# Detection and Localization of Multiple Damages through Entropy in Information Theory
## 4 The Finite Element Models
### Boundary Conditions and Soil-Structure Interactions  
The boundary conditions for the steel pipeline can be divided between the internal and the external boundaries. On the inside, the pipeline was assumed to be empty and have no internal pressure due to the reasons discussed in Section 2.1.  
Concerning the soil-structure interactions, the following assumptions have been made:  
* The mass and stiffening effects of the surrounding soil volume are emulated by a set of spring elements;
* The SP boundaries are connected to these springs, and the edges of the springs opposite to those connected to the SP are fully restrained, i.e., they do not vibrate independently;
* The random noise forces transmitted by the surrounding soil are directly applied on the SP external edge;
* A Rayleigh damping model is assumed, with mass and stiffness multipliers equal to 2.954362 and 0.000304, respectively, which corresponds to 3% damping ratio on the first two modes of the SP.  
Indeed, the FE model aimed to reproduce the soil effects in a realistic yet simplified way without explicitly modeling the ground surrounding the pipeline. The rationale, as introduced before, was that the focus of this research was on the structural, and not geotechnical, vibrational response of the model. To this goal, a set of properly calibrated springs can effectively reproduce the static and dynamic soil-structure interactions (SSIs) of underground structures (see e.g., [44] for an example of a wind turbine mono-pile foundation). Specifically, 21,600 spring elements were utilized.  
The stiffness coefficients of the springs were obtained by static analyzing a dedicated plane strain model of the soil surrounding the SP. Specifically, the spring calibration procedure followed four modeling phases:  
1. The whole soil was modeled with 4-node, 2-DoFs-per-node plane elements (as can be seen in Figure 2a), considering a cylindric volume of radius \(R\) and centered on the pipeline centroid. To avoid edge effects, \(R\) was increased until \(R_{e}=10\) m, which was found to be large enough to eliminate any effect of the external boundaries at the soil-structure interface. This procedure followed the current standard practices in the geotechnical analysis of ground excavations. Values of \(R<R_{e}\) caused the strain field close to the pipe hole to be biased from boundary conditions. Clean sand was assumed for the soil, with a single layer for the whole volume and linear elastic behavior. The mechanical parameters, retrieved from [41], were: density 1850 kg/m\({}^{3}\), Young's modulus 24 MPa, and Poisson's ratio 0.20. No water table was considered.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Carbon Steel Type 235 (EN 1993-1-1 [42])**} \\ \hline Density & 7850 \((SP)+1060\) * & kg/m\({}^{3}\) \\ [PERSON]’s modulus & 210.7 & GPa \\ Poisson’s ratio & 0.30 & – \\ \hline \hline \end{tabular}  
* to account for the equivalent soil contribution.  
\end{table}
Table 1: Mechanical properties of the FE models.  
The external boundaries of the soil volume were defined with no external loads or constraints applied on the flat, perfectly horizontal ground surface. Fixed (zero displacements) constraints were set on the remaining sides. This model was used to define the stiffness of the soil affecting the response of the SP, which has been modeled with independent springs in the third and fourth FE models.
2. The second model is identical to the first, except for the presence of the SP (Figure 2b) (3) The third (yet plane) model added to the SP the presence of the springs. Together with the second FE model, this model was used to calibrate the mass participation of the soil to the dynamics of the soil-SP system.
4. Finally, the fourth reference, the FE model, is constituted by a 3D representation of the pipelines (20 m long), which was modeled with shell elements. A series of springs, calibrated in the previous steps, were used to simulate the contribution of the soil to the dynamics of the soil-SP system in the 3 spatial directions effortlessly.