The Dynamic and Strength Analysis of an Articulated Covered Wagon with the Circular Pipe Design

S S symmetry

Article The Dynamic and Strength Analysis of an Articulated Covered Wagon with the Circular Pipe Design

Oleksij Fomin 1 , Alyona Lovska 2, Juraj Gerlici 3 , Yuliia Fomina 3,*, Ján Dižo 3 and Miroslav Blatnický 3

1 Department of Cars and Carriage Facilities, Faculty of Infrastructure and Rolling Stock of Railways, State University of Infrastructure and Technologies, Kyrylivska Str. 9, 04071 Kyiv, Ukraine; [email protected] 2 Department of Wagon Engineering and Product Quality, Faculty of Mechanical and Energy, Ukrainian State University of Railway Transport, Feuerbach Sq. 7, 61050 Kharkiv, Ukraine; [email protected] 3 Department of Transport and Handling Machines, Faculty of Mechanical Engineering, University of Zilina, Univerzitna 1, 01026 Zilina, Slovakia; [email protected] (J.G.); [email protected] (J.D.); [email protected] (M.B.) * Correspondence: [email protected]

Abstract: An articulated covered wagon design was developed. The wagon feature is that the body-bearing elements are made of circular pipes. This technical solution made it possible to reduce the tare weight of the wagon while ensuring the strength conditions. Mathematical simulation of the dynamic loading of the developed articulated covered wagon design was carried out under the main operating conditions. In the calculations of the observed quantities, an application of symmetry with regard to the longitudinal axis of the wagon was used. The accelerations, as the components of the dynamic load acting on the wagon, were determined. The dynamic loading computer simulation Citation: Fomin, O.; Lovska, A.; results of the developed wagon design are also presented. The strength analysis of the articulated Gerlici, J.; Fomina, Y.; Dižo, J.; covered wagon supporting structure made it possible to conclude that the strength indexes were Blatnický, M. The Dynamic and within the allowed limits. The wagon bearing structure was analyzed for fatigue strength. The Strength Analysis of an Articulated weld strength analysis results of the most loaded part of the wagon-bearing structure are presented. Covered Wagon with the Circular The results obtained for the desired quantities revealed their symmetrical distribution in the wagon Pipe Design. Symmetry 2021, 13, 1398. structure. This research will contribute to improving the efﬁciency of railway transport operation. https://doi.org/10.3390/sym13081398

Keywords: transport mechanics; railway transport; covered wagon; innovative design; dynamics Academic Editors: Juan Luis García Guirao and Sergei D. Odintsov and strength

Received: 7 July 2021 Accepted: 27 July 2021 Published: 1 August 2021 1. Introduction Articulated wagons are used to increase the volume of freights transported by rail. A Publisher’s Note: MDPI stays neutral feature of such wagons is that their design consists of two sections, which are supported with regard to jurisdictional claims in on three bogies. The sections interact with each other through an articulated coupling. published maps and institutional affil- At present, flat wagons are the most widespread among such wagons. The variety iations. of articulated wagons is very large [1], which is due to a number of advantages over other types: • Savings in manufacturing costs; • Reduction of vertical loading on a frame; Copyright: © 2021 by the authors. • Savings in cargo transportation, especially for long-haul traffic; Licensee MDPI, Basel, Switzerland. • Reduction of required wagon fleet; This article is an open access article • Decrease in the cost of the wagon life cycle, etc. distributed under the terms and conditions of the Creative Commons The main distinguishing features of articulated wagons are their design, method of Attribution (CC BY) license (https:// loading, types of bogies, articulated coupling, etc. The articulated parts of the frame and creativecommons.org/licenses/by/ the wagon are symmetrical about the center bogie axis. 4.0/).

Symmetry 2021, 13, 1398. https://doi.org/10.3390/sym13081398 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 1398 2 of 20

To increase the volume of transportation of goods requiring protection from atmospheric precipitation, it is necessary to create innovative articulated covered wagon designs. Such a technical solution will reduce the wagon manufacturing cost at the design stage, as well as increase the efficiency of railway transport operation (reduction of maintenance and repair costs). In this context, it is urgent to undertake research on the creation of an algorithm for the design and calculation of articulated wagons using technical solutions that are non-trivial for car building while meeting the strength and operational reliability conditions. Mathematical simulation of the interaction between a railway wagon and rails has been carried out in a previous study [2]. The motion equations were compiled by linking the linear and nonlinear dynamic characteristics of the system. The equilibrium method was used to derive the load vectors. The results were confirmed by experimental studies. It is important to note that attention was not paid to the determination of the longitudinal loading of articulated wagons under operational loading conditions in this work. The basic requirements for the supporting structures of modern rolling stock are described in [3]. These requirements are applicable for modernized rolling stock, as well as for the designed rolling stock in the conditions of car-building facilities. However, this study did not present the requirements applicable for articulated wagons. The determination of the wagon dynamic loading under operational loading conditions was carried out in [4]. The calculation was carried out for the most unfavorable conditions, namely a shunting collision. Studies of vehicle dynamic loading have been carried out in [5–7]. The mathematical models were confirmed by computer simulations. However, an examination of the longitudinal dynamics of articulated wagons was not conducted in the study. The procedure for designing and manufacturing a scalable railway wagon was described in [8]. In this case, modern software tools were used. The structural concepts used in the wagon’s design ensured compliance with dynamic indexes when interacting with the rail track within the acceptable range. Measures to improve the stability of the rolling stock under operational loading conditions are given in [9]. The research results were confirmed by experimental data. However, this study did not pay attention to the articulated wagon dynamics. The determination of wagon shock-loading was carried out in [10]. In the quality of the studied parameter, accelerations were considered as dynamic loading components. The calculation of the dynamic characteristics of freight wagons with various bogies is provided in [11]. It was found that the indicators of movement dynamic qualities were almost the same when using the designs of freight wagon bogies under consideration. The determination of the dynamic characteristics of articulated wagons was not carried out in this work. Numerical simulation and experimental studies of the rolling stock structural components strength are described in [12]. In this case, the normative values of operational loads were taken into account. The results of the theoretical studies were confirmed by the method of electrical strain measurement. Strength analysis of the articulated wagons’ supporting structures was not carried out in this study. Features of multi-objective crashworthiness optimization are described in [13]. The optimal parameters of the pipes’ cross-sections, ensuring their strength under the influence of shock load, were determined. It is important to note that the determination of the rolling stock body strength using pipes as bearing elements is not given in the study. The purpose of the present article is to cover issues in the dynamics and strength analysis of a circular pipe design for an articulated covered wagon. To achieve this goal, the following tasks were defined: Symmetry 2021, 13, 1398 3 of 20

• Creation of an articulated covered wagon design, the body-supporting elements of which are made of circular pipes. The model was symmetrical with regard to the longitudinal axis; • Conduction of a mathematical simulation of the dynamic loading of the articulated covered wagon supporting structure; • Conduction of a computer simulation of the dynamic loading of the articulated covered wagon supporting structure; • Checking the adequacy of the developed dynamic loading models of the articulated covered wagon supporting structure; • Carrying out of the strength analysis of the articulated covered wagon supporting structure; • Calculation of the fatigue strength of the articulated covered wagon body-supporting structure and determination of its design life; • Calculation of the strength of the weld in the interaction zone of the center sill and the bolster beam.

