# Protection of Buried Pipelines from High-Speed Train Vibrations with Expanded Polystyrene Geofoam
## 2 Materials and Methods
### Description of the HST Moving Loads and Buried Pipeline Numerical Model  
In the sequence, the dynamic response of a steel pipeline located under an HSR line has been thoroughly investigated numerically by properly modifying the validated computational approach in order to implement the HST passing loads [26]. For this purpose, an advanced 3D numerical model consisting of finite elements and infinite elements was developed in order to simulate the problem at hand. In addition, the geometry and loads of a popular HST in Europe, i.e., Thalys train, were used. The steel pipeline was embedded to cross vertically at a shallow depth under the HSR track, as depicted in the sketch of Figure 3. Figure 4 presents the finite/infinite element model that has been developed. Due to the symmetry of the examined track along its longitude axis, only a part of the domain has been analyzed, as shown in Figure 4. The passage of the Thalys HST was implemented by imposing on the railway track multiple moving loads. For this purpose, a user-developed Vdload subroutine was used to input the multiple HST loads into the ABAQUS numerical model. A typical at-grade European HSR configuration has been used for the conducted simulations [5]. This standard three-layered subgrade and foundation system consists  
Figure 4: Numerical model: finite elements at the central part and surrounding infinite elements. Figure 3: Sketch showing the track components, the soil, and the steel pipe. of ballast, sub-ballast, and subgrade components based on the surrounding soil. Table 1 presents the mechanical properties of the track components. The mechanical properties of the single-layered soft clay subsoil are as follows: unit weight \(\gamma\) = 18 kN/m\({}^{3}\), [PERSON]'s modulus \(E\) = 40 MPa, Poisson's ratio \(\
u\) = 0.3, friction angle \(\varphi\) = 30\({}^{\circ}\), and cohesion \(c\) = 10 kPa. The surrounding soil and all track components, except for the rail, have been simulated using 8-noded solid elements. In order to ensure the accuracy of the numerical simulations, the size of the finite elements was chosen to be equal to 20 cm. On the other hand, linear beam elements with 10 cm length were used for the simulation of the rail. The central part of the subsoil model has a total length equal to 50 m and is discretized via finite elements. Moreover, the subsoil layer has a depth of 30 m and a width of 50 m, as shown in Figure 4, and it is surrounded by infinite elements, aiming to prevent wave reflections along the edges of the FE model. HST moving loads quasi-static simulation was performed as time-depending stationary forces at the rail nodes. Each axle load (\(f_{i}\)) is constant, and its position was calculated at each time step (\(t\)) via its initial position (\(x_{0}\)) and the HST passing velocity (\(V_{O}\)). In the present investigation, three passing velocities are examined: quite low (240 km/h), typical operational (300 km/h), and high (360 km/h) velocities. At any time step in which the position of any axle load coincides with the position of a rail node, the whole axle force is considered to act on this node. On the other hand, at time steps in which the position of the moving load is acting on the span between two rail nodes, the moving load is divided into two parts acting on the two nodes of this rail span. According to this process, a user-developed subroutine was created in Fortran77 in order to incorporate train moving loads into the FE model. As previously mentioned, a commonly used train in central Europe, the Thalys HST, has been modeled in the present investigation. The total length of the Thalys HST is close to 200 m and consists of 26 axles and 10 bogies. Thalys bogies are categorized into three types: Y230A locomotive bogies and Y237A and Y237B passenger bogies. Figure 5 illustrates the geometry of the Thalys HST and its axle loads, with data taken from [PERSON] et al. (2017).