SENSITIVITY OF STRESSES TO THE FORCES ACTING ON THE CAST PARTS OF FREIGHT-CAR BOGIE

Dep. «Cars and Car Facilities», Dnipropetrovsk National University named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 373 15 04, e-mail reidemeister.a@gmail.com, ORCID 0000-0001-7490-7180 Dep. «Cars and Car Facilities», Dnipropetrovsk National University named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 373 15 04, e-mail tri_s@ua.fm, ORCID 0000-0002-8256-2634


Introduction
The absence in the three-piece bogies of freight cars of devices preventing from falling onto the track its main supporting elements such as the bolster and the side frame sharply raise the reliability requirements imposed on these elements.However, a fairly large part of the freight train derailment at the current moment is caused precisely by defects in the cast parts of the bogies, namely, the side frame fractures.
The railway enterprises have been facing the problem of mass fracture of side frames since 2006 [3,5,16] and two main directions can be identified as the main possible causes.
The first one is low quality of casting.Work in this direction revealed two possible causeshidden defects in casting [6] and non-compliance with the established steel viscosity requirements KCV (-60°C) [2].To solve these problems, the works are in progress to improve the casting technologyregulating the metal cooling by updating the moulding flasks or the design of the side frame itself.Besides, monitoring more strictly the quality of metal and casting methods by destructive and non-destructive controls.
The second possible cause of mass fractures is structural.In 2001, there was a transition from the I-section of the arch bar on the supra-box area to the box-like section [14].It was suggested that the I-section of the side frame extension acted as an elastically deforming quencher of torque, which occurs during critical loosening of side frames relative to each other and the corresponding angular displacements of the set of wheels in the pedestal jaw opening.In other words, the I-section design was less rigid in the horizontal plane and had a slightly loaded positionthe I-section pedestal, which had a compensating effect of the bending moments of the set of wheels, and in abnormal operation modes, all defects in the side frame operation were visible.The solution to this problem is related to the development of methods for strengthening the side frame structure on the basis of calculations and tests to determine its strength [7,13,15,17,23], reliability, as well as the dynamic characteristics of the car as a whole [4,8,9,16,19,21,22].
Possible solutions are attempts to create welded structures of side frames [1,11].This solution, although it will eliminate the problems of casting, however raises many questions related to the reliability of welded joints.
To increase the reliability of the bogie cast parts, it is also possible to consider the possibility, at the design stage, of the influence of the spring suspension parameters on the stresses arising in the parts.

Purpose
To determine the effect of the force components acting in the axle box and the central spring suspension on the stresses occurring in the side frame of the three-piece bogie.

Methodology
The sensitivity of the indicator  to the value j P ( j -index that separates the considered quan- tity from the set of values i P , which depends on) means the ratio of the change in the indicator  to the change in the argument In the case of a linear dependence of the exponent on all arguments, this definition is unambiguous in the sense that the sensitivity value () j k  does not depend on the fact at which values of the arguments the indicator  was calculated, as well as on the selected increment j P  .In the general case it is necessary somehow to characterize the totality of the values () j k  in a compact form.It is common to use the Morris method for this purpose [20], which consists in the fact that the sensitivity values are calculated at random points in the domain of definition with specially chosen increments of arguments (to reduce the amount of calculations), and then the mathematical expectation indicates a substantially nonlinear character of the dependence.
Since the dependence of the stress tensor components on the loads is linear, and the change in the parameters of the running parts causes a relatively small change in the loads, the nonlinearities (in determining the equivalent stresses and the number of cycles before the appearance of fracture) can hardly be regarded as essential, which eliminates the need to apply more sophisticated methods of sensitivity analysis such as Sobol's indices [18].
Stresses that arise in the bogie side frame under the action of a static load are shown in Figure 1.The heavily loaded areas near the lower corner of the central spring opening, on the lower side frame member and at the point R55 of the pedestal jaw opening are clearly visible.The stress levels in these places for various combinations of loads are given in Table 1 and approach the maximum permissible values for 20GL steel.The І calculation mode corresponds to the loading variants a and b, and III to c...f.Also, the stress concentration is noticeable in the corners of the process window (101 MPa) and the upper corners of the central spring opening (114 MPa).The stress concentration at these points is expected from the point of view of the geometry of the model.Also these points are checkpoints according to [12].Taking into account the obtained results, the points shown in Figure 2 were chosen to estimate the stressed state of the side frame.Their total number is 43, they are located in the middle of the lower side frame member (1-5, 18, 19, 25), in the upper (8,9) and the lower (6, 7, 26, 27) corners of the central opening for spring suspension, in the opening between the diagonal tension member and the column (10-13), in the inner corner of the pedestal opening (14,15), in the middle (20-24) and the cantilever part (16,17) of the arch bar.The points located symmetrically are not shown in Fig. 2.And they are assigned with the numbers 4'... 27'.
The side frame perceives the forces from the central spring suspension and the axle box.To assess the effect of these forces on the stresses arising in the side frame, we sequentially determined the stresses from unit forces acting in three directions at the appropriate places.To balance the action of unit forces on the side frame, we applied inertia forces and a moment of inertia correspond-ing to the acting forces.The points of application of unit forces and the direction of the local coordinate axes are shown in Figure 3 Fig. 2. Checkpoints .

