Research into the transverse loading of the container with sandwich-panel walls when transported by rail

The article presents the results of the research into the transverse loading of the container during rail transportation. The peculiarity of the container is its walls made of sandwich panels. Such a solution will help to reduce the dynamic loading on the container in operation and, accordingly, improve its strength. Appropriate calculations have been carried out in order to determine the optimum (minimum) sheet thickness, provided that the permissible deflection in operation is ensured. The transverse loading on the container placed on a flat wagon at the side rolling has been investigated. It is found that the acceleration value obtained is almost 5% lower than that acting on the container of a typical design. The strength of the container with sandwich-panel walls is calculated using the finite element method. The results of the calculation show that the maximum stresses are 6% lower than those occurring in the container of a typical design. The research may be of values for those who develop recommendations for designing advanced modular vehicles and improving the efficiency of transportation.


Introduction
The efficient performance of the machine-building industry requires development and putting into operation modern structures of vehicles.For a long time railway transport has been the most competitive industry of machine engineering, which accounts for a large proportion of freight transportation [1].
The experience of development of the wagon fleet in the UIC member countries indicates a steady trend in improving the level of traffic safety, which ensures operational reliability and environmental compatibility of vehicles.At the same time, one of the most optimal solutions to achieve these requirements is the introduction of modular vehicles, in particular, containers, into operation.
In addition to a number of advantages of these vehicles in operation compared to others, there are also significant disadvantages associated with insufficient strength of their structures.The reason for it is significant operational loads, including dynamic sign-variable ones affecting the vehicles.These loads affect not only the container, but also the freight inside.Due to the fact that this freight has its own degree of freedom, the container structure can have additional loading.And this loading can damage the container (figure 1 [2]).
This requires additional maintenance costs.Besides, it can also affect the traffic safety and cause ecological damage.Therefore it is important to research into structural improvements for containers in order to reduce their dynamic loading at operational modes.
Research devoted to the structural improvements in the container is quite relevant, which is confirmed by a large number of publications.For example, in article [3] the authors propose and substantiate the design of a container for fruit and vegetable products.All structural solutions regarding the improvements in the container are confirmed by appropriate strength calculations.The study presents main operational loading diagrams for containers.
Also, the design of a container for transportation of fruit and vegetable products is proposed in [4].The study presents the results of strength calculation and the operational requirements for the container.At the same time, when designing these container structures, the authors did not propose any solution to improve the strength of container walls.
Special features of designing the ISO container are studied in [5].The main loading diagrams for the container in operation are analysed, as well as the structural resistance to external loads.However, it should be noted that the improvements proposed in this work do not increase the strength of container walls when containers are transported either by rail or by other transport mode.
To reduce the dynamic loads on vehicles and, thereby, improve their strength characteristics, it is advisable to use sandwich panels in their structures.The use of sandwich panels in the rail vehicle design is substantiated in [6].The study presents an algorithm for optimizing the bearing structure of a vehicle.The results of the calculations demonstrate that this solution contributes to a 16.36% reduction in the bearing structure if compared to the prototype.
The use of sandwich panels in the vehicle body structure is also substantiated in studies [7,8].The research was carried out on the example of a rail car.These solutions are implementable in both manufacture and modernization of rail cars.The feasibility of the proposed solutions has been proved by theoretical calculations of the strength for the bearing rail car structure.
A similar solution is proposed in [9].The author's team focus on improving the reliability of the bearing structure of a vehicle by introducing composite panels.It was proven that the use of such panels could improve the endurance of vehicle bodies.
However, it should be noted that the authors of these studies did not consider the use of sandwich panels for the structure of removable vehicles, in particular, containers.
The review of literature allows us to conclude that the issues of improvements for the container are quite relevant.At the same time, their higher strength by means of sandwich panels in the bearing structures has not yet been thoroughly studies.In this regard, there is a need to conduct research in this area.
The objective of the study is to highlight the results of determining the transverse dynamics and the strength of a container with sandwich panel walls when transported by rail.To achieve the objective the following tasks were set: • to substantiate the structural solutions to make the container sidewalls of sandwich panels; • to determine the transverse dynamics of the container; • to calculate the container strength.
2. The substantiation of the structural solution to make the container sidewalls of sandwich panels It is proposed to introduce sandwich panels as the components of container sidewalls to ensure their strength.The sandwich panels are composed of two metal sheets with an energy-absorbing material between them (figure 2).Such a solution will improve the strength of the container by reducing its loading.Appropriate calculations were carried out to determine the thickness of the cover sheet of the container.The cover sheets for each section were considered as thin-walled slabs with a width of 6.058 m and a height of 2.591 m.
The sheet thickness was determined by the formula [10]: where P -the pressure acting on the sheet area; σ -the allowable stresses of the material of the cover; a -the sheet width; b -the sheet height; µ -Poisson's ratio; π -to constant, which is equaled to 3.14.
On the basis of the calculations it was found that at [σ] = 210 MPa, µ = 0.28 (steel), the transverse force value 0.6•P k •g, where P k -the carrying capacity of the container, the value δ = 1.6 mm.Taking this into account, the thickness of the energy-absorbing layer can be taken as 32.8 mm, based on the condition of maintaining the wall size within that of a typical container.

