Investigation of the Strength and Dynamic Load on a Wagon Covered with Tarpaulin for 1520 mm Gauge Lines

Abstract

Higher efficiency of rail transportation at the present stage of development of the transport industry necessitates the creation and introduction of rail vehicles with improved technical and economic characteristics among which is reduced tare weight. The issue of reducing the tare weight of wagons is quite urgent. It deals not only with the sprung mass of the wagon but also with the load on the rail track, which is under the influence of constant cyclic loads. Therefore, the present study deals with the development of a wagon covered with tarpaulin for carrying goods requiring protection against the environment. The loads inherent for operation on 1520 mm gauge lines are considered. The covered wagon mod. 11-217 is chosen as a prototype. The profiles of the covered wagon frame components are selected according to the moment of resistance of their cross-sections. It is found that the proposed design has a 16% lower tare weight than that of the prototype. The results of the strength calculation for the wagon under the main design operating modes have proved the feasibility of its structural design. The motion of the covered wagon over a track irregularity has been assessed as ‘excellent’. The results of the study will contribute to the creation of recommendations for the development of modern structures of covered wagons as well as improve the efficiency of railway transportation.

1. Introduction

The efficiency of rail freight transport directly depends on the technical level of rolling stock. One of its most important segments is freight wagons, which directly provide the transportation process of goods [1,2]. Currently, the wagon fleet includes a very large number of wagon designs: open wagons, covered wagons, flat wagons, hopper wagons, tank wagons, etc. This allows for the transportation of various types of cargo by rail [3]. The goods that need protection against the environment are transported in wagons equipped with a roof (covered wagons).

Most covered wagons are used for transporting containers and piece goods, which are secured in the body with special lashing devices.

The better technical and economic indicators of modern covered wagons are achieved by reducing their tare weight and increasing the payload capacity. One such solution can be the use of tarpaulin. Wagons covered with tarpaulin have been successfully operated on 1435 mm gauge lines for a long time which indicates the feasibility of such a design [4]. However, for 1520 mm gauge lines, such wagons are not widespread, since the main trend is a universal wagon. At present, a high demand for the transportation of piece goods including those in international traffic, demonstrates that research in this area is rather urgent.

Studies were constantly carried out to improve the operational efficiency of wagons, including covered wagons. Thus, in [5], the authors analysed the existing structures of covered wagons and identified the main areas for their improvement. One of the key research issues was the facilitation of loading/unloading operations. The authors proposed a peculiar covered wagon design with increased loading space. The work highlighted the features of determining the stress state of this wagon and presented the results of the appropriate calculation. However, the reduction in the tare weight was not provided, notwithstanding that it was one of the most promising areas for improving the design.

The analysis of the load on the body of a special covered wagon for carrying goods requiring special climatic conditions is presented in [6]. The authors identified the influence of refrigeration equipment on the frequencies and shapes of the wagon body oscillations. However, solutions to improve the wagon design are not been presented.

Measures to improve the body of a specialized covered wagon were described in [7]. The points of load applications were determined using a 3D spatial model. The wagon was tested on a special stand and the results demonstrated that the strength of the proposed body structure was provided. However, the issue of increasing the technical and economic indicators of the wagon was not considered.

Publication [8] gives an analysis of the fatigue strength of the body of a covered wagon for special cargo. The authors used a rigid flexibility model and conducted the corresponding simulation. The results were confirmed by field experiments on a vibration stand. The conclusions and the prospects for further research were presented. However, the authors did not propose solutions aimed at improving the technical and economic indicators of the wagon.

The issues of modernization of wagons to increase the demand for them were discussed in [9]. The main damages to the wagons in operation were identified. The analysis of statistical data was presented; it included the correlation between failures and their significance, as well as the impact of the modernization of wagons on these failures. However, the authors did not propose solutions for wagon modernization.

Of special interest is work [10], which describes the rational distribution of packaged goods in covered wagons. As noted, it significantly affects the efficiency of the transportation process. A mathematical tool was proposed that allowed optimizing the placement of goods in the wagon. However, the author did not consider the issue of improving the wagons to increase the efficiency of the transportation process.

