**1. Introduction**

The problem of locally variable stiffness of the railway surface within the transition zones of an engineering facility has been identified, which in turn may lead to accelerated degradation of the structure and the need to incur increased expenditures on maintaining the infrastructure in proper condition.

The aim of the research and computational work was to analyze the dynamic impact of the rail vehicle on various solutions of the railway surface structure, with particular emphasis on the phenomenon of the threshold effect within the transition zones of the engineering facility.

The basic task of the surface is to enable safe and stable driving of a rail vehicle on a specific trajectory and to take over loads from the wheels of a rail vehicle and transfer them to the subtrack. Two basic types of surface can be distinguished: (1) classic surface (ballast) and (2) unconventional surface (ballastless).

In ballastless surfaces, rubble has been replaced with layers of materials with different modulus of elasticity. They are arranged in such a way that materials with lower modules are built into the lower layers of the structure and the higher layers have the higher modulus

**Citation:** Idczak, W.; Lewandrowski, T.; Pokropski, D.; Rudnicki, T.; Trzmiel, J. Dynamic Impact of a Rail Vehicle on a Rail Infrastructure with Particular Focus on the Phenomenon of Threshold Effect. *Energies* **2022**, *15*, 2119. https://doi.org/10.3390/ en15062119

Academic Editors: Larysa Neduzha and Jan Kalivoda

Received: 15 February 2022 Accepted: 9 March 2022 Published: 14 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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of elasticity. In this way, each subsequent layer of the structure (sub-track, frost-resistant layer, asphalt-stabilized layer/hydraulically stabilized layer, concrete support layer, primer, rail) has increasing rigidity. The thicknesses and materials of individual layers should be selected so that the structure works in the field of elastic deformation [1]. Constructions of ballastless surfaces can be divided as follows: (1) surfaces in which the supporting layer consists of prefabricated slabs (IPA, Bogl, VA Shinkansen, OBB-PORR), (2) surfaces in which the supporting layer is a concrete slab (Rheda, EBS System) or bituminous layer (Getrac) laid directly on the construction site and (3) surfaces in which the rail is located in specially prepared rail channels (ERS, Infudo system, BBEST system) [2–7].

Due to a number of advantages of ballastless surfaces (including lower thickness and lower weight), they are often used in engineering structures such as bridges, viaducts, as well as in tunnels. If there is a ballast surface on the trail and on the engineering object a ballastless surface, one may observe the so-called threshold effect on the access to the facility and behind it.

The phenomenon of the threshold effect is due to the different parameters of adjacent different surfaces. This is especially noticeable in the vicinity of engineering facilities. In places of changing the type of surface, in the zones in front of and behind the object, vertical irregularities of rails (basins) are formed, which increase during operation and cause further deformations. As a result of dynamic loads, the track grate often rises, the track twist increases, as well as uneven wear of the rails and damage to the fasteners on both types of surfaces. Gaps may form under the sleepers, which threatens the stability of the structure. The threshold effect has a negative impact not only on the railway surface, but also on the object that is exposed to excessive loads and vibrations.

The causes of the threshold effect occurrence can be divided into primary (mechanical) and secondary (geometric). Among the primary causes, the following stand out: a change in the elasticity of the ground substrate, a change in bending stiffness of the supporting system, a change in the mass of the surface and a change in the surface damping value. The secondary causes include: the constantly occurring primary effect, the settling of a backstrip and a subtrack as a result of dynamic loads and vibrations and the unevenness and geometric defects of the rails [8].

The threshold effect causes an increase in the wheel-rail interactions and overloads of the bottom and subtrack. In the zones in front of and behind the object, vertical irregularities of rails (basins) are formed, which increase during operation and cause further deformations. As a result of dynamic loads, the track grate often rises, the track twist increases, as well as uneven wear of the rails and damage to fasteners on both types of surfaces. Gaps may form under the sleepers, which threatens the stability of the structure.

A significant impact on the threshold effect is the phenomenon of different effective stiffness of rail bending in the case of ballast and ballastless surfaces. Fastened with fastenings, the rail is subject to cyclic pressing and lifting. In the case of ballastless surfaces, the force needed to raise the rail is much greater, thanks to which there is no local loss of mutual contact between individual elements of the surface. The essence of greater effective bending rigidity is particularly visible: (1) in the case of the rail systems in the sheath, where the rail is embedded in an elastic mass that limits its freedom of bending and (2) in the case of direct attachment of the rail to the support plate or fastening the rail to bridges [9].

The threshold effect has a negative impact not only on the railway surface, but on the object that is exposed to excessive loads and vibrations, as well [8–10]. It is assumed that the difference in track stiffness on and off the object should not be more than 30%. The transition from a ballast to a ballastless surface is shown in Figure 1.

In the course of the topic implementation, for the purpose of performing this analysis of the dynamic impact of a vehicle on the railway surface, a computational model was created, allowing determination of the impact of differentiated rail support on the dynamic response of the entire structure. The starting point for the considerations was the Bernoulli-Euler beam, located on the elastic Winkler substrate. The dynamic load, caused by the passage of a multiaxle rail vehicle and the different parameters of different types of surfaces, were

taken into account. As a consequence, a fourth-order differential equation was obtained. It was solved by the finite differences method. A script in MATLAB was developed for a numerical solution of the problem. At the same time, it should be emphasized that development of an algorithm using the finite differences method does not involve the need to use complicated and expensive computer software.

**Figure 1.** Transition from a ballast to a ballastless rail surface as a cause of the threshold effect.

In order to verify and validate the created algorithm, in situ studies of vertical displacement of a dynamically loaded railway rail were carried out. The research was carried out using laser scanning technology.