2. Materials and Methods In order to reduce the material consumption of the bearing structure of the covered wagon, it was optimized according to the criterion of minimum material consumption. When doing this, the method of optimization for strength reserves was used. To determine the strength reserves of a typical bearing structure of a covered wagon, a 3D model was built in the SolidWorks Simulation software package (Vélizy-Villacoublay, France). Strength calculation was carried out using the finite element method. Isoparametric tetrahedrons were taken into account when compiling the finite element model of the covered wagon supporting structure. The optimal number of model elements was determined by the graphical–analytical method. Based on the results of the calculations, the optimal geometric parameters of circular pipes were determined, taking into account the minimum material consumption. It was proposed to manufacture the supporting structure of a covered wagon from the circular pipes. On the basis of the proposed supporting structure for a covered wagon, an articulated wagon was created. To determine the dynamic loading of the supporting structure of a covered wagon, mathematical simulation was carried out. The calculation was carried out in the MathCad software package. The articulated coupling was simulated by using a rigid connection. The magnitude of the longitudinal force that acts on the wagon bearing structure was assumed to be 2.5 MN [14,15]. Bogies 18–100 were used as running gears. Differential equations were solved in MathCad software. The Runge–Kutta method [16,17] was used as a calculation method. In order to use the standard function rkfixed in MathCad, a transition was made from second-order equations to first-order equations. The initial velocities and displacements were taken to be equal to zero. To determine the acceleration fields in relation to the improved supporting structure for a covered wagon, a computer simulation of its dynamic loading was carried out. The calculation was carried out in CosmosWorks software [18,19]. This computational software was chosen because it is currently widely used in the calculations of wagon structures. For example, Uralvagonzavod has designed and put into operation a series of tank wagons, which are designed specifically in this software product. The finite element method [20,21] was used for the calculation. While compiling the finite element model, isoparametric tetrahedrons were used. The graphoanalytical method was used to determine the optimal number of model elements. The number of model nodes was 822,453 and the number of elements was 2,487,734. The maximum element size was 35 mm and the minimum one was 7 mm. The percentage of elements with an aspect ratio of less than three was 72.5 and the percentage of elements with an aspect ratio of more than 10 was 1.43. The minimum number of elements in the circle was 22 and the ratio of the increase in the elements size was 1.8. Symmetry 2021, 13, x FOR PEER REVIEW 4 of 21

ments. The number of model nodes was 822,453 and the number of elements was 2,487,734. The maximum element size was 35 mm and the minimum one was 7 mm. The

Symmetry 2021, 13, 1398 percentage of elements with an aspect ratio of less than three was 72.5 and the percentage4 of 20 of elements with an aspect ratio of more than 10 was 1.43. The minimum number of elements in the circle was 22 and the ratio of the increase in the elements size was 1.8. To test the adequacy of the developed dynamic loading models of the articulated coveredTo testwagon the adequacybearing structure, of the developed the calc dynamiculation was loading carried models out ofaccording the articulated to the coveredF-criterion wagon [22–24]. bearing structure, the calculation was carried out according to the F- criterionA calculation [22–24]. was performed to test the strength of the improved supporting structure Aof calculationthe covered was wagon. performed The finite to test elemen the strengtht method of thewas improved used as asupporting calculation structure method, ofimplemented the covered in wagon. the SolidWorks The ﬁnite Simulation element software method waspackage. used as a calculation method, implementedThe developed in the SolidWorkssupporting Simulation structure of software the articulated package. covered wagon body was calculatedThe developed for the fatigue supporting strength. structure The calculation of the articulated was carried covered out with wagon CosmosWorks body was calculatedsoftware. The for test the fatiguebase amounted strength. to The 107 calculationcycles [25,26] was in carriedthis case. out The with fatigue CosmosWorks curve was 7 software.obtained by The dividing test base each amounted numerical to 10 stresscycles value [25 ,26of ]the in thisreference case. Thecurve fatigue SN by curve the elas- was obtainedticity modulus by dividing of the each reference numerical material stress AS valueME and of the then reference multiplying curve SNthe byobtained the elasticity value modulusby the elasticity of the referencemodulus materialof the current ASME material. and then multiplying the obtained value by the elasticityTo determine modulus ofthe the design current service material. life of the supporting structure of a covered wagon, the methodTo determine devised the by design Ustich service was used. life of the supporting structure of a covered wagon, the methodTo ensure devised the strength by Ustich of the was supporting used. structure of the covered wagon in the zones whereTo the ensure individual the strength components of the supporting interact with structure each other, of the the covered calculation wagon of inthe the welds zones in wherethe most the loaded individual nodes components was carried interact out. with each other, the calculation of the welds in the most loaded nodes was carried out. 3. Creation of an Articulated Covered Wagon Design, the Body-Supporting Elements 3. Creation of an Articulated Covered Wagon Design, the Body-Supporting Elements of Which Are Made ofof CircularCircular PipesPipes The use ofof circularcircular pipespipes inin thethe coveredcovered wagonwagon supportingsupporting structurestructure waswas proposedproposed toto reducereduce itsits emptyempty weight.weight. We havehave alreadyalready carriedcarried outout workwork onon thethe introductionintroduction ofof circular pipespipes asas elementselements ofof wagonwagon supportingsupporting structures.structures. When doingdoing so,so, thethe mainmain typestypes ofof wagonswagons were were taken taken into into account: account: open open wagons, wagons, ﬂat flat wagons, wagons, covered covered wagons wagons and hopperand hopper wagons. wagons. On the On basis the basis of the of created the created designs, designs, articulated articulated wagons wagons were proposedwere pro- andposed patented. and patented. However, However, in previous in previous publications, publications, open wagons open wagons were taken were into taken account. into Inaccount. this regard, In this in regard, this article, in this we article, describe we the describe features the involved features in involved creating andin creating calculating and thecalculating bearing structurethe bearing of anstructure articulated of coveredan articulated wagon, covered designed wagon, for the designed hauling of for goods the requiringhauling of protection goods requiring from atmospheric protection from precipitation. atmospheric precipitation. A coveredcovered wagonwagon modelmodel 11-21711-217 (Figure(Figure1 ),1), built built by by PJSC PJSC Altaivagonzavod Altaivagonzavod (Novoal- (Novo- taisk,altaisk, Russia), Russia), was was considered considered as as a prototype.a prototype.

Figure 1. A covered wagon model 11-217.

Optimization was carried out on thethe strengthstrength reservesreserves ofof typicaltypical elementselements ofof thethe covered wagonwagon bodybody bearingbearing structure.structure. The spatial model of thethe coveredcovered wagonwagon bodybody made of roundround pipespipes isis shownshown inin FigureFigure2 2.. When When creating creating the the model, model, structural structural elements elements thatthat rigidlyrigidly interactinteract withwith eacheach otherother werewere takentaken intointo account.account. The center sill of suchsuch aa wagonwagon cancan bebe mademade eithereither ofof oneone pipepipe (Figure(Figure3 3a)a) or or two two pipespipes (Figure(Figure3 3b).b). ItIt isis possiblepossible toto seeseethat thatthe the created created structures structures are are symmetrical symmetrical to to the the longitudinallongitudinal axis.axis. The proposed technical solution made it possible to reduce the mass of the covered wagon body-supporting structure compared to the prototype wagon by 4% (one pipe) and 4.84% (two pipes). In other words, this wagon design can reduce the manufacturing cost during production in the car-building facilities. To increase the volume of cargo transportation, an articulated wagon was developed on the basis of the proposed covered wagon design (Figure4). Symmetry 2021, 13, x FOR PEER REVIEW 5 of 21

Symmetry 2021, 13, 1398 5 of 20 SymmetrySymmetry 2021 2021, 13, ,x 13 FOR, x FOR PEER PEER REVIEW REVIEW 5 of 521 of 21

Figure 2. The supporting structure of the covered wagon made of round pipes.

FigureFigure 2. The 2. The supporting supporting structure structure of the of thecovered covered wagon wagon made made of round of round pipes. pipes.pipes.

(a) (b) Figure 3. The frame of a covered wagon made of round pipes: (a) a center sill made of one pipe; (b) a center sill made of two pipes.

The proposed technical solution made it possible to reduce the mass of the covered wagon body-supporting structure compared to the prototype wagon by 4% (one pipe)

and 4.84% (two pipes).(a) (a In) other words, this wagon design can reduce(b) (b) the manufacturing cost during production in the car-building facilities. FigureFigure 3.To The increase 3. The frame frame ofthe a ofcoveredvolume a covered wagon of wagoncargowagon made transportation, mademade of round ofof round round pipes: pipes: pipes: an ( aarticulated) ( a( a)center) a a center center sill wagon sill madesill made made of was ofone of one developed onepipe; pipe; pipe; (b ()b ()b a) a center sill made of two pipes. a centeroncenter the sill basis sill made made of ofthe oftwo twoproposed pipes. pipes. covered wagon design (Figure 4).