The peculiarities of improving the covered wagon body by using a new design of side doors are highlighted in [11]. The authors believed that it could contribute to improving the efficiency of loading/unloading operations. Moreover, it could help to reduce the time of the loading/unloading of the wagon. However, this solution does not contribute to the improvement of the technical performance of the wagon.

Study [12] proposed an improved sectional covered wagon. The sectional design allowed simultaneous transportation of various types of cargo. The publication highlights the features of the strength calculation of this wagon under the main operating loads. It should also be noted that the operation of such a wagon has certain disadvantages, for example, when transporting different cargo, the wheelset axles of bogies must be uniformly loaded. In addition, such a wagon design has an increased tare weight, which does not contribute to the improvement of its technical and economic indicators.

Publication [13] proposes the covered wagon design consisting of a body, mounted on the running gears, with sliding doors, operated by the lower mechanism, and a roof. The roof was fixedly connected to the upper rail of the wagon and was made as a frame with evenly spaced transverse arches. At the top, the wagon was covered with corrugated sheets with solid corrugations along the wagon. The upper guides of the doors were fixedly connected to the upper body rail. The roof was made of at least two parts connected to each other. Its frame contained two longitudinal beams, with which the ends of the transverse arches were connected, and two cross-bearers. These beams were installed at the ends of the frame and were combined with longitudinal beams. Each end wall of the roof was provided with a ventilation device. However, the technical result of this utility model was to reduce the loading/unloading time and improve the conditions of cargo transportation. That is, this improvement did not contribute to reducing the tare weight of the wagon.

The wagon design proposed in [14] had a similar drawback. The covered wagon contained the body that was mounted on bogies and consisted of a centre sill, side and end walls, a flooring, a roof with loading hatches, and side wall doors. The floor of the wagon on both sides of the centre sill had openings equipped with grating installed at the floor level and equipped with unloading devices with opening and closing mechanisms.

Publication [15] proposes the covered wagon design with a frame, installed on the running gear, with anti-slip flooring, a body with sliding doors and a mechanism to move the doors. The roof with ventilation devices at its ends was made as a frame sheathed externally with rolled sheets and formed by bows. To improve the maintainability and provide the ease of wagon maintenance, the sidewalls, end and inclined parts of the roof were sheathed externally with smooth sheet metal. That is, the technical result of this design was to improve the repairability and wagon maintenance.

An ingenious covered wagon design is proposed in [16]. Its peculiarity was that the centre sill, side and longitudinal beams were made of corresponding hexagonal hollow profiles with cables stretched in them. This wagon design had a lower tare weight compared to those of the existing prototypes. However, its significant drawback was the complexity of manufacturing, maintenance and repair.

The design of the universal covered wagon proposed in [17] had the same disadvantage. The peculiarity of the wagon structure was that it had a hinge element in the middle part of the centre sill. There were hinge elements at the junctions of the centre sill with the end, bolster and main beams. Hinge elements were also provided at the junctions of the side wall posts with the lower rail and at the junction of the roof with the side wall posts.

Publication [18] proposes the design of a wagon for carrying goods requiring protection against atmospheric agents. The peculiarity of this design was that behind the sliding door leaves, the doorways were equipped with barrier doors. The improvement could reduce the cost of manual labour, resource and downtime of the wagon. However, such a structural solution did not reduce the wagon tare weight.

The analysis of literary sources has shown that the issues of improvements of wagons for carrying goods requiring protection against atmospheric agents are quite relevant. At the same time, the issues of how to improve their operational efficiency require further research.

In this regard, the purpose of this article is to highlight the results of the improvement in the wagon covered with tarpaulin and determination of its strength and dynamic load when operating on 1520 mm gauge lines. To achieve this purpose, the following tasks were set:

-

To determine the profile of the arches of the frame and build its spatial model;

-

To calculate the strength of the wagon frame and make its modal analysis;

-

To assess the motion of the wagon when moving over a joint irregularity.

The article includes five sections under which such issues have been addressed. Section 1 presents the relevance of the study and the analysis of publications dealing with the problem addressed in this article. In Section 2, the authors provide materials and methods used in their study. Section 3 highlights the results obtained by the authors. These results are discussed in Section 4. The conclusions on the results of this study are given in Section 5.

