- Email: [email protected]
switch contact
Wear 265 (2008) 1439–1445
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Assessment of methods for calculating contact pressure in wheel-rail/switch contact M. Wiest a,b , E. Kassa c,∗ , W. Daves a,b , J.C.O. Nielsen c , H. Ossberger d a
Materials Center Leoben Forschung GmbH, Franz-Josef-Straße 13, A-8700 Leoben, Austria Institute of Mechanics, Montanuniversit¨ at Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria c CHARMEC/Department of Applied Mechanics, Chalmers University of Technology, SE-41296 Gothenburg, Sweden d VAE GmbH, Alpinestraße 1, A-8740 Zeltweg, Austria b
a r t i c l e
i n f o
Article history: Accepted 14 February 2008 Available online 27 May 2008 Keywords: Wheel-rail contact Crossing Contact pressure and contact forces Finite element modelling Multibody dynamics
a b s t r a c t Two models for wheel-rail rolling contact that are based on the half-space assumption are compared in this paper: Hertz and the non-Hertzian method implemented in the computer program CONTACT. These two models are further restricted by the assumption of linear-elastic material behaviour. Moreover, one elastic and one elastic–plastic finite element model of the contact are investigated with the commercial code ABAQUS. The finite element method is not limited by the half-space assumption or applicable to a linear-elastic material model only. The objective is to assess the four methods based on calculated contact pressure, contact patch size and penetration depth. Contact loads and contact locations, used as input data in the analysis, are taken from a vehicle dynamics simulation in the software GENSYS. The comparison is performed at a given cross-section in the crossing panel of a selected turnout design. To fulfil the requirements of the half-space assumption, the dimensions of the contact area must be small compared to the radii of curvature of the bodies in contact. On the selected cross-section, however, the half-space assumption does not hold since the smallest radius of rail curvature at the contact point is 13 mm, which is comparable to the largest semi-axis of the contact area. Nevertheless, it is found that the contact pressure distributions calculated using Hertz and CONTACT correlate well with those results obtained from the finite element method as long as no plastification of the material occurs. © 2008 E. Kassa. Published by Elsevier B.V. All rights reserved.
1. Introduction Several numerical tools are available to study rolling contact problems. The wheel-rail rolling contact problem includes elastic and plastic contact of two bodies with similar material properties. A survey of computational methods for studies of contact mechanics, including elastic contact problems and plastic rolling contact theory, is performed by Kalker [1]. In the literature, a couple of attempts exist to analyse discrepancies and conformities between the different methods. As various authors arrive at different conclusions regarding the agreement or disagreement of results between the several available calculation methods, the authors of the present paper perform calculations with four different methods to receive a conclusive result. Yan and Fischer studied the applicability of Hertz contact theory to wheel-rail contact problems for a standard rail, a crane rail and a crossing nose in [2]. They found that for elastic material prop-
∗ Corresponding author. Tel.: +46 31 772 1499; fax: +46 31 772 3827. E-mail address: [email protected] (E. Kassa).
erties, and if the surface curvature does not change within the contact area, the contact pressure distributions from their threedimensional finite element (FE) calculations agreed well with those from Hertz contact theory. Their results are especially interesting as the agreement between the elastic FE method and Hertz theory remained for cases where the half-space assumption was violated. In addition, Telliskivi and Olofson [3,4] used the FE method to investigate the contact conditions for measured wheel-rail profiles of an ordinary track. Two methods based on the half-space assumption, Hertz analytical method and Kalker’s program CONTACT, were compared with the FE method for two test cases. For the case where the contact condition satisfied the half-space assumption, the difference in maximum contact pressure between the FE method and Hertz/CONTACT was negligible, but large discrepancies were observed for the case where the radius of one of the profiles was small compared to the dimensions of the contact area. Pau et al. [5] investigated the distribution of contact pressure in the wheelrail contact area by an ultrasonic technique. Substantial agreement between their results and Hertz theory was found, although significant differences were observed in terms of regularity of the shape of the contact pressure ellipsoid. Good conformity of contact stresses
0043-1648/$ – see front matter © 2008 E. Kassa. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2008.02.039
1440
M. Wiest et al. / Wear 265 (2008) 1439–1445
Fig. 1. Example of sampled rail cross-sections in the crossing panel. Dimensions are in millimeters.