#### **2. Materials and Methods**

## *2.1. Materials*

The railway surface is affected by technical and operational factors. Among the technical factors, one should mention such parameters of the structure itself as: the type of rail steel used and its longitudinal elastic modulus; the type of rail and the geometric moment of inertia of its cross-section and the type of surface; vibration-damping coefficient; type of sleepers, their spacing; the subtrack structure, and the resulting modulus of elasticity of the substrate. Operational factors include the type of rolling stock and its chassis layout, the wheelbase of the vehicle and the load on its single axle, as well as the speed of passage on the surface.

In structural calculations of the railway surface, an important parameter is the longitudinal elastic modulus (or Young's modulus). For the rail steel, its value is 210 GPa [11].

The study assumes a geometric moment of inertia of the cross-section of the 60E1 rail equal to 3038.3 cm<sup>4</sup> with a mass of 60.21 kg/m.

In the literature of the subject for calculations, a model of the railway surface is adopted, in which the rail is based on the elastic Winkler substrate characterized by the modulus of elasticity of the substrate. The validity of this assumption has been confirmed by previous research and analytical work—among others, in paper [12]—the sufficiency of the elastic substrate model for the analysis of such issues was confirmed. The value of the elastic modulus of the substrate is influenced by the materials from which the subsequent layers of the surface were made, and their thickness. These are a rail washer, sleeper, ballast, subtrack, a concrete or asphalt slab (instead of a ballast) in the case of unconventional surfaces, and, in the case of a surface on an engineering object, the structure of the object

itself (instead of a subtrack). The value of the elastic modulus of the entire surface "k" can be calculated from the following relationship [13]:

$$\frac{1}{\mathbf{k}} = \sum\_{\mathbf{i}} \frac{1}{\mathbf{k}\_{\mathbf{i}}} \,' \,' \tag{1}$$

where: k—modulus of elasticity of the entire surface, ki—modulus of elasticity of layer "i".

The elastic modulus of the rail washer is at the level of (90–100) MPa [14], while the value of the modulus of elasticity of the prestressed concrete primer is at the level of 31 GPa and the value of the wooden primer is at the level of (9.4–10) GPa [15]. For the concrete supporting layer, it is about 34 GPa; for the asphalt-stabilized layer, 5 GPa; and for the hydraulically stabilized layer, 10 GPa [1]. For the ballast layer, the modulus of elasticity is (250–300) MPa [15], and for the subtrack (40–120) MPa [16,17]. For a reinforced concrete engineering object, the modulus of elasticity is at the level of 28.5 GPa [17]. Taking into account the above values and relation (1), as well as using the data provided in [18,19], elastic moduli for various structures of the railway surface were determined. The results are presented in Table 1.

**Table 1.** Modulus of elasticity for different types of railway surface.


Precisely determining the value of the elastic modulus of the substrate is difficult. Its size can be influenced by many factors, such as: ambient temperature, soil humidity, infrastructure maintenance status, surface age, or the current transferred load [19].

With the passage of time after applying the load, the amplitude of the system vibrations decreases its value. This is due to the phenomenon of vibration damping. The vibrationdamping force is an action inside the structure that opposes the load. The following are distinguished in the constructions: (1) structural damping and (2) material damping. Structural damping is caused by the connection and cooperation of individual elements of the structure. Material attenuation is caused by the structure of the material and internal friction [20]. The value of the damping force C is described by the relation:

$$\mathbf{C} = \mathbf{c} \cdot \frac{\mathbf{dw}}{\mathbf{dt}}\,,\tag{2}$$

where: C—damping force, c—vibration-damping coefficient, dw/dt—change of vertical displacement of the rail as a function of time.

Based on the literature for the ballast railway surface, a vibration-damping coefficient of 22.6 MNs/m<sup>2</sup> [17] was assumed. The analysis of the literature shows that ballastless surfaces are characterized by worse damping properties due to greater rigidity. For the purposes of calculations, a vibration-damping coefficient was assumed for this type of construction with a value 15% lower than for the classic surfaces.

The chassis system determines the way in which the dynamic loads generated by the rail vehicle are transmitted to the surface. The load is transmitted pointwise, in the places of contact of the wheel with the rail. The static scheme of this system is a series of concentrated forces, applied to the rail in the spacing defined by the chassis design. The chassis of locomotives, as a rule, consists of two trolleys in a system of two or three axles each. The wheelbase in the trolley is (2.60–4.15) m. The wheelbase of the wagon bogie is equal from 1.5 m to even 8.0 m.

The axle load of a rail vehicle must not exceed the limit values specified by the Railway Infrastructure Manager. Depending on the type and condition of the surface, these values are determined individually for each section of the railway line. In Poland, on railway lines managed by Polskie Linie Kolejowe, the axle load of a rail vehicle may not exceed 221 kN/axle [21].

Currently, the world record for instantaneous speed developed by a conventional rail vehicle is 574.8 km/h. It was established by the French TGV V150 train on 3 April 2007 on the TGV Est line between Strasbourg and Paris [22]. However, commercial travel speeds are much lower. For shunting driving, this can be a speed of even less than 10 km/h, and for high-speed passenger trains (200–300) km/h. The higher the speed of rail vehicles, the higher the requirements for proper diagnostics and maintenance of the surface are greater and more restrictive. Driving at a higher speed causes greater dynamic effects on the surface and subtrack. This results in accelerated wear, to which rails, fastenings and bottom rails are particularly exposed. In rails, cracks, wavy wear, and contact-fatigue damage are more common, in the attachments there is a lower clamping force, and in the ballast there are also crushed cavities and its crushing. The speed of travel should be predicted at the planning and design stage, so as to counteract the abovementioned phenomena already at these stages [23].