TheThe proposed proposed technical technical solution solution made made it po itssible possible to reduce to reduce the themass mass of the of thecovered covered wagonwagon body-supporting body-supporting structure structure compared compared to theto theprototype prototype wagon wagon by 4%by 4%(one (one pipe) pipe) andand 4.84% 4.84% (two (two pipes). pipes). In other In other words, words, this this wagon wagon design design can can reduce reduce the themanufacturing manufacturing costcost during during production production in the in thecar-building car-building facilities. facilities. To increaseTo increase the thevolume volume of cargo of cargo transportation, transportation, an articulatedan articulated wagon wagon was was developed developed on theon thebasis basis of the of theproposed proposed covered covered wagon wagon design design (Figure (Figure 4). 4).

FigureFigure 4. 4.AnAn articulated articulated covered covered wagon. wagon.

Symmetry 2021, 13, x FOR PEER REVIEWThe The section section frame frame had had a amodified modiﬁed design, design, in in contrast contrast to to the designs shown inin FigureFigure6 of3 21. 3.On On the the side side of of the the section section support support on on the th middlee middle bogie, bogie, the the bolster bolster beam beam was was replaced replaced by a bybeam a beam made made of aof round a round pipe pipe (Figure (Figure5). 5).

FigureFigure 4. An 4. articulatedAn articulated covered covered wagon. wagon.

TheThe section section frame frame had had a modified a modified design, design, in contrast in contrast to the to thedesigns designs shown shown in Figure in Figure 3. On3. Onthe theside side of the of thesection section support support on thone thmiddlee middle bogie, bogie, the thebolster bolster beam beam was was replaced replaced by aby beam a beam made made of a of round a round pipe pipe (Figure (Figure 5). 5).

FigureFigure 5.5. AA frameframe ofof anan articulatedarticulated coveredcovered wagon.wagon.

For the interaction between the sections, a typical articulated coupling type SAC-1 was used from the company WABTAC (Figure 6) [27,28].

Figure 6. SAC-1 articulated coupling.

However, it is possible to use articulated couplings from other manufacturers.

4. Mathematical Simulation of Dynamic Loading of the Articulated Covered Wagon Supporting Structure To determine the dynamic loading of an articulated covered wagon under one of the most loaded operating conditions—that is, a jerk—a mathematical model was created, which is represented by Equations (1)–(6): М ′′⋅+x М ⋅⋅ϕ+hkxx () − =Р , SS11 S 1 S 1 S 1 S 2 l (1)

I ⋅ϕ +М ⋅hx ⋅ − g ⋅ϕ ⋅М ⋅h = SS11 S 1 S 1 S 1 S 1 (2) =⋅()()ΔΔSS11 − + ⋅ ΔΔ  SS 11 − ⋅ lFfr sign121122 sign lk k ,

Mz⋅=⋅+⋅− kΔΔSS11 k F()sign Δ S 1 − sign Δ S 1 , SS11 11 2 2 fr 1 2 (3)

М ′′⋅+x М ⋅⋅ϕ−hkxx () − =0, SS22 S2 S 2 S 1 S 2 (4)

I ⋅ϕ +М ⋅hx ⋅ − g ⋅ϕ ⋅М ⋅h = SS22 S 2 S 2 S 2 S2 (5) =⋅()()ΔΔSS22 − + ⋅ ΔΔ  SS 22 − ⋅ lFfr sign121122 sign lk k ,

Mz⋅=⋅+⋅− kΔΔSS22 k F()sign Δ S 2 − sign Δ S 2 , SS22 11 2 2 fr 1 2 (6)

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Figure 5. A frame of an articulated covered wagon.

ForFor the the interaction interaction between between the the sections, sections, a a typical typical articulated articulated coupling coupling type type SAC-1 SAC-1 waswas used used from from the the company company WABTAC WABTAC (Figure6 6))[ 27[27,28].,28].

FigureFigure 6. 6. SAC-1SAC-1 articulated articulated coupling. coupling.

However,However, it it is is possible possible to to use use articula articulatedted couplings couplings from ot otherher manufacturers.

4.4. Mathematical Mathematical Simulation Simulation of of Dynamic Dynamic Loading Loading of of the the Articulated Articulated Covered Covered Wagon Wagon Supporting Structure Supporting Structure To determine the dynamic loading of an articulated covered wagon under one of the To determine the dynamic loading of an articulated covered wagon under one of the most loaded operating conditions—that is, a jerk—a mathematical model was created, most loaded operating conditions—that is, a jerk—a mathematical model was created, which is represented by Equations (1)–(6): which is represented by Equations (1)–(6): .. .. M0 · x + M · h · ϕ + k0x − x = P , (1) S1М ′′S⋅+1x МS1 ⋅⋅ϕ+hkxxS1 ()S1 −S2 =Р , l SS11 S 1 S 1 S 1 S 2 l (1) .. .. IS · ϕ + MS · h · xS − g · ϕS · MS · h = 1 S1⋅ϕ +1 ⋅ ⋅1 − ⋅ϕ1 ⋅ ⋅1 = ISS. S М S.hxS  S g S.МS Sh . S 111 11 1 11 1 1 (2) l · Ff r sign∆1 − sign∆2 + l k1 · ∆1 − k2 · ∆2 , (2) =⋅()()ΔΔSS11 − + ⋅ ΔΔ  SS 11 − ⋅ lFfr sign121122 sign lk k , .. . S1 . S1 M · z = k · ∆S1 + k · ∆S1 − F sign∆ − sign∆ , (3) S1 S1 1 1 SS2 2 f r S1 S2 Mz⋅=⋅+⋅− kΔΔ11 k F()sign Δ 1 − sign Δ 1 , SS11 11 2 2 fr 1 2 (3) .. .. M0 · x + M · h · ϕ − k0x − x = 0, (4) S2 S2 S2 S2 S1 S2 М..′′⋅+x М ⋅⋅ϕ−hkxx..  () − =0, SS22 S2 S 2 S 1 S 2 (4) IS · ϕ + MS · h · xS − g · ϕS · MS2 · h = 2 S2 2 2 2 . S2 . S2 . S2 . S2 (5) l · Ff r signI ∆⋅ϕ1 − +signМ ∆ ⋅hx2 ⋅ + −l g ⋅ϕk1 · ⋅∆М1 − ⋅hk2 =· ∆2 , SS22 S 2 S 2 S 2 S2 (5) =⋅()()ΔΔSS22 − + ⋅ ΔΔ  SS 22 − ⋅ .. lFfr sign121122 sign lk. S2 k ,. S2 · = · S2 + · S2 − − MS2 zS2 k1 ∆1 k2 ∆2 Ff r sign∆1 sign∆2 , (6)

SS S S i Mz⋅=⋅+⋅−i kΔΔ22 k0 F()sign Δ 2 − sign Δ 2 , = − · SS22= 11+ · 2 2 fr 1 2 (6) where ∆1 zSi l ϕSi ; ∆2 zSi l ϕSi , M Si is the gross weight of the i-th section; MSi

is the weight of the bearing structure of the i-th section; ISi is an inertia moment of the i-th section; Pl is a value of the longitudinal force acting on the automatic coupling; Ff r is an absolute value of the dry friction force in the coil spring group; k0 is the stiffness of connection between sections; k1, k2 are the springs stiffness of the bogies spring groups; and xi, ϕi, zi are coordinates that determine the section movements relative to the corresponding axes. The study of the oscillatory process was carried out in a ﬂat coordinate system. At the same time, it was taken into consideration that the wagon had three degrees of freedom: vibrations of jerking, chattering and pitching [29]. The design scheme is presented in Figure7. Due to the fact that the wheelsets had an axial load of 23.5 t/axle (230.535 kN/axle), the gross weight of the wagon was intended to not exceed 1383.21 kN. It is also possible to use bogies with increased axial load under such a wagon. The accelerations acting on the articulated covered wagon are shown in Figure8. Symmetry 2021, 13, x FOR PEER REVIEW 7 of 21

