## 2 Materials and Methods
### The Probabilistic Approach
#### 2.2.2 Dynamic Soil Characteristics and Nonlinear Soil Behavior Curves  
The variations in the dynamic soil characteristics can have a major effect on how the structure behaves under seismic loads. For example, if a section of the soil profile at a certain depth is softer, the following two phenomena can occur:  
* If the pile forces are mainly generated by the global behavior of the structure (such as rocking), the forces in the piles will increase, because the soil offers a weaker support. * If the pile forces are generated from local effects, such as local soil pressure or friction due to the seismic ground motion, the pile forces may be reduced, as the soil will apply a smaller force. Figure 11: Example of generated time histories using (**a**) Method 1 and (**b**) Method 2. The two main dynamic characteristics of the soil are the S-wave velocity (Vs) and damping (D). Due to their soil shear strain dependency, they exhibit negative correlation as follows: damping is large for soft soil and very small for stiff rocks. For the probabilistic approach, the values of Vs and D for each soil layer are a pair of statistically dependent random variables that share the dependency described above. The soil profile is usually divided in segments of soil or rock with similar properties. Furthermore, each of these segments will be divided into multiple layers, due to the  
Figure 12: Response spectra of generated time histories for the three main directions: (**a**) X; (**b**) Y; (**c**) Z.  
requirements of the site response analysis. The correlation length may be considered constant along such a segment. In ACS SASSI, the probabilistic Vs and D profiles can be generated using two methods [60]:  
* Method 1 uses a single stochastic field as follows: the generated profiles have large variations around the mean value. * Method 2 uses the following two stochastic fields: a large wavelength, which generates slow-amplitude variation, and a short wavelength with rapid-amplitude variation. To better understand the differences between the two models, Figure 13 presents an example of just two soil profiles generated using each of the methods described above. For both methods, the inverse variation of Vs and D is highlighted; for larger Vs values, the damping is small and vice versa. Except for extreme situations, which would have been highlighted by the geological and geotechnical studies, the variations in soil properties from Method 1 are too large and sharp, making them unlikely to happen in reality. On the other hand, Method 2 produces very realistic profiles, with variations similar to those found in usual geotechnical studies. For the present study, Method 2 with a lognormal distribution has been used. The soil profile has been divided in the following 3 segments:  
* 0-6 m: sand;
* 6-15 m: gravel;
* 15-30 m: gravel. The choice of segment length depends on the intervals where a relevant change in soil behavior actually happens:  
* The limit between the sand and gravel layers is at a depth of 6 m;
* The Seed-Idriss soil behavior curves (Figure 6) are also defined in intervals; the limit between the second and third interval is at a depth of 15 m, which falls inside the very thick gravel layer and divides it in two segments. The parameters used for the generation of the soil profiles (coefficients of variation and correlation lengths) are summarized in Table 2.  
Figure 13: Example of generated soil profiles using (**a**) Method 1 and (**b**) Method 2. The G-\(\gamma\) and D-\(\gamma\) linear-equivalent soil behavior curves are each generated using a one-dimensional random field, with large correlation lengths (slow variations) [55].