2. Materials and Methods

A covered wagon mod. 11-217 was chosen as a prototype for developing a wagon covered with tarpaulin [19]. The load-bearing structure of this wagon has a metal frame, the walls of which have a 1.4 mm thick sheathing. On the inside, the wagon body is sheathed with 8 mm thick plywood. The wagon is loaded through the side doors moving on special rollers along the wagon body. Due to the two-leaf doors of the wagon, the loading/unloading process can be increased with forklifts.

The frame with end walls of this covered wagon is shown in Figure 1. It consists of upper rail (1), body posts (2), door posts (3), corner posts (4) and two end walls (5) with a sheathing and a framework that includes upper rail (6) and posts (7). Also, the frame includes arches (8) that form the roof. Loading hatches are provided on the roof of the covered wagon. Due to them, the wagon can be used for carrying bulk cargo (e.g., grain).

The frame of the wagon mod. 11-217 is shown in Figure 2. It is formed by a centre sill (1), bolster beams (2) side beams (3), cross bearers (4), longitudinal beams (5), main transverse beams (6), end beams (7), braces (8), short stub beams (9) and long stub beams (10).

The centre sill consists of two Z-profiles No. 31. The bolster beam has a box section and is formed by four sheets. The vertical plates have a thickness of 6 mm, the upper plate is 8 mm, and the lower plate is 10 mm.

The side beams of the wagon frame are formed by channel No. 20. It should be noted that a Z-shaped profile is welded to these beams in the doorway areas. The end beams are formed by a U-shaped profile, which is formed by 6 mm thick plates. The main cross bearers are welded and have an I-beam cross-section formed by 6 mm thick plates. The braces are made of channel No. 14. Transverse and longitudinal beams are designed to support the floor and are made of bent channels 100 × 80 × 5 mm. The floor of this wagon is formed by boards with a thickness of 55 mm, which are quarter-joined. In the areas of the doorway, the floor is reinforced with a 4 mm thick metal sheet. This solution is provided in order to ensure sufficient strength of the floor when forklifts are used.

On the basis of the typical wagon frame, the design of a wagon covered with tarpaulin is proposed (Figure 3). It includes arches installed on the typical covered wagon frame; the arch profile is similar to the body of the prototype wagon. The arches have a degree of freedom in the longitudinal plane. The wagon is loaded after the arches have moved towards the end wall (Figure 4). The posts are fixed relative to the frame by a typical mechanism used on similar wagons. The end walls are formed by vertical and transverse beams, and covered with a horizontal sheet.

The red arrow in Figure 4 shows the direction of movement of the arches.

The profiles of the frame arches are selected according to the moments of resistance of their components using a rod system of the wagon frame (Figure 5); it is calculated using Lira CAD [20]. This software allows for determining the internal factors acting in the rods when an external load is applied to them. The external loads include the lateral loads, the wind load (Figure 6), the centrifugal load (Figure 7), and the vertical load applied to the wagon frame.

The wind load P_w is calculated by the formula:

$$P_w=S\cdot \omega ,$$

(1)

where S is the area of the side surface of the body; ω is the wind pressure (ω = 500 Pa).

The value of the centrifugal force is calculated as

$$P_{cf}={\displaystyle \frac{P_{br}\cdot V^2}{g\cdot R}},$$

(2)

where P_br is the gross weight of the wagon body; V is the speed; R is the curve radius.

The calculation is made for a speed of 80 km/h and a curve radius of 250 m. A vertical load is applied to the frame at the full payload capacity of the wagon.

The next stage of the study includes the calculation of the strength of the bearing structure of the wagon in SolidWorks Simulation using the finite element method [21,22,23]. The spatial model of the wagon is created using the options of SolidWorks. The optimal number of model elements is determined by the graphic analytical method based on building dependence of the maximum stresses on the number of finite elements [24,25,26]. When this dependence begins to be described by a horizontal line, this is the optimum number of finite elements.

The body is calculated for two main operating modes (I and III) and for the 1520 mm track gauge [27].

