WHEEL/RAIL TANGENTIAL CONTACT STRESS EVALUATION BY MEANS OF THE MODIFIED STRIP METHOD WHEEL/RAIL TANGENTIAL CONTACT STRESS EVALUATION BY MEANS OF THE MODIFIED STRIP METHOD

The contact area assembled from strips and normal stress the strips is calculated with the Strip method. The output parameters from the Strip method that come into the modified procedure are: The article deals with the way of calculating tangential stresses over non-elliptical contact patch where Kalker’s simplified method FASTSIM can be used advantageously. This method named FASTSTRIP is adapted for non-elliptical contact area calculated by means of the Strip method. This method is almost as quick as FASTSIM and the results are similar to the CONTACT results. This method may be useful for rail vehicles in track dynamics computation. computation procedure.


Introduction
The rail /wheel contact relations for purposes of rail vehicles dynamics are often calculated by means of Hertz method [1] and Kalker´s simplified method applied in the program code FASTSIM [2]. Kalker's variation method [3] used to be considered as an etalon for contact patch and contact stress between railway wheel and rail calculation. Normal stresses and contact patches areas are assessed with the program code NORM [3], tangential stresses and tangential forces with the program code TANG [3]. The computation with Kalker's variation method takes longer time than the computation with the simplified method. This is the reason why the variation method is not widely used for rail vehicles dynamics computation and the simplified method is preferred for this purpose. The results gained with the simplified method are partially different (but acceptable) from the results gained with the variation method. The most significant difference consists in the contact area shape and size calculated in program FASTSIM that always presupposes to be elliptical. The Strip method procedures [4] give more opportunities to solve the contact with respect to non-elliptical contact patch. Our aim is to create the calculation procedure of "FASTSIM" type -we can name it "FASTSTRIP" for calculation of stresses over a non-elliptical contact area. We derived the procedures for fast non-elliptical contact patch calculation [5] as a presupposition of tangential forces computation. The resultant values of our brand new procedure presented in this article are closer to calculation results achieved by Kalker's variation method [3], while the computation speed is similar to the computation speed of FASTSIM [2]. The authors' initial remarks on the wheel/rail tangential contact stress evaluation by means of the modified Strip method can be found in [6] and first article in [7].

Prerequisites
To begin with, we calculate the moduli of shear elasticity for wheel and rail materials [3]:  The contact area assembled from strips and normal stress above the strips is calculated with the Strip method.
The output parameters from the Strip method that come into the modified procedure are: N -number of strips, y i -centre of i-th strip coordinate,. y d -half-length of the strips, x di -half-length of the i-th strip, p oi -normal stress in the middle of i-th strip, A NS -area of all strips.
We determine the number of splitting up the strips in the longitudinal direction: The tangential forces , T T x y and the spin moment Mz are set to be zero at the beginning.

Mathematical computational alghoritm
The mathematical computational algorithm is schematically depicted in the flow chart in Fig. 2.

Calculation at the i-th strip (A):
For the i-th strip position the coordinate in the imaginary ellipse area is as follows: and for its width the following holds: For tangential maximum stress the relation mentioned below holds: where: n is the friction coefficient.
We will calculate the following constants: , , , , For the half-length of the strip the following relation holds: For a calculating step we have: n a x x ei d = (13) and for the area element: A y e e d x $ d = (14) For the strip slip in the x-axis direction we have: We used the modified "FASTSTRIP" method for the tangential stresses computation. Figure 1 shows the program dialog with graphical output of results.
will be used.
For the semi axes proportion the following relation is valid: We will set , , C C C 11 22 23 constants for the given D parameter and n friction coefficient.

Calculation over the length of i-th strip (B):
The , p p x y tangential stresses are set to be zero at the beginning of the calculation over the strip length.
For the current slip in the y direction following holds:  n is a friction coefficient, N is a normal force. Fig. 2 Flow chart of the procedure method is that the computation is executed along the strip separately, regardless of the strip size -whether it is smaller or longer than the virtual ellipse border.

Fig. 3 Plot of Pmax CONTACT-NORM, Pmax NFASTSTRIP and Pmax
HERTZ against wheelset treads profiles lateral movement

Results and validation
We analyzed the contact patch area, contact stress between the wheel equipped by S1002 tread profile and UIC60 rail head profile inclined by 1:40. The lateral shift is in interval of (cca -5mm to 5mm).

Input parameters
The wheel force is Q = 100.000N. We used our fast strip method [5] for the computation of contact patches and contact stresses and we compared our results with the results obtained by Kalker's Contact-NORM method [3].
In Table 1

Results of tangential stresses calculation
Tangential stresses, T x , T y forces and M z moment for separate strips are computed by means of the T x , T y and M z computed by "CONTACT-TANG", "FASTSIM" and "FASTSTRIP" methods for the purpose of comparison.

Conclusions
Our aim is to create the calculation procedure of "FASTSIM" type. We named this procedure for calculation of stresses over non-elliptical contact area "FASTSTRIP". The result values are closer to the results achieved by Kalker's variation method results and the computation speed is similar to the computation speed of FASTSIM. This method is adapted for a non-elliptical contact area calculated by means of the Strip method [5]. This method utilizes the FASTSIM theory [2] as a calculation engine for tangential stress assessment. The calculation procedure is outlined in Fig. 2 in which a flowchart with two program loops is illustrated. These loops are in detail described in the part "Mathematical model". Results and validation follow. In Table 1 are some input parameters, Figs. 3, 4, 5 and 6 compare the results gained by means of NORM [3] and our calculation procedure [5] shift, area and pmax. They express the reality that the ground input parameters for tangential forces calculations are mutually very close. Figures 7, 8, 9 and 10 give results of tangential stresses calculation for input parameters. The curves of dependencies (Tx, Ty Mz) calculated with TANG [3] and FASTSTRIP are shown in graphs. For better resolution are these curves shown in Figs. 11 and 12 as comparative proportional curves. The meaning or importance of the procedure FASTSTRIP for us or anybody who writes his/her own code is in the fact that this procedure can be implemented into the code for computation of rail vehicles dynamics offering the advantage of fast computations. Of course, the field may be analysed from other points of view [8, 9 and 10] where multi-software platforms are presented. The field of wheel /rail contact task is very close to the computational analysis of contact stress distribution in the case of mutual slewing of roller bearing rings [11 and 12], or from the more sophisticated field [13, 14 and 15].