Acta Polytechnica CTU Proceedings https://doi.org/10.14311/APP.2022.35.0023 Acta Polytechnica CTU Proceedings 35:23–26, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague RAILWAY TRACK DEFLECTION ANALYSIS BY USING EVOLUTIONARY ALGORITHMS Pavel Kulich∗, Otto Plášek Brno University of Technology, Antonínská 548/1, 601 90 Brno, Czech Republic ∗ corresponding author: pavel.kulich@vutbr.cz Abstract. In contrast to numerical methods, analytical modelling of the railway track is one of the less time-consuming and computationally demanding methods which, in combination with the computing power commonly available today, can form an effective tool for analysing the behaviour of the railway track. This paper deals with the use of iterative methods of evolutionary algorithms, together with analytical modelling, for the purpose of reverse analysis of the measured deflection caused by moving loads acting on the railway track. The theoretical assumptions of the analytical model used, the data collection methodology and the method used for the reverse analysis are presented. The results of the analysis are also presented. Keywords: Railway track deflection, reverse analysis, evolutionary algorithms. 1. Introduction The analytical model presented in [1] was used for the reverse analysis. The model consists of two infinitely long beams lying on the Pasternak foundation, which, in contrast with the Winkler foundation, considers the shear interaction of adjacent elements. Graphical model interpretation is shown in Figure 1. The model can be interpreted so that the first layer represents the rail; the second represents the sleeper with substructure. The influence of dynamic variables is taken into account in the same way as in the Frýba model [2]. The model can be described by the following differ- ential equations: EI1 d4w1(x, t) dx4 + m1 d2w1(x, t) dt2 + c1 dw1(x, t) dt + k1[w1(x, t) − w2(x, t)] = 0, (1) EI2 d4w2(x, t) dx4 − GA dw2(x, t)2 dx2 + m2 d2w2(x, t) dt2 + c2 dw2(x, t) dt + (k1 + k2)w2(x, t) − k1w1(x, t) = 0, (2) where EI1 is upper beam bending stiffness, m1 is the mass of one meter of the upper beam. According to the paragraph above, the upper beam represents the rail, but the values of EI1 and m1 do not necessarily correspond to the mechanical parameters of the rail itself due to the interaction of the rail with other elements of the railway track. The parameters k1 and c1 are parameters that act between the first and second layers of the structure and can be interpreted as properties of the fastening system. The bottom layer Figure 1. Schematic of the two-layer model on Paster- nak foundation. of the model corresponds to the sleepers and partly to the substructure with parameters EI2 - bending stiffness of the second layer, m2 - mass of one meter of the second layer, GA - shear stiffness of the bottom layer. Sleepers are not capable of transferring bending loads in the longitudinal direction of the track, and therefore the bending stiffness EI2 can be understood as the residual ability of the sleeper layer to transfer (when the axle passes) the bending load by horizontal action on the trackbed. This stiffness, therefore, takes on a non-zero but negligible value. The parameters k2 and c2 represent the properties of the substructure. It is necessary to introduce the relative coordinate s to solve the above differential equations. The effect of the wheel load Q is introduced during the solution through boundary conditions. By substituting the parameters EI1 = 4500 kN m2, EI2 = 0.1 kN m2, GA = 6000 kN , k1 = 250000 kN m−2, k2 = 40000 kN m − 2, v = 30 ms − 1, c1 = 90 kN sm − 2, c2 = 120 kN sm − 2, m1 = 60 kg, m2 = 300 kg, Q = 100 kN into the solution matrix, we obtain the deflection curve for the upper layer w1 and the deflection curve for the bottom layer w2. 23 https://doi.org/10.14311/APP.2022.35.0023 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Pavel Kulich, Otto Plášek Acta Polytechnica CTU Proceedings Figure 2. Output of the two-layer model. The horizontal axis of both deflection curves is in the dimensionless relative coordinate s. 2. Methods 2.1. Track Vertical Displacement Measuring Method The input data for the reverse analysis are the track vertical displacements measured in-situ. Vertical dis- placement of railway track is measured relatively to a plain of substructure. The displacement sensors were positioned in the monitored section in such a way that it was possible to detect the pushing of the rails and sleepers. The rail foots and the sleepers in four points along longitudinal axis were monitored. The sensors were positioned as follows: • S0 – right sleeper head; • S1, S2 – right rail; • S3, S4 – sleeper in thirds; • S5, S6 – left rail; • S7 – left sleeper head. Using this sensor set, it is possible to obtain the deflection value of the rail unloaded by the twisting of the cross-section. The rail deflection is easily calcu- lated as the average of the values measured by the pair of sensors S1, S2 and S5, S6. Using sensors placed on the sleeper, the sleeper deflection curve can be calculated for each measured moment using an inter- polation polynomial (cubic spline). It is also possible to calculate the deflection value of the sleeper under the rail at a point that is otherwise inaccessible for sensor placement. Figure 3. Schematic of sensor placement [3]. 