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The vibrations induced by railway traffic can affect the stability of structures, buildings and buried structures. To evaluate this impact, this study was carried out considering the case of the Regional Express Train which will connect Dakar to Blaise Diagne International Airport. For that, the modeling software Plaxis dynamic [ 1], able to generate harmonic loads, is used and permitted to have a dynamic analysis and comparison between static and dynamic load for one passage of the train for 2.56 s. In the modeling, two behavior laws were used those of Mohr Coulomb for the layers of soil, embankments and the form layer, and then the linear elastic model for the rest of the elements. The results obtained showed extreme vertical displacements 40.18 mm for the building and when no load is applied on the track, there was 40.24 mm for a static load, and 40.17 mm for a dynamic load. Also, it was observed for the track a displacement of 33.73 mm for a static load and 19.83 mm for a dynamic load. However, further studies are necessary to take into account the permanent deformation after an accurate cycle of train passage in order to better evaluate the railway traffic impact.

Several authors have worked on the response of structures and soil subjected railway traffic solicitation in particular to consider it as dynamic and cyclic. The most studied models seem to be those using the finite element method as they make it possible to model complex geometries, the modeling of vibratory phenomena in the soil taking into account the problems of reflection on the boundaries of the mesh domain. In order to remedy this, authors have found alternatives either by using the absorbing boundaries or by coupling it to the methods of the border elements [

When a train moves along the track, each of its wheels in contact with the rail interacts with the track. The defects on the wheel and on the rail produce a force of dynamic interaction between the rail and the wheel, of frequency corresponding to the speed of passage over the size of the defect [

The actions acting on the structures can be classified in deterministic and random solicitations. Deterministic solicitations are periodic (harmonic or anharmonic), impulsive or maintained according to their form of variation over time. As for the case of railway stresses, they are modeled in the form of periodic harmonic solicitations characterized by several loading cycles [

y ( t ) = A sin ( ω 0 t + θ ) (1)

y ( t ) : simple harmonic function;

A: represents the amplitude. This is the maximum value of the harmonic function;

t: time variable (s);

θ: phase angle at the origin of the function (rad);

T: period;

ω: pulsation (rad/s).

The empirical laws of two-parameter cyclic behavior traditionally used by the engineer are insufficient to model the dynamic behavior of the soil in large deformations. If one tries to simulate the classical results in the laboratory, the Hardin-Drnevitch hysteretic cyclic law [_{o}, α), where G_{o} is the shear modulus at small deformations and α a non-linear parameter of the model, proves to be inadequate for medium and strong deformations. To overcome this limitation, an empirical law with three parameters (G_{o}, α, β) is proposed where β is the damping parameter. As in the initial law, the parameter G_{0 }characterizes the small deformations and the parameter a reports the secant modulus. The additional parameter β makes it possible to correct the Masing’s law [

The field of very small deformations (ε < 10^{−}^{5});

The field of small deformations (ε < 10^{−}^{4});

The field of mean deformations (ε < 10^{−}^{3});

The field of large deformations (10^{−}^{3} < ε).

In the literature, the structure of vehicles is often modeled by a complex multi-axle system of masses-springs-dampers. The calculation of lane structures can be simplified by considering the vehicle as a load or a group of loads [_{1}, x_{2}, x_{3}) and the movable reference (P, x, y, z), where P is the point of application of the load and x = x_{1} − ct, y = x_{2}, z = x_{3} (

Different models for railway structures given by the literature are presented below:

・ The rails: The simplest model (and also the most used) is Euler Bernouilli’s beam [

・ The soles: They are not always taken into account in the calculations. If they are to be taken into account, they can be considered as a shock absorber [

・ The sleepers: They are quite rigid compared to the other components. They can be a point mass in a 1D calculation or a rigid body of 3 or 6 depending on the 2D or 3D case. In the 3D case, the sleepers of 2 blocks are linked by a beam which represents the spacers [

・ Ballast: This is a very difficult point in the modeling of railways because of its discrete properties. For a simpler 1D model, it can be modeled by a continuous system of springs whose mass is uniform [

・ The platform and the ground: The 1D models consider the infrastructure as a system of springs, says the foundation of Winkler. In 2D or 3D calculations, they are often modeled by an infinite multilayer medium. The behavior of the soil can be linear or non-linear depending on the type of materials (Tresca, Coulomb, Drücker etc.) [

In the literature, the most widely used models for soil modeling are: the finite element method (FEM), the boundary element method and the coupling between the two finite element and boundary element methods [

The train has a speed of 160 km/h with a capacity of 500 people/ram. The axle load is 225 kN with a Track spacing of 1.435 m. The sandy context is composed

of 2 major layers: a layer of sand dune of 10 m thick resting on a large layer of silty clay or “Formation de l’hôpital”. The materials used for the different layers are composed of: a form layer of gravel lateritic soil, a Basalt layer of 0/31.5 and a Ballast layer of 31.5/50 (where d/D represents the granular class in terms of the lower (d) and upper (D) dimensions of the sieve expressed in millimeters.). _{x} and k_{y} are given by k_{x} = k_{y} = 1.157 × 10^{−13} m/s for clay and marl a nd k_{x} = k_{y} = 1 × 10^{−7}