Symmetry 2021, 13, x FOR PEER REVIEW 7 of 21

і Δ=і −⋅ϕ Δ= +⋅ϕ zl 2 zlSS where 1 SSіі; іі, М ′ is the gross weight of the i-th section; М Sі Sі і Δ=і −⋅ϕ Δ= +⋅ϕ is the weight of the zlbearing structure2 zlSS of the i-th section; I is an inertia moment of the where 1 SSіі; іі, ′ is the grossSі weight of the i-th section; М S М S i-th section; is a value of the longitudinal force і acting on the automatic coupling; і Pl Ffr is the weight of the bearing structure of the i-th section; IS is an inertia moment of the is an absolute value of the dry friction force in the coil spring group;і k′ is the stiffness of i-th section; is a value of the longitudinal force acting on the automatic coupling; Pl Ffr connection between sections; k1 , k 2 are the springs stiffness of the bogies spring is an absolute value of the dry friction force in the coil spring group; k′ is the stiffness of groups; and x , ϕ , z are coordinates that determine the section movements relative to connectionі і betweenі sections; k , k are the springs stiffness of the bogies spring the corresponding axes. 1 2 groups; and x , ϕ , z are coordinates that determine the section movements relative to The study of the oscillatoryі і і process was carried out in a flat coordinate system. At the samethe time, corresponding it was taken axes. into consideration that the wagon had three degrees of freedom: vibrationsThe ofstudy jerking, of the chattering oscillatory and process pitching was [29]. carr Theied designout in schemea flat coordinate is presented system. At in Figurethe 7. same time, it was taken into consideration that the wagon had three degrees of free- Duedom: to the vibrations fact that ofthe jerking, wheelsets chattering had an and axial pi loadtching of [29].23.5 Thet/axle design (230.535 scheme kN/axle), is presented the grossin weight Figure of7. the wagon was intended to not exceed 1383.21 kN. It is also possible to use bogiesDue with to increased the fact that axial the load wheelsets under such had a an wagon. axial load of 23.5 t/axle (230.535 kN/axle), Symmetry 2021, 13, 1398 7 of 20 the gross weight of the wagon was intended to not exceed 1383.21 kN. It is also possible to use bogies with increased axial load under such a wagon.

Figure 7. The design scheme of the articulated covered wagon.

TheFigure Figureaccelerations 7. 7. TheThe design design acting scheme scheme on the of of articulatedthe the articulated articulated covered covered covered wagon wagon. wagon. are shown in Figure 8.

The accelerations acting on the articulated covered wagon are shown in Figure 8.

(a) (b)

Figure 8. AccelerationsFigure 8. acting on the articulated covered wagon: (a) the first section;a (b) the second b Accelerations(a) acting on the articulated covered wagon: ((b)) the first section; ( ) the section. second section. Figure 8. Accelerations acting on the articulated covered wagon: (a) the first section; (b) the second Thesection. researchThe researchresults made results it possible made it possibleto conclude to conclude that the accelerations that the accelerations acting on acting the on the bearing structurebearing structure of the first of wagon the first section wagon from section the fromside theof the side longitudinal of the longitudinal force action force action were 28.4were m/s²,The 28.4 and research m/s the2 ,accelerations and results the accelerationsmade acting it possible on acting the to second conclude on the wagon second that thesection wagon accelerations were section 27.9 were acting m/s² 27.9 on m/s the2 (Figure bearing8).(Figure 8structure). of the first wagon section from the side of the longitudinal force action were 28.4 m/s², and the accelerations acting on the second wagon section were 27.9 m/s² (Figure5. The Dynamic8). Loading Computer Simulation of the Articulated Covered Wagon Supporting Structure As the frame is the most loaded wagon unit, its frame calculation is given below for a better visualization of the obtained results. When creating the strength model, it was taken into account that the vertical static load st Pv , which is caused by the wagon dead weight and by its loading level, acts on the wagon bearing structure (Figure9). This load is distributed symmetrically to the longitudinal axis, which is depicted by violet arrows in Figure9. It was also taken into account that the longitudinal force Pl acts on the front coupler horns. It was assumed that the wagon would be loaded with a conditional load using the maximum permissible load capacity. The possible movements of the load in the body were not taken into account for the action of the longitudinal load on the wagon. The model was fixed in the areas of support on the chassis. The steel 09G2S was used as a construction material. Based on the calculations, it was found that the maximum acceleration acting on the first section from the side of the longitudinal force action was about 32.4 m/s2, and the maximum acceleration acting on the second section was 31.7 m/s2 (Figure 10). Symmetry 2021, 13, x FOR PEER REVIEW 8 of 21

5. The Dynamic Loading Computer Simulation of the Articulated Covered Wagon Supporting Structure

Symmetry 2021, 13, x FOR PEER REVIEW As the frame is the most loaded wagon unit, its frame calculation is given below8 of for 21 a better visualization of the obtained results. When creating the strength model, it was taken into account that the vertical static load Pvst, which is caused by the wagon dead weight and by its loading level, acts on the 5. The Dynamic Loading Computer Simulation of the Articulated Covered Wagon wagon bearing structure (Figure 9). This load is distributed symmetrically to the longi- tudinalSupporting axis, Structurewhich is depicted by violet arrows in Figure 9. It was also taken into account that theAs thelongitudinal frame is the force most Pl loadedacts on wagon the front unit, coupler its frame horns. calculation It was isassumed given below that thefor wagona better wouldvisualization be loaded of the with obtained a conditional results. load using the maximum permissible load capacity.When The creating possible the movements strength model, of the loadit was in ta theken body into were account not takenthat the into vertical account static for theload action Pvst, which of the islongitudinal caused by theload wagon on the dead wagon. weight and by its loading level, acts on the wagon bearing structure (Figure 9). This load is distributed symmetrically to the longitudinal axis, which is depicted by violet arrows in Figure 9. It was also taken into account that the longitudinal force Pl acts on the front coupler horns. It was assumed that the Symmetry 2021, 13, 1398 wagon would be loaded with a conditional load using the maximum permissible 8load of 20 capacity. The possible movements of the load in the body were not taken into account for the action of the longitudinal load on the wagon.

Figure 9. The design scheme of the articulated covered wagon.

The model was fixed in the areas of support on the chassis. The steel 09G2S was used as a construction material. Based on the calculations, it was found that the maximum acceleration acting on the first section from the side of the longitudinal force action was about 32.4 m/s2, and the maximumFigureFigure 9. 9. TheThe acceleration design design scheme scheme acting of of the the on articulated articulated the second covered covered section wagon. wagon. was 31.7 m/s2 (Figure 10).

The model was fixed in the areas of support on the chassis. The steel 09G2S was used as a construction material. Based on the calculations, it was found that the maximum acceleration acting on the first section from the side of the longitudinal force action was about 32.4 m/s2, and the maximum acceleration acting on the second section was 31.7 m/s2 (Figure 10).

FigureFigure 10. 10. TheThe design design scheme scheme of of the the articulated articulated covered covered wagon. wagon.

TheThe maximum maximum accelerations accelerations were were concentrat concentrateded in in the the middle middle parts parts of of the the sections. sections. NearNear the the console console parts parts of ofthe the sections, sections, the theaccelerations accelerations decreased decreased and andamounted amounted to 23.4 to 2 2 m/s23.42 m/s(first section)(ﬁrst section) and 22.6 and m/s 22.62 (second m/s (secondsection).section). It could be It couldobserved be observedthat the structure that the wasstructure loaded was symmetrically loaded symmetrically to the longitudinal to the longitudinal axis. axis. FigureA 10. modal The design analysis scheme was of carried the articulated out to determine covered wagon. the limiting frequencies and vibration modes of the articulated covered wagon frame. The numerical values of the limiting vibrationThe maximum frequencies accelerations are given in were Table 1concentrat. ed in the middle parts of the sections. Near the console parts of the sections, the accelerations decreased and amounted to 23.4

m/sTable2 (first 1. Numerical section) values and 22.6 of the m/s critical2 (second vibration section). frequencies It could of the be articulated observed covered that thewagon structure frame. was loadedVibrational symmetrically Mode to the longitudinal Frequency, axis. rad/s Frequency, Hz 1 41.16 8.55 2 76.48 12.17 3 116.46 18.54 4 121.18 19.29 5 187.11 29.78 6 187.2 29.79 7 199.67 31.78 8 224.27 35.69 9 258.28 41.11 10 261.69 41.65

The calculations showed that the critical frequencies were within acceptable limits. Symmetry 2021, 13, x FOR PEER REVIEW 9 of 21

A modal analysis was carried out to determine the limiting frequencies and vibration modes of the articulated covered wagon frame. The numerical values of the limiting vibration frequencies are given in Table 1.