The operating conditions in design mode I include pushing heavy wagons back and moving from standstill, collisions of wagons during shunting operations, including during rolling from humps, emergency braking at low speeds, or collisions of wagons in emergency situations, as well as emergency jerk (push).

The operating conditions in design mode III include the movement of a wagon at the full payload capacity as part of a train on straight and curved track sections, over switches of the corresponding design; they also include the movement at speeds from permissible to design, during periodic service control braking, periodic moderate jerks and pushes, and normal operation of mechanisms and the wagon components.

The longitudinal forces regarding these modes are summarized in

Table 1. Values of longitudinal forces acting on the wagon in operation.

The Longitudinal Force, MN
Design Modes
I		III
Quasi-static force	Impact, jerk	Quasi-static force	Impact, jerk
–2.5 +2.0	–3.5 +2.5	–1.0 +1.0	–1.0 +1.0

[27].

The sign “–“ means the compressive force (impact), the sign “+” means the tensile force (jerk).

Longitudinal forces are applied to the front (tensile/jerk) or rear (compression/impact) stops at the level of the axis of the automatic coupling equipment of wagons [27].

The design diagram of the wagon body in the example of design mode III is shown in Figure 8. It is assumed that the vertical load P_v (in red) acts on the wagon. The wind load P_w (pink) is applied to the outer surface of the arches. Also, the centrifugal load P_cf (blue) is applied to the arches. In design mode I, the load on the arches is not taken into account, but the action of longitudinal forces on the frame is included.

The vertical load P_v is calculated with the maximum permissible load of the wagon body. For the prototype wagon, this load is 68 tonnes.

The longitudinal load P_l is applied to the front or rear coupler draft lugs, depending on the design diagram. For example, during an impact, the longitudinal load P_l is applied to the rear draft lug on one side of the wagon, and on the opposite side it is balanced by the inertia force of the wagon mass.

The model is fixed using rigid connections on the centre plates of the body [25]; that is, the model does not include the frictional forces between the centre plates and centre bowls.

It is taken into account that the vertical arches do not have their own degrees of freedom and are rigidly fixed to the frame. An impact on the rear coupler draft lug is absolutely rigid. The material of the body structure is a typical low-alloy Steel 09G2S, which is used for the manufacture of load-bearing structures operating on 1520 mm gauge lines.

This design diagram is also used in the modal analysis of the wagon-bearing structure in SolidWorks Simulation.

The next stage of the study includes the calculation of the main indicators of the wagon dynamics, which are used to estimate its motion. The vertical dynamics of the wagon when it moves over a joint irregularity are investigated.

The design diagram of the wagon is shown in Figure 9.

The equation of motion of the wagon can be written as [28].

$$\begin{array}{l}{M}_1\cdot {\displaystyle \frac{d^2}{d{t}^2}}{q}_1+2\cdot k_b\cdot q_1-k_b\cdot q_3-k_b\cdot q_5=\\ =-F_{FR}\cdot \left(sign\left({\displaystyle \frac{d}{dt}}{\delta }_1\right)+sign\left({\displaystyle \frac{d}{dt}}{\delta }_2\right)\right),\end{array}$$

(3)

$$\begin{array}{c}{M}_2\cdot {\displaystyle \frac{d^2}{d{t}^2}}{q}_2+\left(2\cdot l^2\cdot k_b\right)\cdot q_2+\left(l\cdot k_b\right)\cdot q_3-\left(l\cdot k_b\right)\cdot q_5=\\ =F_{FR}\cdot l\cdot \left(sign\left({\displaystyle \frac{d}{dt}}{\delta }_1\right)+sign\left({\displaystyle \frac{d}{dt}}{\delta }_2\right)\right),\end{array}$$

(4)

$$\begin{array}{c}{M}_3\cdot {\displaystyle \frac{d^2}{d{t}^2}}{q}_3-k_b\cdot q_1+\left(l\cdot k_b\right)\cdot q_2+\left(k_b+2\cdot k_1\right)\cdot q_3+2\cdot {\beta }_1\cdot {\displaystyle \frac{d}{dt}}{q}_3=\\ =F_{FR}\cdot sign\left({\displaystyle \frac{d}{dt}}{\delta }_1\right)+k_1\left({\eta }_1+{\eta }_2\right)+{\beta }_1\left({\displaystyle \frac{d}{dt}}{\eta }_1+{\displaystyle \frac{d}{dt}}{\eta }_2\right),\end{array}$$

