## 4 Results of Computational Modeling  
We also computed strength factors for the carrying structures of circular-tube wagons. Let us consider some peculiarities of strength calculations for the carrying structure of an articulate flat wagon with the finite element method realized in CosmosWorks software [19; 20; 21; 22; 23; 24]. The design diagram of the carrying structure of a flat wagon for design mode I (jerk) is given in Figure 10. A longitudinal load of 2.5 MN was applied to the front stops.  
Figure 8: Accelerations on the carrying structure of the articulated boxcar: (**a**) first and (**b**) second sections of the flat wagon from longitudinal force.  
Figure 9: Accelerations on the carrying structure of the articulated hopper wagon: (**a**) first and (**b**) second sections of the flat wagon from longitudinal force.  
We considered that each section of the flat wagon was loaded with two 20-foot containers. The vertical load from the containers was applied to the horizontal surfaces of fitting stops in the form of the remote load, with consideration of the center of gravity of the container.  
A finite element model of the carrying structure of a flat wagon was built with spatial isoparametric tetrahedrons. The optimal number of elements in the mesh was determined with the graph-analytical method. The number of elements in the mesh was 5,406,526 and the number of nodes was 1,538,366. The maximum size of an element in the mesh was 15 mm, the minimum size was 3 mm, and the maximum element side ratio was 3078.9. The percentage of elements with a side ratio less than 3 was 87.6% and the percentage with more than 10 was 0.212%. The number of elements in the circle was 8 and the element size gain ratio was 1.7. The model was fixed in the zones where the carrying structure rested on the gearing parts.  
The results of the strength calculation for the carrying structure for a flat wagon are given below. The maximum equivalent stresses were concentrated in the end parts of the center sill and accounted for about 180 MPa; thus, they did not exceed the admissible values (Figure 11).  
The maximum displacements in the structural nodes were taken in the middle sections; they accounted for 3.4 mm (Figure 12). The maximum strain was \(2.3\times 10^{-3}\).  
Strength calculations were also conducted for other types of articulated wagon (Table 2).  
The method described in [25] was used to determine the design life of wagons designed using the developed computational models:  
\[T_{n}=\frac{\left(\sigma_{-1D}/[n]^{m}\cdot N_{0}\right)}{B\cdot f_{e}\cdot \sigma_{a}^{m}}, \tag{13}\]  
where \(\sigma_{-1D}\) is the average value of strength endurance of a detail, \(n\) is the admissible safety factor, \(m\) is the degree of fatigue curve, \(N_{0}\) is the number of tests, \(B\) is the coefficient characterizing a period of continuous work of an object, \(f_{e}\) is the effective frequency of dynamic load, and \(\sigma_{a}^{m}\) is the amplitude of equivalent dynamic loads. The calculation results showed that the design lifetime of the proposed carrying structure models of the wagons exceeded 32 years.  
Figure 11: Stress state of the articulated flat wagon section.  
Figure 10: Design diagram of the carrying structure of the flat wagon.