Table 1. Numerical values of the critical vibration frequencies of the articulated covered wagon frame.

Vibrational Mode Frequency, Rad/s Frequency, Hz 1 41.16 8.55 2 76.48 12.17 3 116.46 18.54 4 121.18 19.29 5 187.11 29.78 6 187.2 29.79 7 199.67 31.78 8 224.27 35.69 9 258.28 41.11 10 261.69 41.65

The calculations showed that the critical frequencies were within acceptable limits.

6. Validation of the Developed Dynamic Loading Models of the Articulated Covered Wagon Supporting Structure for Adequacy The input parameter of the model was the longitudinal force acting on the front coupler horn, and the output was the acceleration acting on the supporting structure (Table 2). Symmetry 2021, 13, 1398 9 of 20 The calculation results are shown in Figure 11. The percentage difference between the results of the mathematical and computer simulation is shown in Figure 12. In this 6.case, Validation the maximum of the Developed divergence Dynamic Loadingwas about Models 12%. of the Articulated Covered Wagon Supporting Structure for Adequacy TableThe 2. input Numerical parameter values of the of model the acceleration was the longitudinal forces forceacting acting on onthe the articulated front covered wagon sup- couplerporting horn, structure. and the output was the acceleration acting on the supporting structure (Table2).

2 Table 2. Numerical valuesLongitudinal of the acceleration forces Force acting Acting on the articulated covered wagon supportingAcceleration structure. Value, m/s

on the Front Coupler Horns, Acceleration Value, m/s2 Longitudinal Force Acting on the Front Coupler Horns, MN Mathematic Simulation Computer Simulation MN Mathematic Simulation Computer Simulation 1.11.1 12.312.3 13.7 13.7 1.3 14.6 16.1 1.51.3 16.814.6 18.6 16.1 1.71.5 19.316.8 21.1 18.6 1.9 21.2 23.6 2.11.7 23.919.3 27.2 21.1 2.3 26.1 29.8 2.51.9 28.421.2 32.4 23.6 2.1 23.9 27.2 The calculation2.3 results are shown in Figure 11. The percentage26.1 difference between the 29.8 results of the mathematical and computer simulation is shown in Figure 12. In this case, the maximum divergence2.5 was about 12%. 28.4 32.4

Symmetry 2021, 13, x FOR PEER REVIEW 10 of 21

FigureFigure 11. Acceleration Acceleration forces forces acting onacting the articulated on the arti coveredculated wagon covered supporting wagon structure. supporting structure.

Figure 12. The divergence between the results of the mathematical and computer simulations. Figure 12. The divergence between the results of the mathematical and computer simulations. The calculations showed that for the adequacy variance of Sa = 44.2 and the repeatability variance Sr = 31.8, the calculated value of the criterion was Fc = 1.4. Moreover, the The calculations showed that for the adequacy variance of Sa = 44.2 and the repeat- tabular value of the criterion was Ft = 3.07. So the hypothesis of adequacy was not rejected. ability variance Sr = 31.8, the calculated value of the criterion was Fc = 1.4. Moreover, the tabular value of the criterion was Ft = 3.07. So the hypothesis of adequacy was not rejected.

7. Strength Analysis of an Articulated Covered Wagon Bearing Structure A calculation was performed using the finite element method to determine the strength indexes of the articulated covered wagon bearing structure. Fragments of the finite element model are shown in Figure 13.

(a) (b)

(c) (d) Figure 13. (a) The interaction zone of the vertical post with the bottom chord; (b) the interaction zone of the cross-bearer with the bolster beam; (c) a fragment of the side wall; (d) a fragment of the roof.

The calculated loads acting on the articulated covered wagon bearing structure were identical to the loads that were taken into account for the study of dynamic loading (Figure 14).

Symmetry 2021, 13, x FOR PEER REVIEW 10 of 21

Figure 11. Acceleration forces acting on the articulated covered wagon supporting structure.

Figure 12. The divergence between the results of the mathematical and computer simulations.

The calculations showed that for the adequacy variance of Sa = 44.2 and the repeat- Symmetry 2021, 13, 1398 ability variance Sr = 31.8, the calculated value of the criterion was Fc = 1.4. Moreover, the 10 of 20 tabular value of the criterion was Ft = 3.07. So the hypothesis of adequacy was not rejected.

7. Strength7. Strength Analysis Analysis of an Articu of anlated Articulated Covered Covered Wagon Bearing Wagon BearingStructure Structure A calculationA calculation was performed was performed using using the finite the ﬁnite element element method method to to determine determine thethe strength strengthindexes indexes of of the the articulated articulated covered covered wagon wagon bearing bearing structure. structure. Fragments Fragments of the of ﬁnite the element finite elementmodel model are shown are shown in Figure in Figure 13. 13.

(a) (b)

Figure 13.Figure (a) The 13. interaction(a) The interaction zone of zonethe vertical of the vertical post with post the with bottom the bottom chord; chord; (b) the (b) interaction the interaction zone zone of theof cross-bearer the cross-bearer with with the thebolster bolster beam; beam; (c) a ( cfragment) a fragment of the of theside side wall; wall; (d) (ad fragment) a fragment of the of the roof. roof. Symmetry 2021, 13, x FOR PEER REVIEW The calculated loads acting on the articulated covered wagon bearing structure11 were of 21 Theidentical calculated to loads the loads acting that on the were articulated taken into covered account wagon for thebearing study structure of dynamic were loading identical(Figure to the 14loads). that were taken into account for the study of dynamic loading (Figure 14).

Figure 14. The design scheme of the articulated covered wagon body-supporting structure. Figure 14. The design scheme of the articulated covered wagon body-supporting structure. The calculation results are shown in Figure 15. The maximumcalculation equivalent results are stress shown arose in Figure in the zone15. of interaction between the center sill and the bolster beam and was about 300 MPa; that is, it did not exceed the allowed limits for the steel grade of a metal structure [14,15,25,26]. The distribution of the maximum equivalent stresses along the length of the center sill of the ﬁrst wagon section from the side of the longitudinal force action is shown in Figure 16. It was thus established that the stress distribution occurred according to the polyno- mial law. In the roof of the section, the maximum equivalent stresses amounted to 84 MPa (Figure 17).

(a) (b)

(c) Figure 15. The stress state: (a) the articulated covered wagon supporting structure; (b) the frame of the articulated covered wagon section (top view); (c) the frame of the articulated covered wagon section (bottom view).

The maximum equivalent stress arose in the zone of interaction between the center sill and the bolster beam and was about 300 MPa; that is, it did not exceed the allowed limits for the steel grade of a metal structure [14,15,25,26]. The distribution of the maximum equivalent stresses along the length of the center sill of the first wagon section from the side of the longitudinal force action is shown in Figure 16.

Symmetry 2021, 13, x FOR PEER REVIEW 11 of 21

Symmetry 2021, 13, 1398 Figure 14. The design scheme of the articulated covered wagon body-supporting structure. 11 of 20

The calculation results are shown in Figure 15.

(a) (b)

Symmetry 2021, 13, x FOR PEER REVIEW 12 of 21

(c) SymmetryFigure 2021, 1315., xThe FOR stress PEER state: REVIEW (a) the articulated covered wagon supporting structure; (b) the frame of the articulated cov- 12 of 21 Figure 15. The stress state: (a) the articulated covered wagon supporting structure; (b) the frame of the articulated covered wagonered wagon section section (top view); (top view); (c) the (c frame) the frame of the of articulated the articulated covered covered wagon wagon section section (bottom (bottom view). view).

Figure 16. The distribution of maximum equivalent stresses along the length of the first section center sill.

It was thus established that the stress distribution occurred according to the poly- nomial law. Figure 16.16. The distribution of maximum equivalent stressesstresses along thethe lengthlength ofof thethe ﬁrstfirst sectionsection centercenterIn sill.sill. the roof of the section, the maximum equivalent stresses amounted to 84 MPa (Figure 17). It was thus established that the stress distribution occurred according to the poly- nomial law. In the roof of the section, the maximum equivalent stresses amounted to 84 MPa (Figure 17).