between finite element simulations and a quasi-Hertz approach was found by Sladkowski and Sitarz for elastic calculations in [6]. In this paper, vehicle motion, magnitudes of the contact forces and locations of the contact patches are predicted for a vehicle that runs through a crossing using the multibody system (MBS) software GENSYS [7]. Then, the contact loads and contact locations are used as input data in the contact pressure analysis. A comparison of four contact models is performed for a rail cross-section in the crossing panel of a turnout, see Fig. 1. The crossing panel is the part of the turnout which allows two tracks to cross each other at the same level. To enable wheel flanges to pass the intersecting rails, the continuous rails are interrupted at the crossing point. Crossings are designed so that the wheel should be continuously supported throughout the crossing, giving a smooth transition. However, this is only true for ideal conditions and with a new wheel running over the crossing. In reality, when a wheelset with worn wheels passes the crossing, high impact loads may arise [8,9]. Consequences are damage, fracture and wear of both the crossing and the wheels leading to high maintenance costs. Therefore, investigations of dynamics and damage in a crossing are of great interest. The basis for the prediction of component life is an adequate determination of stresses and strains. Even with the computer power available today, this still remains a challenging task. The present paper is starting a promising approach in this direction. Two contact models based on the half-space assumption are compared: Hertz and the non-Hertzian method implemented in the computer program CONTACT. Moreover, one elastic and one elastic–plastic finite element model of the contact are investigated. The objective is to assess the four methods based on the results of calculated contact pressures, contact patch sizes and penetration depths for a selected cross-section where the half-space assumption is violated. The future goal of the authors work is to develop an interface between the MBS model and the FE model to be able
to use results from the MBS simulation as input to the FE analysis. By performing the present comparison for a selected cross-section, we further succeeded in figuring out the reasons for possible problems which can arise when driving a finite element model using results from a multibody system simulation, e.g. different contact point locations resulting from the two algorithms. During the passage of the wheel over the crossing, situations with several simultaneous contact points between wheel and double rail arrangements, e.g. wing rail with through rail and stock rail with check rail, are possible. Two contact points on the wheel are common at the crossing, with one contact point with the wing rail and the other with the crossing nose [8,10]. At some stage, two contact points may occur on the crossing nose. The contact point locations in crossings are extremely sensitive to the displacements and rotations of the two bodies in contact. The FE method and the MBS method are based on quite different algorithms for locating the contact points. In contrast to the FE method, in most MBS analyses, deformations of the wheel and the rail are not taken into account. Therefore dissimilar results may arise when comparing the MBS and the FE method. A correct interpretation of the discrepancies is necessary when the results are used for damage, wear and fracture investigations. The described problem is most significant for applications in the switch and crossing panels, and it does not seem to appear when the method of driving a FE analysis with results from a MBS simulation is applied for a normal track where there are two standard rails on each side of the track centreline. 2. Vehicle-track interaction Simulation of dynamic interaction between train and railway turnout involves modelling of vehicle, track and wheel-rail contact. A freight vehicle travelling along the standard UIC60-760-1:15 turnout geometry (curve radius 760 m, turnout angle 1:15) in the
M. Wiest et al. / Wear 265 (2008) 1439–1445