(5)

$$\begin{array}{l}{M}_4\cdot {\displaystyle \frac{d^2}{d{t}^2}}{q}_4+\left(2\cdot a^2\cdot k_1\right)\cdot q_4+2\cdot {\beta }_1\cdot {\displaystyle \frac{d}{dt}}{q}_4=\\ =-k_1\left({\eta }_1-{\eta }_2\right)-{\beta }_1\cdot a\cdot \left({\displaystyle \frac{d}{dt}}{\eta }_1-{\displaystyle \frac{d}{dt}}{\eta }_2\right),\end{array}$$

(6)

$$\begin{array}{c}{M}_5\cdot {\displaystyle \frac{d^2}{d{t}^2}}{q}_5-k_b\cdot q_1-\left(l\cdot k_b\right)\cdot q_2+\left(k_b+2\cdot k_1\right)\cdot q_5+2\cdot {\beta }_1\cdot {\displaystyle \frac{d}{dt}}{q}_5=\\ =F_{FR}\cdot sign\left({\displaystyle \frac{d}{dt}}{\delta }_2\right)+k_1\left({\eta }_3+{\eta }_4\right)+{\beta }_1\left({\displaystyle \frac{d}{dt}}{\eta }_3+{\displaystyle \frac{d}{dt}}{\eta }_4\right),\end{array}$$

(7)

$$\begin{array}{c}{M}_6\cdot {\displaystyle \frac{d^2}{d{t}^2}}{q}_6+\left(2\cdot a^2\cdot k_1\right)\cdot q_6+2\cdot {\beta }_1\cdot {\displaystyle \frac{d}{dt}}{q}_6=\\ =-k_1\cdot a\cdot \left({\eta }_3-{\eta }_4\right)-{\beta }_1\cdot a\cdot \left({\displaystyle \frac{d}{dt}}{\eta }_3-{\displaystyle \frac{d}{dt}}{\eta }_4\right),\end{array}$$

(8)

where M₁, M₂ are the mass coefficients of the body (mass and moment of inertia) during translational and angular displacements, respectively; M₃, M₄ are the mass coefficients (mass and moment of inertia) of the first bogie facing the engine during translational and angular displacements, respectively; M₅, M₆ are the mass coefficients (mass and moment of inertia) of the second bogie facing the engine during translational and angular displacements, respectively; k_b is the stiffness of the springs forming spring suspension; k₁ is the track stiffness; β₁ is the damping coefficient of the track; a is half the bogie; q_i is the generalized coordinates corresponding to translational and angular displacements of the wagon; F_FR is the frictional force in the spring suspension; δ is the deformation of the spring suspension; η_i is the joint irregularity set as the harmonious law of motion.

The design diagram includes three bodies: the wagon body and two bogies. It is taken into account that the oscillatory system has 6 degrees of freedom, which characterize translational displacements (bouncing oscillations) and angular displacements (galloping oscillations). Bouncing oscillations in Figure 9 are indicated by vertical arrows, and galloping by circular arrows.

Equations (3), (5) and (7) describe the translational displacements of the body and two bogies, respectively, and Equations (4), (6) and (8) describe their angular movements. The interaction of the body and the unsprung masses is described through an elastic connection that forms the spring suspension of the bogies. A track irregularity is included as a disturbing action, which generates the oscillatory process, and, thus, the frictional forces in the spring suspension of the wagon. When setting the disturbing effect on the running gear of the wagon, the parameters of the elastic-viscous track are also taken into account. It should be noted that the reactions of the track are proportional to its deformations and the rates of these deformations.

The calculation is carried out provided that the body rests on bogies mod. 18-100, which are the most common in operation for 1520 mm gauge lines. Mathematical model (3)–(8) is solved in MathCad [29,30] using the classical method of step-by-step Runge–Kutta iteration [31,32]. The following initial conditions are taken into account [28]:

-

The initial displacement of the body is 0.004 m;

-

The initial displacement of the bogies is 0.003 m; and

-

The initial speeds of the body and bogies are zero.