Figure 17. TheThe stress stress state of the articulated covered wagon section roof.

In the side wall of the section the maximum equivalent stress was 98 MPa and was concentrated in the zone of interaction of the bottom chord with the bolster post (Figure 18).Figure 17. The stress state of the articulated covered wagon section roof.

In the side wall of the section the maximum equivalent stress was 98 MPa and was concentrated in the zone of interaction of the bottom chord with the bolster post (Figure 18).

Symmetry 2021, 13, x FOR PEER REVIEW 12 of 21

Figure 16. The distribution of maximum equivalent stresses along the length of the first section center sill.

It was thus established that the stress distribution occurred according to the poly- nomial law. In the roof of the section, the maximum equivalent stresses amounted to 84 MPa (Figure 17).

Symmetry 2021, 13, 1398 12 of 20 Figure 17. The stress state of the articulated covered wagon section roof.

In the side wall of the section the maximum equivalent stress was 98 MPa and was concentratedIn the side in wallthe zone of the of section interaction the maximum of the bottom equivalent chord with stress the was bolster 98 MPa post and (Figure was concentrated18). in the zone of interaction of the bottom chord with the bolster post (Figure 18).

SymmetrySymmetry 2021 2021, 13, 13, x, xFOR FOR PEER PEER REVIEW REVIEW 1313 of of 21 21

FigureFigure 18. 18. The The stress stress state state of of the the side side wall wall of of the the articulated articulated covered covered wagon wagon section. section. Figure 18. The stress state of the side wall of the articulated covered wagon section. InIn thethe sectionsection endend wallwall thethe maximummaximum equivalentequivalent stressstress waswas concentratedconcentrated inin thethe bottombottomIn part the part section and and amounted amounted end wall theto to 184 maximum184 MPa MPa (Figure (Figure equivalent 19). 19). The The stress stress stress was decreased concentrateddecreased towards towards in the the bottomthe top top ofpartof the the and wall. wall. amounted to 184 MPa (Figure 19). The stress decreased towards the top of the wall.

FigureFigureFigure 19. 19.19. The TheThe stress stressstress state statestate of ofof the thethe articulated-type articulated-typearticulated-type covered coveredcovered wagon wagonwagon section sectionsection end endend wall. wall.wall. The maximum displacement was found for the main longitudinal beams of the sections TheThe maximum maximum displacement displacement was was found found for for the the main main longitudinal longitudinal beams beams of of the the sec- sec- and was about 4 mm (Figures 20 and 21). tionstions and and was was about about 4 4 mm mm (Figures (Figures 20 20 and and 21). 21).

FigureFigureFigure 20. 20.20. Displacements DisplacementsDisplacements in inin the thethe articulated articulatedarticulated covered coveredcovered wagon wagonwagon supporting supportingsupporting structure structure.structure. .

(a(a) ) (b(b) ) FigureFigure 21. 21. Displacements Displacements in in the the frame frame of of the the articulated articulated covered covered wagon wagon section: section: ( a(a) )top top view; view; ( b(b) )bottom bottom view. view.

TheThe maximum maximum displacement displacement was was 2.6 2.6 mm mm in in the the middle middle parts parts of of the the sections’ sections’ center center sillssills and and 1.5 1.5 mm mm in in the the cantilever cantilever parts. parts. TheThe distribution distribution of of maximum maximum displacements displacements al alongong the the length length of of the the center center sill sill of of the the firstfirst wagon wagon section section from from the the side side of of the the longitudinal longitudinal force force action action is is shown shown in in Figure Figure 22. 22.

Symmetry 2021, 13, x FOR PEER REVIEW 13 of 21

Figure 18. The stress state of the side wall of the articulated covered wagon section.

In the section end wall the maximum equivalent stress was concentrated in the bottom part and amounted to 184 MPa (Figure 19). The stress decreased towards the top of the wall.

Figure 19. The stress state of the articulated-type covered wagon section end wall.

The maximum displacement was found for the main longitudinal beams of the sections and was about 4 mm (Figures 20 and 21).

Symmetry 2021, 13, 1398 13 of 20 Figure 20. Displacements in the articulated covered wagon supporting structure.

(a) (b)

FigureFigure 21. 21. DisplacementsDisplacements in in the the frame frame of of the the articulated articulated covered covered wagon section: ( (aa)) top top view; view; ( (bb)) bottom bottom view. view.

TheThe maximum maximum displacement displacement was was 2.6 2.6 mm mm in in the the middle middle parts parts of of the sections’ center sillssills and and 1.5 1.5 mm mm in in the the cantilever cantilever parts. parts. Symmetry 2021, 13, x FOR PEER REVIEW 14 of 21 TheThe distribution distribution of of maximum maximum displacements displacements al alongong the the length length of of the center sill of the firstﬁrst wagon wagon section section from from the side of the longitudinal force action is shown in Figure 2222..

FigureFigure 22.22. DistributionDistribution ofof maximummaximum displacementsdisplacements alongalong thethe lengthlength ofof thethe ﬁrstfirst sectionsection centercenter sill.sill.

InIn thethe roofroof nodesnodes thethe maximummaximum displacementdisplacement waswas 1.31.3 mmmm (Figure(Figure 23 23).).

FigureFigure 23.23. DisplacementsDisplacements inin thethe roofroof nodesnodes ofof thethe articulatedarticulated coveredcovered wagonwagon section.section.

In the side wall the maximum displacement was 0.9 mm (Figure 24). The displacement was 0.5 mm in the area of interaction between the bolster post and the main longitudinal beam of the section frame.

Figure 24. Displacements in the side wall nodes of the articulated covered wagon section.

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Figure 22. Distribution of maximum displacements along the length of the first section center sill.

In the roof nodes the maximum displacement was 1.3 mm (Figure 23).

Symmetry 2021, 13, 1398 14 of 20 Figure 23. Displacements in the roof nodes of the articulated covered wagon section.

In thethe sideside wall wall the the maximum maximum displacement displacement was was 0.9 0.9 mm mm (Figure (Figure 24). 24). The The displacement displace- wasment 0.5 was mm 0.5 in mm the in area the of area interaction of interaction between between the bolster the bolster post and post the and main the longitudinal main longi- tudinalbeam of beam the section of the frame.section frame.

Symmetry 2021, 13, x FOR PEER REVIEW 15 of 21

Figure 24. Displacements in the side wall nodes of the articulated covered wagon section.

InIn the the end end wall wall the the maximum maximum displacement displacement was was 1.2 1.2 mm mm and and the the displacements displacements were were concentratedconcentrated in in the the bottom bottom part part of of the the wall wall (Figure (Figure 25).25).

Figure 25. Displacements in the end wall nodes of the articulated covered wagon section. Figure 25. Displacements in the end wall nodes of the articulated covered wagon section. Towards the top of the wall, the displacements were reduced to about 1 mm. Towards the top of the wall, the displacements were reduced to about 1 mm. 8. The Fatigue Strength Analysis of the Articulated Covered Wagon Body-Supporting 8.Structure. The Fatigue Determination Strength Analysis of the Designof the Articulated Life Covered Wagon Body-Supporting Symmetry 2021, 13, x FOR PEER REVIEW 16 of 21 Structure.The fatigue Determination curve of of the the covered Design wagon Life supporting structure material is shown in FigureThe 26 fatigue. curve of the covered wagon supporting structure material is shown in Figure 26. The calculations made it possible to conclude that the fatigue strength of the articulated covered wagon body-supporting structure could be determined from a given test base. The technique described in [30,31] was used to determine the design life of the articulated covered wagon body-supporting structure. In this case, the design life of the wagon was determined by Equation (7):