facing move is considered. Rail cross-sections are sampled at several locations along the turnout and are used as input, see Fig. 1. The numerical simulation is performed using the commercial MBS software GENSYS. The specific features of a turnout, such as rails with no inclination, check rail and variations in rail cross-section, are considered in the simulation [8,10]. To reduce differences in the GENSYS and FE models, the track model in GENSYS is taken as almost rigid with a track stiffness of 600 MN/m both in vertical and lateral directions. The corresponding track damping coefficients amount to 250 kNs/m. The track model is rigid in the longitudinal direction. This enables a fair comparison of calculated contact pressures from Hertz, CONTACT and the FE models. The contact geometry problem is solved in advance. The precalculated contact point functions are collected in tables and used in the subsequent time integration analysis. The elastic contact problem is solved based on Hertz theory for the normal contact and Kalker’s simplified theory (FASTSIM) for the tangential contact problem. In these solutions, several simplifications are necessary to arrange fast calculations, and plastic deformations in and near the contact zone are not considered. Despite the fact that assumptions made in Hertz theory are often violated, this approach is normally adopted in vehicle-track interaction simulations because it is simple and fast. The magnitude and orientation of the normal contact force depends on the curvature difference of the two contacting surfaces and the contact angle at the contact point. Often a small shift in the location of the contact point is significant since this could lead to major differences in the contact geometry. To determine the elastic deformation in the contact, the contact geometry and the relative displacement between the wheel and the rail are used. The deformation ın normal to the contact plane is computed at the contact point. Then, the normal force can be calculated based on the linear Hertzian contact stiffness CH . The relation between the normal force Fn and the normal deformation ın is given by
Fn =
CH ın , if ın > 0 . 0, else
1441
Fig. 2. Contact point location on the sampled rail cross-section and wheel crosssection.
faces and that no plastic deformation occurs in the contact patch. However, for the general problem, including situations of conformal contact and plasticity, the FE method is the only available tool. FE models are able to describe the contact behaviour of two bodies in greater detail, including their real geometry without the need of making any assumptions on constant curvature, for example. Other major strengths of the FE method are the ability to describe accurately the inelastic deformation of the material and to calculate residual stresses and strains [2,9,13]. Multiple point contacts are also accounted for. Routines for investigation of damage can be included, such as demonstrated, e.g. in [14]. Further, monitoring of the material behaviour due to cyclic loading is possible. 3.1. Hertz contact theory
(1)
The coefficient CH depends on the contact geometry of the contact point and on the material properties of the two contacting bodies. A look-up table based on Hertzian formulae is used to determine this coefficient. Vehicle motion, magnitudes of forces and creepages in the wheel-rail contacts, and dimensions and locations of the contact patches are predicted when the vehicle reaches the turnout crossing. The resulting displacements of the leading wheelset of the leading bogie for the time during which the wheel passes the crossing nose are used as input to the subsequent FE analysis in the commercial code ABAQUS [11]. Based on contact forces and contact positions from GENSYS, a detailed finite element investigation of the wheel-rail/switch contact is performed. Here, a comparison of contact mechanics methods is carried out for a rail profile located in the crossing panel of the turnout at the time when the wheel is passing it, see Fig. 2. 3. Wheel-rail contact models Based on the elastic half-space assumption, a number of theories have been derived for the normal wheel-rail contact problem [12]. Wheel and rail cross-sections are profiled, and the normal and tangential contact stresses are highly concentrated on the contact region. Hence, contact stresses can be approximated by considering the wheel and the rail as semi-infinite elastic solids, i.e. elastic half-spaces. This means that the dimensions of the contact patch are small compared to the radii of curvature of the contacting sur-