The calculation is carried out on the example of the wagon movement at a speed of 80 km/h.

3. Results

Using the design diagrams shown in Figure 6 and Figure 7, a diagram of the bending moments acting in the framework is obtained (Figure 10).

The analysis of this diagram shows that the maximum bending moment in the frame is 4.45 kN · m and has a negative value. This moment occurs in the bolster beam. In the arches, the maximum moment is 3.44 kN · m and it is positive. According to this bending moment M, the moment of resistance W and the known permissible stresses [σ], the profile of the frame arches is selected. The following classic expression is used: W = M/[σ]. Taking into account that [σ] for Steel 09G2C is 210 MPa [27], then W = 16.38 cm³. Provided that the fatigue safety factor, which for the elements of the wagon bodies in accordance with [33] is n = 1.8, then W = 29.48 cm³.

According to this moment of resistance, the use of a rectangular tube is proposed as a profile of the frame. Technologically, this profile is quite convenient for installation and maintenance [34]. It should also be noted that this profile is more durable and lighter, thus it is more economical.

In accordance with [33], several tube variants are selected with the parameters given in

Table 2. Parameters of tubes for arches.

Width and Height, cm	Wall Thickness, mm	Moment of Resistance, cm3	Mass of 1 m, kg
70 × 70	8	29.74	13.85
80 × 80	4.5	30.4	10.26
90 × 90	3.5	32.24	9.26
100 × 100	3	35.41	8.96

.

Thus, according to Table 2, the use of a 100 × 100 tube is the most rational, since it has the smallest mass. The proposed design has a 16% smaller tare weight than the prototype.

Taking into account the selected profile of the arches, a spatial model of the wagon body is built and its strength is calculated. When building the model, the welds between the individual components of the structure are not taken into account. The finite-element model of the framework is formed by tetrahedra (Figure 11) since the mesh is created on a solid body [35,36,37]. The number of nodes is 93,440, and the number of elements is 263,648. The largest element size is 100 mm and the smallest is 20 mm.

The results of the calculations are given in

Table 3. Results of the strength calculation of the body.

Strength Index	Loading Mode
	Mode I				Mode III
	Impact	Compression	Jerk	Tension	Impact/ Compression	Jerk/ Tension
Stress, MPa	308	278.6	281.4	275.2	203.5	201.4
Displacements in units, mm	4.8	4.7	4.7	4.6	4.6	4.6

.

By analysing the data given in Table 3, it can be concluded that the maximum stresses in the wagon body occur during an impact due to the maximum longitudinal load acting on the body. In other diagrams, the stresses in the body have lower values. For example, in design mode I (jerk), the stress is slightly greater than that during tension. This can be explained by the fact that when jerking, a large longitudinal force acts on the frame. In design mode III, the stresses in the body are almost the same.

The maximum stresses during an impact occur in the area of interaction between the centre sill and the bolster beam (Figure 12). The minimum stresses occur in the frame mounting areas, i.e., in the areas where it rests on the bogies. This occurs due to rigid connections installed here. In the end walls of the body, the stresses are about 125 MPa, while in the middle part of the frame, they are about 145 MPa.

In the areas of interaction between the centre sill and the bolster beam, the maximum stresses are 308 MPa (Figure 13). Moreover, these stresses are concentrated in the area of the centre sill, which is located closer to the striker. In the area of interaction between the centre sill and the bolster beam, which is located on the opposite side, these stresses amount to 304.8 MPa.

It should be noted that the maximum stresses do not exceed the permissible values, which, for design mode I, are taken to be 0.9σ_y [27], where σ_y is the yield strength of the material (σ_y = 345 MPa). This distribution of stresses can be explained by the fastening diagram selected.

The maximum displacements occur in the upper parts of the arches located behind the centre of the wagon body and are about 4.8 mm (Figure 14).

To ensure the safety of the wagon movement, a modal analysis of its design. The calculation was made for the wagon as part of a train when moving on the track. The design diagram is shown in Figure 8. In accordance with this diagram, the natural forms (Figure 15) and frequencies of oscillations (

Table 4. Values of natural frequencies of wagon body oscillations.