The determination of the transverse dynamics of the container with sandwich panel walls
To substantiate the use of sandwich panels in the sidewalls of the container, mathematical modelling of its dynamic loading was carried out, provided that the container was placed on a flat wagon at side rolling oscillations.The design diagram of the container placed on the flat wagon is shown in figure 3.For this purpose, mathematical model (2) was formed.
where I F M -the inertia moment of the flat wagon; c B -the stiffness of the springs of the suspension group of the bogie; b -the half-width of the flat-wagon frame; F C -the moment of forces occurring between the container and the flat wagon frame; I C -the inertia moment of the container relative to the longitudinal axis; g -the free fall acceleration; M k -the mass of the container; z -the half-height of the container; F F W -the moment of forces occurring between the flat wagon and the container; F f -the moment of forces occurring between the container and the freight; c -the stiffness of the energy-absorbing material; β -the viscous resistance coefficient of the energy-absorbing material; I f -the inertia moment of the freight; q 1 , q 2 , q 3 -the generalised coordinates that determine the movement of the flat wagon, container and freight, respectively.
Mathematical model (2) was solved in MathCad at the initial conditions equalling zero [11][12][13].The model was reduced to the normal Cauchy form with a subsequent solution according to the Runge-Kutta method [14][15][16].On the basis of the calculation it was found that the maximum accelerations to the container were 1.7 m/s 2 (figure 4).
The value of the acceleration obtained was almost 5% lower than that to the container of a typical design.The calculation was made at the stiffness coefficient of the energy-absorbing material 1.5 kN/m and the viscous resistance coefficient 2.0 kN•s/m.These parameters were determined by a sequential selection, provided that the accelerations were within the permissible values.

The strength calculation for the container with sandwich-panel walls
The resulting acceleration value was included in the strength calculation of the container.The spatial model of the container was built in SolidWorks (figure 5).It included the structural elements rigidly interacting with each other.The strength calculation was made in SolidWorks Simulation with the finite element method.The Mises criterion was used as the calculation criterion [17,18].
The viscoelastic connection in the sandwich panels that form the sidewalls was included as the spring-damper connection by means of the software complex options (figure 6).
The design diagram of the container included the following loads (figure 7): vertical static P v , transverse P t applied to the sidewall.The transverse loading included the pressure from the freight (grain), as well as the dynamic loading.The dynamic loading was applied from the container slope side.
The pressure from the freight to the container walls was determined with the Coulomb method using Sinelnikov's adjustment: where γ -the volumetric mass of the freight; h -the container height; ρ -the angle of the internal friction; α -the angle of inclination of the container (taken equal to about 6°).When building the finite-element model of the container, spatial tetrahedrons were used.The model consisted of 373575 nodes and 11119509 elements.The maximum element size was 80 mm and the minimal element size was 16 mm.
The model was secured by fittings.Steel 09C2Cu was used as the structural material [19,20].The results of the calculation are given in figure 8.
The maximum stresses were recorded in the contact areas between the bottom side rail and the fitting stops; they amounted to about 180 MPa (stress in the external layer).The resulting stresses were 6% lower than those in a typical container design.
The research conducted will be of value for those who develop recommendations for designing advanced structures of modular vehicles and improving the efficiency of transportation.

Conclusions
The structural solution to make the container sidewalls of sandwich panels has been substantiated.In this case, the thickness of the sheets that form the sandwich panel should be 1.6 mm, thus ensuring their strength.The thickness of the energy-absorbing layer can be taken as 32.8 mm, while observing the wall size within that of a typical container.
The research deals with determination of the transverse dynamics of the container with sandwich-panel walls.The maximum accelerations on the container are 1.7 m/s 2 .The value of the acceleration obtained is almost 5% lower than that acting to the container of a typical design.The calculation is made at the stiffness coefficient of the energy-absorbing material 1.5 kN/m and the viscous resistance coefficient 2.0 kN•s/m).
The research also includes the strength calculation for the container with sandwich-panel walls.The results of the calculation show that the maximum stresses occur in the contact areas between the bottom side rail and the fitting stops; they are equal to about 180 MPa.It should be noted that the resulting stresses are 6% lower than those occurring in a typical container design.

Figure 1 .
Figure 1.Damage to containers in operation: (a) design diagram of a flat wagon; (b) deformation and cracks in the cover.

Figure 3 .
Figure 3.The design diagram of the container placed on the flat wagon.

Figure 6 .
Figure 6.Diagram of viscoelastic bonds in the container walls.

Figure 7 .
Figure 7. Calculation diagram of the container.

Figure 8 .
Figure 8. Stress state of the container.