In Hertz contact theory, due to the elastic deformation of the two profiled bodies, the common surface will attain an elliptic boundary [15]. Hertz contact theory is based on a number of assumptions. The two bodies in contact are regarded as half-spaces, where the contact area is small compared to the minimum dimensions of the bodies in contact. In addition, no plastic deformation in the contact patch is assumed, and the radii of curvature of wheel and rail profiles in the contact patch are assumed to be constant. It is clear that these are rough assumptions in cases of conformal contact and two-point contacts, which are common in turnout applications. For example, for contacts near the gauge corner of the crossing rail, there is a significant variation of the wheel and rail profile curvatures within the contact patch and the dimensions of the contact patch may not be small compared to the radii of curvature of the contacting surfaces. In some cases, the half-space assumption is not valid when contact occurs close to the gauge corner. In this case, a non-elliptical contact area may ensue. According to Hertz theory, the normal pressure is distributed as an ellipsoid over the elliptic contact area with semi-axes a and b. The ellipsoidal contact pressure distribution p(x,y) is expressed by
3Fn p(x, y) = 2ab
1−
x 2 a
−
y 2 b
,
(2)
where Fn is the dynamic normal contact force obtained from the GENSYS simulation, while x and y are local coordinates with the origin at the centre of the contact ellipse. The x coordinates are bounded by the semi-axis a, |x| ≤ a, and the y coordinates are bounded by the semi-axis b, |y| ≤ b. The maximum pressure is at
1442
M. Wiest et al. / Wear 265 (2008) 1439–1445
the centre of the contact ellipse. The semi-axes of the contact area are calculated as 1/3
a = ah Fn b=
,
(3)
1/3 bh Fn ,
(4)
where ah and bh are parameters that depend on the contact geometry and on the material properties of the two contacting surfaces. The two bodies that are in contact are assumed to have identical material properties with Young’s modulus 200 GPa and Poisson’s ratio 0.3. ¨ According to the method in Lundberg and Sjovall [16], the contact geometry is defined by a parameter . This parameter is given by
1
2
1 1 − + = r rx1 ry1 1 1 1 1 1 + + + = r rx1 rx2 ry1 ry2
1 1 − rx2 ry2
2
+2
1 1 − rx1 ry1
(A2 + 1)E − 2K , (A2 − 1)E
(6)
where E and K are the complete elliptic integrals with the modulus k2 = 1 − A−2 . The semi-axes of the contact area, a and b, are then given as
a=
3A2 E(1 − ) G
b=
1/3
· Fn
,
(7)
,
(8)
1/3
3E(1 − ) AG
1/3
1/r
1/3
1/r
· Fn
where G is the shear modulus and is Poisson’s ratio. The elastic deformation (penetration depth) at the centre of the contact patch is computed by ı=
K
1/r
2A2 E
1/3
3(1 − ) 2G
2/3
2/3
· Fn
.
3.3. Finite element model Detailed investigations on stresses and strains in the contacting bodies require a very dense mesh when using the finite element method. Thus, because of limited computational power, only sections of the wheel and the crossing are modelled, see Fig. 3. The length of the sections of both wheel and crossing is 250 mm, while
1 1 − rx2 ry2
where rx1 and ry1 are the principal radii of curvature in the x- and y-directions of the first body, and rx2 and ry2 are the principal radii of curvature in the x- and y-directions of the second body. The angle between the principal planes is denoted ϕ. The ratio of the semi-axes, A = a/b, depends on the curvature of the two bodies in contact and on the angle between the principal axes. To determine the semi-axes of the contact area, first the ratio of the semi-axes A is determined by the following equation =
However, like the Hertz contact solution, the CONTACT algorithm is based on the half-space assumption. Wheel and rail are considered as elastic half-spaces. In addition, CONTACT is restricted to elastic problems, which means that plastic deformation in the contact patch is not accounted for.