Mode
1	2	3	4	5	6
Frequency, Hz
14.801	14.816	16.612	18.849	19.34	21.996

) of the covered wagon are calculated.

The safety of the wagon movement is assessed by the first natural frequency of oscillations in accordance with the regulatory document [27]; it should be at least 8 Hz. In the case under study, it is 14.801 Hz. Therefore, in terms of the modal analysis, the safety of the wagon movement is ensured.

Also, this study includes the determination of the dynamics of the proposed wagon design. Taking into account the results of the calculations, the indicators of the wagon dynamics were obtained. They were used to estimate the wagon motion: acceleration in the centre of mass and the coefficient of vertical dynamics. Figure 16 and Figure 17 show the results of the calculation for the movement of the unloaded wagon. The maximum acceleration acting in the centre of mass when the wagon moves over a rail joint is 4.3 m/s² (0.43 g). This acceleration occurs when the wagon passes a joint irregularity. Further, the acceleration value decreases and amounts to about 3.5 m/s² (0.35 g). This acceleration corresponds to ‘excellent’ motion in accordance with [27], since it does not exceed a value of 0.5 g. The coefficient of vertical dynamics of the wagon is 0.49. At the moment when the first bogie facing the engine passes a joint irregularity, this coefficient is positive, and it is negative for the second bogie. This coefficient has a maximum value when the bogie passes an irregularity; after, that it decreases slightly and amounts to about 0.48. This value also corresponds to ‘excellent’ motion, since it does not exceed 0.5. Therefore, the motion of the unloaded wagon is assessed as ‘excellent’.

If the wagon is loaded, the maximum acceleration acting in its centre of mass is about 1.5 m/s² (0.15 g), (Figure 18) and the vertical dynamics coefficient is 0.19 (Figure 19). It should be noted that the accelerations in the centre of mass of the loaded wagon are lower than for an unloaded wagon since the body has a greater mass. Therefore, the coefficient of vertical dynamics is lower.

Consequently, when the loaded wagon moves over a rail joint, the motion is also assessed as ‘excellent’.

4. Discussion

To reduce the tare weight of the covered wagon, it is proposed to improve its design by changing the body frame and covering it with tarpaulin. The profiles of the frame arches are selected according to the moments of resistance of their components. At the same time, the body is considered as a rod system and the corresponding calculation is carried out. Based on this calculation, the use of square-section tubes is proposed as the profile for the arches (Table 2). The choice of the profile is justified not only by a sufficient moment of resistance at the given operating loads but also by the manufacturability of installation, as well as wagon maintenance.

The strength of the wagon body under the main loading modes, which are inherent for operation on 1.520 mm gauge lines, is calculated.

It is found that the maximum stresses occur during design mode I (impact); the results are presented in Table 3. They are concentrated in the zones of the interaction of the bolster beams with the centre sill and amount to about 308 MPa (Figure 12 and Figure 13). Thus, these stresses do not exceed the permissible values, which are assumed to be equal to 310.5 MPa for this mode in accordance with [27]. Also, the results of the calculation have made it possible to determine the maximum displacements in the body. Such displacements occur in the upper parts of the arches located behind the centre of the wagon body and amount to about 4.8 mm (Figure 14). This is due to the fact that the body is secured at the centre sills. That is, their middle parts are not fastened. This is accompanied by the largest displacements.

The study also includes a modal analysis of the wagon body for assessing the safety of the wagon operation in terms of frequency analysis. It is found that the first natural oscillation frequency of the body is higher than 8 Hz (Table 4). This demonstrates that, according to this criterion, the safety of the wagon operation is ensured.

Also, this study includes an assessment of the wagon moving over a joint irregularity using mathematical modelling of its oscillations in the vertical plane (Figure 9). The calculations show that the motion of the unloaded wagon can be assessed as ‘excellent’ (Figure 16 and Figure 17). The same result is obtained for the loaded wagon (Figure 18 and Figure 19).

