologies. In Figure 3, the acceleration predictions from non-linear methods show similar behaviors, but the maximum acceleration predictions, especially at the surface, are significantly different. In this case, the proposed  
Figure 1: Top: Acceleration time history for Diamond Heights station during Loma Prieta earthquake (1989); bottom: Fourier transform. Figure 2: Time evolution of the displacement at the surface using the truncated Extended Harmonic-Modal Hybrid (XHMH) method. approach is closer to the EERA solution. Figure 4 the shear strain time history from nonlinear methods are reasonably close for most cases compared to EERA analysis, but the peak strain predictions are significantly different. As previously discussed the XMHM addresses the real nonlinear problem, while EERA performs on an equivalent linear behavior. The differences between both nonlinear solutions are due to the different soil properties considered and also to the different mesh resolution (being finer the one associated with the XMHM). The XMHM solution was validated by performing a standard time-integration of the considered nonlinear model. Figure 5 shows the modulus reduction, damping, and shear stress curves for the three different methodologies. It can be noticed that the results for the modulus reduction curve are similar. The damping and shear stress curves exhibit differences, particularly at high strains, depending on the considered model, which involve slightly different damping modeling. Figure 6 shows PGA, the maximum displacements at the top of each layer and maximum strain at the mid-depth of each layer for the three considered solution techniques. Certain differences can be observed for the PGA values between the DEEPSOIL nonlinear model and the XHMH model, however, the maximum displacements are nearly exact matches. For the maximum values of shear strain, the three approaches show variations. These variations may be due to the different damping models for each method or the selection of curve fit values. Figure 3: Acceleration time history of layer 1, layer 3, and layer 5 for the Equivalent Linear Response Analysis (EERA), DEEPSOIL, and the truncated XHMH. Finally, it can be observed in Figures 3-6, which despite the differences that the three methods exhibit, the global behavior remains similar. The computational methodology here proposed, which is the truncated XHMH technique, proceeds extremely fast in the nonlinear case, even when addressing a stratified soil with different material properties, enabling the response evaluation instantaneously (real-time). The difference between the methods is the way to address the nonlinear behaviors. The proposed approach does not pretend to discuss on the different models' accuracy. It simply aims at proving that general nonlinear soil models can be solved with the same computational performances than linear models. Figure 4: Shear strain time history of layer 1, layer 3, and layer 5 for the EERA, DEEPSOIL, and the truncated XHMH. Figure 5: (**a**) Normalized modulus reduction, (**b**) material damping, and (**c**) shear stress curves for the three approaches used in the numerical example for layer 1.