2.2. Evolutionary Algorithm Method Reverse analysis of measured data can be classified as an optimization problem. The quantities to be optimized are the parameters of the analytical model. When solving the problem, we look for a combination of model parameters that leads to the best possible solution. The quality of the solution is assessed by the so-called fitness function, which is a mathemati- cal expression of the similarity of the measured data and the output of the analytical model for the given parameters. In general, the higher the quality of the solution, the higher the value of the fitness function. For reverse analysis, algorithms have been devel- oped that use as input a set of random combinations of parameters of the analytical model, successively test each set of parameters, evaluate their quality and from the best combinations create (by precisely defined methods) combinations of new parameters, which test repeatedly. This method of evaluation 24 vol. 35/2022 Railway Track Deflection Analysis by Using Evolutionary Algorithms Figure 4. Measured in-situ deflection leads to better quality solutions with each subsequent iteration. The number of iterations and the number of elements of the parameter set are the input param- eters of the algorithm. The optimisation is completed when the required number of iterations is reached. 3. Model results The optimization algorithms were tested on data mea- sured in Planá nad Lužnicí, km 74,978 - section with- out under sleeper pads. Track superstructure consists of 60 E1 rails, W14 fastening system and B 91S/1 sleepers. Assessed data trace is measured during the passage of the passenger train. Data traces from sen- sors S2 and S6 were selected for further processing, and their values were averaged at the corresponding time points. The resulting data trace was used for comparison with the deflection w1 of the analytical model. All axles of the passenger train were consid- ered. The analytical model presented above gives the de- flection for only one axle. To obtain a total deflection line for multiple axles is necessary to introduce the superposition principle for several deflection lines. Fig- ure 4 shows that the axles exert different forces on the superstructure, which is also taken to account during superposition calculation. The displacement line obtained by the presented analytical model captures the main characteristics of the measured signal, such as the travel wave in front of the first axle, the travel waves between axles and the steepness of the drop of the track under the passing axle. Parameters of the analytical model are shown in Table 1. The track critical speed is also calculated. 4. Conclusion This article discussed reverse analysis theoretical pre- sumptions, data collection methodology, data pro- cessing methodology. The algorithm output is also compared with in-situ measured data. From the text above it can be concluded that the presented analyti- cal model can be used for further analyses of the rail track dynamic behaviour. Despite the convincing qual- itative results, the evolutionary algorithms, because of its stochastic character, can lead to misleading quan- titative outputs. Examination of the applicability of evolutionary algorithms for the purposes of reverse analysis, or the incorporation of other methods, will be the subject of the author’s further work. List of symbols EI1 Upper layer bending stifness [N m2] EI2 Bottom layer bending stifness [N m2] GA Bottom layer shear stifness [N] k1 Fastening system spring stifness [N m−2] k2 Foundation spring stifness [N m−2] c1 Fastening system damping [Ns m−2] c2 Foundation damping [Ns m−2] m1 Upper layer mass [kg m−1] m2 Foundation layer mass [kg m−1] Q Wheel force [N] v Vehicle speed [km h−1] vcr Critical speed [km h−1] x Longitudinal coordinate [m] t Time [s] s Relative coordinate [–] w1 Rail layer deflection [m] w2 Sleeper layer deflection [m] 25 Pavel Kulich, Otto Plášek Acta Polytechnica CTU Proceedings Figure 5. Analytical model comparison with measured data EI1 EI2 GA k1 k2 c1 c2 [kN m2] [kN m2] [kN ] [kN m−2] [kN m−2] [kN s m−2] [kN s m−2] 5428 0,739 549 403287 66903 17 190 m1 m2 v vcr [kg m−1] [kg m−1] [km h−1] [km h−1] 292 515 90 1018 Table 1. Analytical model parameters Acknowledgements The paper was prepared within the project "Affordable Railroad Smart Sensing System 4.0", project number TM01000016, which is co-financed with the state sup- port of the Technology Agency of the Czech Republic within the 1st public competition of the Support Pro- gramme for Applied Research, Experimental Development and Innovation DELTA 2 2019. References [1] P. Kulich. Dynamic Analysis of Track. Diploma thesis, Brno University of Technology, 2017. [2] C. Esveld. Modern Railway Track. MRT productions, Zaltbommel, second edi edn., 2001. [3] O. Plášek, J. Smutný, R. Svoboda, M. Hruzíková. Analysis of railway track dynamic parameters. In EVACES 09. Experimental Vibration Analysis for Civil Engineering Structures, pp. 239–240. Wroclaw University of Technology, Wroclaw University of Technology, 2009. 26 Acta Polytechnica CTU Proceedings 35:23–26, 2022 1 Introduction 2 Methods 2.1 Track Vertical Displacement Measuring Method 2.2 Evolutionary Algorithm Method 3 Model results 4 Conclusion List of symbols Acknowledgements References