However, the authors believe that this study has certain advantages over the known ones. For example, unlike the results presented in [5], the proposed improvement can reduce the tare weight of the wagon as one of the most important technical and economic indicators. In comparison with the results presented in [6,9,11], the authors proposed solutions aimed at improving the wagon body to reduce its tare weight. The advantage of this study in comparison with the studies presented in [7,8] is that the proposed improvement of the body can reduce the tare weight. In contrast to [10], an increase in the efficiency of rail transportation can be achieved by improving the wagon body, rather than optimizing the location of freight in the wagon. The solutions proposed in [12] increase the tare weight of the wagon in contrast to those considered in the study presented in this article.

A significant advantage of the results obtained over those covered in [13,14,18] is that the solutions proposed in this study can improve the technical and economic performance of wagons, not only loading/unloading operations. In contrast to the technical results regarding the covered wagon structure, given in [15], the improvement discussed in this article can reduce the tare weight of the wagon. Compared to the technical results in [16,17], the improved design of the wagon presented here does not cause significant difficulties in manufacturing, maintenance and repair.

The results of the analysis are different from those obtained for known analogs.

As part of the research, the authors have proposed a scientific approach to the design of bearing structures of tarpaulin-covered wagons, namely, the solutions to reduce the tare weight of the existing covered wagons used on the 1520 mm gauge. Also, these solutions will make it possible to make situational adaptations of flat wagons for carrying goods requiring protection from atmospheric agents by equipping them with a protective tent. The features of its design are identical to those considered in this work. This will expand the range of goods transported on flat wagons, as well as reduce the possible shortage of covered wagons in operation. The authors have applied for patents on these wagon designs.

The obtained dependencies of forces in the bearing structure of the wagon (Figure 10) will make it possible to further optimize its design. For example, it can be the use of profiles with different geometric parameters along the length. In this case, the strength of such a structure can be calculated using the design diagram developed during this study (Figure 8). The indicators of wagon dynamics can also be determined using mathematical modela (3)–(8) when taking into account the corresponding input data. Therefore, the proposed design approach is also appropriate for further stages of wagon improvement.

The main limitation of this study is that the design diagram of the wagon does not include the frictional forces between the centre plates and centre bowls.

The disadvantage of this study is that when determining the strength of the wagon body, welds in the areas of interaction of its individual components were not included. Thus, the model was considered one piece.

The further development can include the determination of the strength of the wagon body for 1435 mm track lines, under excessive loading conditions, in particular when transported by train ferries. It is necessary to provide special units on the bearing structure of the wagon for securing it on the ferry deck. In addition, during this study, attention should be paid to how reliable the freight is secured in the wagon. This is due to the need to ensure the stability of the wagon on the deck. These tasks will definitely be considered in the further research of the authors.

5. Conclusions

1. The arch profile of the wagon body frame covered with tarpaulin is determined. The profiles are selected according to the moment of resistance of the cross-section of the arches, taking into account the known value of the bending moment and the structural material. Rectangular tubes are proposed for this profile. The technological reasons, in particular the convenience of mounting on the wagon, as well as its maintenance, are also considered when choosing this profile.

2. The strength calculation of the wagon frame and its modal analysis are performed for design modes I and III. The maximum stresses are in design mode I (impact) and concentrated in the areas of interaction between the bolster beams and the centre sill; they amount to about 308 MPa. Their value is lower than permissible ones. The maximum displacements occur in the upper parts of the arches located behind the centre of the wagon body and are about 4.8 mm.

The results of the modal analysis of the wagon body show that the first natural oscillation frequency has a value of 14.8 Hz. That is, traffic safety in terms of frequency analysis is ensured.

3. The estimation of the motion of the wagon moving over a joint irregularity is determined. The calculation is carried out for the unloaded wagon as well as the loaded wagon. The maximum acceleration acting in the centre of mass of the unloaded wagon moving over a joint irregularity is 4.3 m/s², and the coefficient of vertical dynamics is 0.49.

For the loaded wagon, the maximum acceleration value is about 1.8 m/s²; the vertical dynamics coefficient is 0.19.

Consequently, the motion can be assessed as ‘excellent’ either for the unloaded wagon or the loaded wagon.

The studies conducted will contribute to the development of recommendations for the modern structures of covered wagons, as well as improve the efficiency of railway transportation.