m ()σ − / []nN⋅ T = 10A (7) d ⋅⋅σ m Bfeae

where σ is the average value of the endurance limit, MPa; n is the permissible as- FigureFigure 26. 26. −The1TheA fatigue fatigue curve curve of of the the covered covered wagon wagon supporting supporting structure structure material. material. surance factor; m is a fatigue curve exponent; N0 is a test base; B is a coefficient that characterizesIn the calculations, the time of the continuous following object input operation parameters expressed were used: in seconds; the average fe is valuean effec- of tive frequency of dynamic stresses, s−1; and σae is an amplitude of equivalent dynamic the endurance limit was determined as 0.5·σE of a material (steels 09G2D and 09G2S) and amountedstresses. to 245 MPa; the test base was taken as 107 cycles (recommended test base for steel);The continuous coefficient operation that characterized time at ϑa = the 33.3 time m/s ofwas continuous taken to beobject 6514.37 operation s; the effectivewas de- frequencytermined usingof dynamic Equation stresses (8): was determined by taking into account the parameters of the spring suspension of bogies 18–100 and amounted⋅⋅3 to 2.7 Hz; the allowable safety 365 10 La factor was taken to be equal to 2; the fatiВ =gue curve exponent, for a welded construction(8) ϑ (1+ 0.34) was taken to be equal to 4; and the amplitudea of equivalent stresses was determined on

thewhere basis La ofis strengththe average analysis daily andwagon amounted mileage, to km 51.6 (≈ MPa.250 km [30]); ϑa is the average wagon speed,Based m/s; on and the 0.34 calculations, is the empty it was run found ratio. that the wagon design life is at least 32 years. The effective frequency of dynamic stresses was calculated with Equation (9): 9. The Strength Analysis of the Weld Joint in the Interaction Area of the Center Sill 1.1 g and the Bolster Beam f = , е π (9) The strength analysis of the articulated2 coveredfst wagon body-supporting structure showed that one of the most loaded nodes was the interaction node of the center sill and where fst is the static spring deflection, mm. the bolster beam. The joining of these structural elements with each other was carried out by welding (Figures 27 and 28). To ensure the strength of the articulated covered wagon body-supporting structure, a strength analysis of the weld joint in the interaction area of the center sill and the bolster beam was carried out.

Figure 27. The interaction node of the center sill and the bolster beam.

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The calculations made it possible to conclude that the fatigue strength of the articulated covered wagon body-supporting structure could be determined from a given test base. The technique described in [30,31] was used to determine the design life of the articulated covered wagon body-supporting structure. In this case, the design life of the wagon was determined by Equation (7): Symmetry 2021, 13, x FOR PEER REVIEW 16 of 21

(σ /[n])m · N = −1A 0 Td m (7) B · fe · σae

where σ−1A is the average value of the endurance limit, MPa; n is the permissible assurance factor; m is a fatigue curve exponent; N0 is a test base; B is a coefﬁcient that characterizes the time of continuous object operation expressed in seconds; fe is an effective frequency of −1 dynamic stresses, s ; and σae is an amplitude of equivalent dynamic stresses. The coefﬁcient that characterized the time of continuous object operation was determined using Equation (8): 365 · 103 · L B = a , (8) ϑa(1 + 0.34) where L is the average daily wagon mileage, km (≈250 km [30]); ϑ is the average wagon a a speed, m/s; and 0.34 is the empty run ratio. FigureThe 26. effective The fatigue frequency curve of ofthe dynamic covered wagon stresses supporting was calculated structure with material. Equation (9):

In the calculations, the following input1.1 parrametersg were used: the average value of fe = , (9) the endurance limit was determined as 0.5·2πσE of fast material (steels 09G2D and 09G2S) and amounted to 245 MPa; the test base was taken as 107 cycles (recommended test base for wheresteel); fcontinuousst is the static operation spring deﬂection, time at ϑa mm. = 33.3 m/s was taken to be 6514.37 s; the effective frequencyIn the calculations,of dynamic stresses the following was determined input parameters by taking were into used: account the the average parameters value of of thethe endurance spring suspension limit was of determined bogies 18–100 as 0.5 ·anσEdof amounted a material to (steels 2.7 Hz; 09G2D the andallowable 09G2S) safety and 7 amountedfactor wasto taken 245 MPa;to be theequal test to base 2; the was fati takengue curve as 10 exponentcycles (recommended for a welded testconstruction base for steel);was taken continuous to be equal operation to 4; and time the at ϑamplitudea = 33.3 m/s of wasequivalent taken tostresses be 6514.37 was s;determined the effective on frequencythe basis of of strength dynamic analysis stresses and was amounted determined to 51.6 by taking MPa. into account the parameters of the springBased suspension on the calculations, of bogies 18–100it was found and amounted that the wagon to 2.7 Hz; design the allowable life is at least safety 32 factoryears. was taken to be equal to 2; the fatigue curve exponent for a welded construction was taken to9. beThe equal Strength to 4; andAnalysis the amplitude of the Weld of equivalent Joint in the stresses Interaction was determined Area of the on Center the basis Sill of strengthand the analysisBolster Beam and amounted to 51.6 MPa. Based on the calculations, it was found that the wagon design life is at least 32 years. The strength analysis of the articulated covered wagon body-supporting structure 9.showed The Strength that one Analysis of the most of the loaded Weld nodes Joint was in the the Interaction interaction Area node of of the the Center center Sillsill and andthe bolster the Bolster beam. Beam The joining of these structural elements with each other was carried out by weldingThe strength (Figures analysis 27 and of 28). the articulated covered wagon body-supporting structure showedTo thatensure one the of strength the most of loaded the articulate nodes wasd covered the interaction wagon body-supporting node of the center structure, sill and thea strength bolster beam.analysis The of joiningthe weld of joint these in structural the interaction elements area with of the each center other sill was and carried the bolster out bybeam welding was carried (Figures out. 27 and 28).

FigureFigure 27.27.The The interactioninteraction nodenode ofof thethe centercenter sillsill and and the the bolster bolster beam. beam.

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FigureFigure 28.28. PlacementPlacement ofof thethe weldweld inin thethe interactioninteraction zonezone ofofthe thecenter centersill sillpipe pipewith withthe the bolster bolster beam beam webweb plate.plate.

ToThis ensure node the underwent strength ofthe the following articulated strains covered during wagon operation: body-supporting tension–compression, structure, a strengthbending analysisand torsion of the [32,33]. weld jointIn light in the of interactionthis, we used area the of method the center for sill calculating and the bolster welds beampresented was carriedin [34–36]. out. Based on our practical experience of carrying out such calculations, the proposedThis node method underwent is simple the following and sufficiently strains reliable. during operation: tension–compression, bendingThe andstress torsion from the [32, longitudinal33]. In light offorce this, action we used was determined the method by for using calculating Equation welds (10) presented[34–36]: in [34–36]. Based on our practical experience of carrying out such calculations, the proposed method is simple and sufﬁciently reliable.F The stress from the longitudinal forceτ = action was determined by using Equation F , (10) (10) [34–36]: 0.7kdπ F where F is a longitudinal force actingτ on= the weld;, k is a weld leg; and d is a pipe diame- (10) F 0.7kπd ter. whereTheF is stress a longitudinal from the forcebending acting moment on the action weld; wask is acalculated weld leg; with and d Equationis a pipe (11): diameter. The stress from the bending moment action was calculated with Equation (11): 4M τ = , M 4M 2 (11) τ = 0.7kdπ , (11) M 0.7kπd2 where М is the bending moment acting on the weld. whereTheM isstress the bendingfrom the moment torque strength acting on action the weld. was calculated with Equation (12): The stress from the torque strength action was calculated with Equation (12): 4T τ = , (12) T 4T π 2 τ = 0.7kd, (12) T 0.7kπd2 where Т is the torque strength acting on the weld. whereTheT is total the torquestress was strength calculated acting with on the Equation weld. (13): The total stress was calculated with Equation (13): ττττ=++22 Σ ()() (13) q FM T τ = (τ + τ )2 + (τ )2 (13) Permissible stresses for theΣ parent metalF Mcan be determinedT using Equation (14): Permissible stresses for the parentσσ metal=−⋅()0.6 can 0.8 be determined , using Equation (14): p Y (14) σ σp = (0.6 − 0.8) · σY, (14) where Y is the yield stress, MPa. Then, at σY = 345 MPa, we have where σY is the yield stress, MPa. σ =−207 276 MPa. Then, at σY = 345 MPa, we havep Taking into account the geometric parameters of the pipe section and the bolster σp = 207 − 276 MPa. beam web plate, τΣ = 192.78 MN was obtained, which was lower than the permissible value; that is, the strength condition of the weld fulfilled. Taking into account the geometric parameters of the pipe section and the bolster beam web plate, τ = 192.78 MN was obtained, which was lower than the permissible value; that 10. DiscussionΣ is, the strength condition of the weld fulﬁlled. To reduce the material consumption of the supporting structure of a covered wagon, it was proposed to construct it from round pipes. On the basis of the proposed structure, the supporting structure of an articulated covered wagon was created. The proposed