Findings
As a result of the calculation, we obtained the tensors of stresses arising from the action of unit loads at the side frame checkpoints.On the basis of the obtained stress components, the corresponding equivalent stresses arising from the action of unit loads, the sensitivity coefficients, were calculated.The intensity of the effect of unit forces on the stresses at the points of the finite element model is shown in Fig. 4.
In Figure 4, the intensity of the shading of individual cells characterizes the degree of the force effect on equivalent stresses at the point of the model.
Analyzing the results of the calculation, we can conclude that the points 6, 8, 9 and symmetrical 6', The points located on the lower member (p. 1 -3, 18, 19, 25), the lower corner of the central spring opening (p.6, 7, 27) are sensitive to vertical loads acting on the central opening roof, the sensitivity factor is 0.411 MPa/kN and 1.154 MPa/kN respectively, and points 6, 7, 27 are also sensitive to vertical loads from the friction keyup to 0.772 MPa/kN.The transition from the lower member to the diagonal tension member (p. 4, 5) is almost equally sensitive to all loading variants except the longitudinal forces in the pedestal opening and from the action of the friction keysup to 0.156 MPa/kN.The upper corner of the central spring opening (p.8, 9) is sensitive to vertical and transverse loads from the action of the friction key up to 1.204 MPa/kN.The arch bar in the middle part (p.20-24) is more sensitive to the transverse loads acting from the vibration dampers and in the axle boxup to 0.166 MPa/kN.For points located in the process window area (p. 10 -13) it is quite difficult to determine a specific group of forces that exert significant influence, since this area experiences a complex loadingup to 0.348 MPa/kN.
The inner corner of the pedestal opening (p.14, 15) is mostly influenced by longitudinal and vertical forces from the nearest pedestal openingup to 0.143 MPa/kN.The arch bar in the pedestal opening area (p.16,17) is mostly influenced by the forces in the places of vertical and transverse axle box intersections as wellup to 0.109 MPa/kN, the influence of the remaining loads is less approximately twofold.
Thus, to reduce, for example, the stresses in the lower corners of the pedestal opening (p.6, 7, 27) by 5% (6.95 MPa, III design mode), it is necessary to reduce the vertical load on the central spring opening by 19.64 kN (average sensitivity coefficient is 0.35 MPa/kN).To reduce stresses in the inner corner of the pedestal opening (p.14, 15) by 5%, it is necessary to reduce the level of longitudinal or vertical load components in the axle box by 52 MPa (average sensitivity coefficient is 0.132 MPa/kN).Reducing the load in this range can be difficult; therefore, along with changes in the parameters of the spring sets, it is necessary to provide for an increase in the strength of the structure due to local reinforcement.
In operation, static forces act on the side frame from the car gross weightabout 220 kN with an axial load of 23.5 tons per axle and the dynamic forces arising from the movement of the car can be from 50% to 80% of the static load [10].Reduction of static loads is not advisable, since their main component is the load-carrying capacity of the car.Dynamic loads, the magnitude of which can reach up to 176 kN, can be reduced through the use of rational parameters of spring suspension and the structure as a whole.At the same time, a decrease in the dynamic component of the loads acting on the side frame by only 5% -8.8 kN, can lead to a decrease in the stress level in the side frame by 1.37-10.60MPa.This decrease is not significant in evaluating the strength of the structure, however, in assessing its durability, reducing the dynamic load amplitude by 5% will cause an increase in longevity by 20% (fatigue curve index 4).

Originality and practical value
For the first time, the effect of forces acting on the three-piece bogie side frame on the stress level arising in it has been estimated.
The obtained sensitivity factors can be used to optimize the parameters of the freight car bogie for increase of durability of its details.
The stress tensors obtained can be used to estimate the effect of complex loading on the side frame strength and durability.

Conclusions
We determined stress sensitivity coefficients on the certain sections of the three-piece bogie side frame to external loads acting on the side frame from the side of the pedestal and spring openings.The stress tensors obtained can be used to estimate the effect of complex loading on the side frame strength and durability.

Fig. 1 .
Fig. 1.Stress distribution in side frame model for the III-design mode, MPa.

8 'Fig. 4 .
Fig. 4 .The intensity of the effect of unit forces on the stresses at the finite element model points