Layer | Behavior Law | Parameters | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

E (kN/m²) | γ h (kN/m^{2}) | γ d kN/m^{2}) | c (kPa) | φ (°) | N [ | ξ [ | α [ | B [ | K x m/s | K y m/s | ||

Ballast | Linear Elastic | 2.105 | 18 | 17 | - | - | 0.1 | 0.01 | 0.25 | 0.0003979 | 10^{−13} | 10^{−13} |

Under Ballast | Linear Elastic | 3.105 | 25 | 24 | - | - | 0.2 | 0.01 | 0.25 | 0.0003979 | 10^{−9} | 10^{−9} |

Form layer | Linear Elastic | 1.5.105 | 20 | 19 | 1 | 32 | 0.2 | 0.01 | 0.25 | 0.0003979 | 10^{−9} | 10^{−9} |

Backfill | Mohr-Coulomb | 5.104 | 20 | 19 | 1 | 32 | 0.3 | 0.03 | 0.7539 | 0.001193 | 10^{−8} | 10^{−8} |

Sand dune | Mohr-Coulomb | 12000 | 17.6 | 15.6 | 2 | 32 | 0.333 | 0.03 | 0.754 | 0.001193 | 3.9 × 10^{−6} | 3.9 × 10^{−6} |

Silty clay | Mohr-Coulomb | 18270 | 17.8 | 15.8 | 8 | 22 | 0.3 | 0.03 | 0.754 | 0.001193 | 10^{−7} | 10^{−7} |

Structures | Parameters | |||
---|---|---|---|---|

EA (kN/m) | EI (kNm^{2}/m) | W (kN/m/m) | d_{eq} (m) | |

Rail | 1.44 × 10^{6} | 4930 | 0.55 | 0.202 |

Building | 5.106 | 9000 | 5 | 0.147 |

m/s for silty clay. For the dynamic parameters, the frequency is determined as follows [

f = v / L ; (2)

v: speed of the train (km/h),

L: distance between the sleepers (m)

The amplitude is calculated by the Nguyen method [

_{ A = P s t a t + α P s t a t }; (3)

P_{stat}_{:} static load of the wheel (P_{stat} = Q/2 avec Q the axle load)

α: representing coefficient of dynamic forces varying from (0.1 to 0.5)

The simulation was carried out considering 3 loading cases:

Case 1: Case where no load is applied. The movements correspond to a settlement of the building under its own weight;

Case 2: Case of a static load corresponding to a parked train;

Case 3: Case of a dynamic load. For the calculation time, the distance 113.7 m (half the length of the train Thalys) on the speed 160 km/h that gives us a passage time of 2.56 seconds.

The first simulation (Case 1) shows a vertical displacement u_{y} of 40.15 mm under building B, 40.16 mm under building A and 18 mm under the track (

In the second simulation (Case 2), the value of the displacement is 40.23 mm for building B and 40.24 mm for building A (

settlement of 33.73 mm is observed.

In the third simulation (Case 3), a displacement of 40.16 mm is noted for the building B and 40.17 mm for building B_{2} (

The maximal vertical accelerations give a value of 8.72 m/s^{2} (

The results showed that depending on the type of loading, the influence is very sensitive at the level of the track. The settlements are more important in the

Type of loading | Displacements (m) | ||
---|---|---|---|

Track | Building A | Building B | |

Without load | 18 × 10^{−3} | 40.16 × 10^{−3} | 40.15 × 10^{−3} |

Static load | 33.73 × 10^{−3} | 40.24 × 10^{−3} | 40.23 × 10^{−3} |

Dynamic load | 19.85 × 10^{−3} | 40.17 × 10-^{3} | 40.16 × 10^{−3} |

Impact | −13.88 × 10^{−3} | −0.07 × 10^{−3} | −0.07 × 10^{−3} |

track than for the surrounding building. A gap of 13.88 mm is observed between static load and dynamic load for one passage of the train (

Post-settlement of a railway structure is mainly caused by the self-weight of the embankment and train traffic loadings, and field measurements show that dynamic loading from train traffic has a greater contribution [

All our thanks give to SETEC for making available the geotechnical data of the project.

The authors declare no conflicts of interest regarding the publication of this paper.

Samb, F., Fanoukoe, F.A., Keno, A.J.N. and Diop, M. (2019) Railway Traffic Vibration Impact Analysis on Surrounding Buildings by FEM―Case Study: TER (Regional Express Train) Dakar―AIBD. Geomaterials, 9, 17-28. https://doi.org/10.4236/gm.2019.91002