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10. Discussion To reduce the material consumption of the supporting structure of a covered wagon, it was proposed to construct it from round pipes. On the basis of the proposed structure, the supporting structure of an articulated covered wagon was created. The proposed technical solution made it possible to reduce the weight of the bearing structure of the covered wagon body in comparison to the prototype wagon by 4% (one pipe) and 4.84% (two pipes). In other words, this wagon design can reduce the cost of manufacturing in the conditions of car-building enterprises. In addition, the creation of articulated wagons makes it possible to reduce the cost of the trucks, since the use of three bogies can be considered for two sections. The dynamic loading of the covered wagon bearing structure was determined by mathematical modeling. In this case, the presence of three degrees of freedom for each section was taken into account. In further research in this direction, we plan to analyses the dynamic loading of the wagon supporting structure in the vertical plane. The results obtained by means of mathematical modeling were confirmed and verified by computer modeling. When carrying out the calculations, the rigid fastening of the supporting structure was assumed; that is, possible clearances between the nodes of interaction between the supporting structure and the bogies were not taken into account. The main indicators of the strength of the proposed covered wagon bearing structure were determined. The strength of the bearing structure of the covered wagon was ensured. However, at this stage, the calculation was performed in quasi-statics. The fatigue strength and design life of the covered wagon bearing structure were investigated. It was found that with a test base of 107, the fatigue strength of the supporting structure was ensured. The design service life of the covered wagon supporting structure was judged to be at least 32 years. We also determined the strength of the welded joints at the most loaded elements of the supporting structure. The results of the analyses of the dynamic loading, strength, fatigue strength and design service life allow us to conclude that the use of circular pipes as elements of an articulated covered wagon supporting structure is expedient. In further research in this direction, it is important to experimentally determine the dynamic loading and strength of the covered wagon bearing structure. This can be done through the method of similarity using an electric strain gauging. Also, the issues that arise in researching the running characteristics of articulated wagons made of round pipes generate interest. The proposed research direction will be developed in a study of the expediency of using fillers for the round pipes from which the supporting structure is made. This will reduce the load on the wagon in operation, as well as improve the anti-corrosion properties. The research carried out will contribute to improving the efficiency of railway transport operations.

11. Conclusions Based on the calculations, we can make the following conclusions: 1. The design of an articulated covered wagon was created, the body-supporting elements of which were made of round pipes. The proposed solution made it possible to reduce the weight of the covered wagon body-supporting structure compared to the prototype wagon by 4% (one pipe) and 4.84% (two pipes). To increase the volume of cargo transportation, an articulated wagon was developed on the basis of the proposed covered wagon design. When using bogies 18–100, its gross weight should not exceed 1383.21 kN. Alternatively, it is possible to use bogies with increased axial load under such a wagon. 2. Mathematical simulation of the dynamic loading of the articulated covered wagon supporting structure was carried out. The calculation was carried out according to the Runge–Kutta method in MathCad software. The study results led to the conclusion that the acceleration acting on the ﬁrst section of the supporting structure from the Symmetry 2021, 13, 1398 18 of 20

longitudinal force action side was 28.4 m/s2, and the acceleration acting on the second section of the supporting structure was 27.9 m/s2. 3. Computer simulation of the dynamic loading of the articulated covered wagon supporting structure was carried out. It was found that the maximum acceleration acting on the first section from the side of the longitudinal force action was about 32.4 m/s2, and the maximum acceleration acting on the second section was 31.7 m/s2. At the same time, the maximum acceleration was concentrated in the sections’ middle parts. Towards the sections’ console parts, the acceleration decreased and amounted to 23.4 m/s2 (first section) and 22.6 m/s2 (second section). 4. The validity of the developed models of the dynamic loading of the articulated covered wagon bearing structure was checked. The F-criterion was used for the calculations. It was determined that with an adequacy variance of Sa = 33.6 and repeatability variance of Sr = 25.8, the criterion design value did not exceed that from the table. Therefore, the adequacy hypothesis was not rejected. 5. A strength analysis of the articulated covered wagon supporting structure was carried out. The finite element method, implemented in CosmosWorks software, was used in the calculation. The maximum equivalent stress arose in the zone of interaction between the center sill and the bolster beam and was about 300 MPa; that is, it did not exceed the allowed values. The maximum displacement arose in the sections’ main longitudinal beams and was about 4 mm. It can be seen that the loads of the structure were distributed symmetrically with respect to the longitudinal structure. 6. The fatigue strength of the articulated covered wagon body-supporting structure was calculated. The fatigue curve was obtained by dividing each numerical stress value of the SN reference curve by the elasticity modulus of the ASME reference material and multiplying the obtained value by the elasticity modulus of the current material. It was established that, for a given test base (N = 107), the strength of the articulated covered wagon body-supporting structure was ensured. The design life of the articulated covered wagon was calculated. The calculation results showed that the wagon design life would be at least 32 years. 7. A strength analysis of the weld joint in the interaction area between the center sill and the bolster beam was carried out. Tension–compression, bending and torsion strains were taken into account. Taking into account the geometric parameters of the pipe section of the center sill and the bolster beam web plate, the total stress value τΣ = 192.78 MN was obtained, which was within the permissible limits. It is important to say that the theoretical studies carried out made it possible to verify the expediency of using circular pipes as bearing structural elements, as well as to form an algorithm for the calculation of the supporting structure of an articulated covered wagon made of circular pipes under the most unfavorable loading mode in operation, which has not been previously covered in scientific publications. The conducted studies will contribute to reducing the cost of manufacturing wagons at the design stage, as well as increasing the efficiency of railway transport operations.

12. Patents 1. Fomin, O.V.; Lovska A.O. Covered wagon: pat. 111572 Ukraine: IPC (2016.01) B61D 3/00 B61F 1/00 B61F 1/02 (2006.01) B61F 1/08 (2006.01) B61D 17/04 (2006.01) B61D 17/08 (2006.01) B61D 17/12 (2006.01); application 18 September 2015; publ. 10 May 2016. Bull. №9. (In Ukrainian) 2. Panchenko, S.V.; Fomin O.V.; Lovska A.O. Covered wagon of articulated type: pat. 145219 Ukraine: IPC (2020.01) B61D 3/00 B61D 17/00 B61F 1/00 B61F 1/02 (2006.01) B61F 1/08 (2006.01); application 07 July 2020; publ. 25 November 2020, Bull. №22. (In Ukrainian)

Author Contributions: Conceptualization, O.F., A.L. and Y.F.; methodology, O.F., A.L. and J.G.; software, A.L. and M.B.; validation, O.F., A.L. and J.D.; formal analysis, O.F. and J.D.; investigation, Symmetry 2021, 13, 1398 19 of 20

O.F. and J.G.; resources, O.F. and A.L.; writing—original draft preparation, O.F. and Y.F.; writing— review and editing, J.D. and M.B.; visualization, A.L.; supervision, J.G.; project administration, O.F. and J.G. All authors have read and agreed to the published version of the manuscript. Funding: This publication was issued thanks to the support of the Cultural and Educational Grant Agency of the Ministry of Education of the Slovak Republic as part of project no. KEGA 036ŽU- 4/2021: Implementation of modern methods of computer and experimental analysis of properties of vehicle components in the education of future vehicle designers. This study was carried out within the scientific theme of “Innovative principles of the construction of resource-saving designs of wagons by taking into consideration the refined dynamic loads and functional and adaptive flash concepts”, which is funded from the State Budget of Ukraine, 2020. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors thank to Ukrainian State University of Railway Transport, State University of Infrastructure and Technologies, and University of Zilina for support. Conflicts of Interest: The authors declare no conflict of interest.

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