(9)
⎫ ⎪ ⎪ cos 2ϕ ⎬ ⎪ ⎪ ⎭
,
the depth of the sections at the contact zone is 35 mm for the crossing and 30 mm for the wheel. Both bodies are modelled with a very fine mesh in the contact zone with an element size on the surface of 0.5 mm × 0.5 mm. Altogether the model consists of 110,000 eightnoded linear hexahedral solid elements with reduced integration. Each element has 24 degrees of freedom. The total number of nodes is 130,000. “Tied contact”-constraints are used for coarsening of the mesh of the wheel and the crossing in regions with sufficient distance from the contact zone, see Fig. 4. In the contact region care was given for the elements to be of good quality, while less effort was spent outside the contact region and distorted elements were accepted. Various algorithms are available to study the contact situation. In ABAQUS/Standard, contact between the two bodies is defined using a strict “master–slave” algorithm. The two surfaces that may interact are called a “contact pair” which consists of a “master” surface and a “slave” surface. For each node of the “slave” surface the closest point on the “master” surface is calculated where the normal to the “master” surface passes through the node on the “slave” surface. Between this point on the “master” surface and the corresponding “slave” node the interaction is then discretized. The “slave” nodes are constrained so they cannot penetrate the “master” surface. As the nodes of the “master” surface are in principle able to penetrate the “slave” surface, special attention has to be paid to the order in which the two surfaces are specified. To enforce the contact constraints, the Lagrange multiplier method is used [18]. Any arbitrary motion of the surfaces is allowed for by the contact
3.2. Contact In the case of conformal or two-point contact, the Hertzian contact model may constitute a source of error. Unlike Hertz solution, Kalker’s programme CONTACT is able to deal with arbitrary surface geometries of the two contacting bodies, which results in a non-elliptical contact patch. The program CONTACT is based on the boundary element (BE) method [17]. Rough surfaces can be considered since an arbitrary discretization of the potential contact region is possible. The potential contact region is divided into a number of rectangular elements of equal size. The exact position of the origin of the contact does not need to be known in advance. Here, the potential contact region is discretized with rectangular elements with 63 elements in the x-direction and 59 elements in the y-direction. The element size is 0.5 mm × 0.2 mm. The magnitude of the normal pressure is positive in the contact area, and it is equal to zero outside of it.
(5)
Fig. 3. Finite element mesh and geometry of the whole analysed model.
M. Wiest et al. / Wear 265 (2008) 1439–1445
Fig. 4. Contact region of the finite element mesh at the studied section of the crossing nose.
1443
formulation. Frictionless contact is assumed for the FE model to simplify comparisons with Hertz and CONTACT. Only the normal contact response is compared. To restrain and move the two parts of the system, the crossing and the wheel are each connected to a steering point by use of rigid connections. Boundary conditions are restricting the movement of the crossing part in vertical and longitudinal directions and by a horizontal spring with spring stiffness 600 MN/m in the lateral direction, see Fig. 3. The vertical track spring included in the MBS model is omitted in the FE model as GENSYS does not consider vertical displacements of the rail when contact point locations are determined. Further, the wheel axle is oriented horizontally in the FE model as GENSYS does not include the roll angle of the wheel, i.e. the angle between the wheel axle and the horizontal plane, when finding the contact point location. The yaw angle of the wheel, which is defined as the angle between the wheel axle and the plane normal to the track centreline in the actual position of the wheelset, is neglected because its influence on the selected cross-section is not significant. The steering point of the wheel is located at the centre of the wheel axle. Loads, displacements and velocities, received from the MBS software, can be applied to this point to guide the wheel.
Fig. 5. Contact area computed using: (a) Hertz, (b) CONTACT, (c) the elastic FE method and (d) the elastic–plastic FE method.
1444
M. Wiest et al. / Wear 265 (2008) 1439–1445
Table 1 Material properties
4. Results and discussion
Property
Manganese steel Mn13 (crossing)
Standard steel (wheel)
Young’s modulus (GPa) Poisson’s ratio Density (kg/m3 ) Yield stress (MPa)
190 0.3 7800 360
210 0.3 7800 –
In the present investigation the wheel is not rolled along the crossing. Instead, the wheel is positioned at one given contact point location calculated by GENSYS for the chosen cross-section and loaded with the normal contact force received from GENSYS for this contact point. The final centre of the contact patch that is calculated by the FE simulation can deviate slightly from the given initial contact point location. Table 1 summarizes the properties for the materials included in the FE model. Manganese steel Mn13 is used for the crossing. Typical for this kind of steel is the very low yield stress and an extremely good hardening behaviour. The elastic–plastic material behaviour of Mn13 is realised by using the von Mises plasticity model. For the elastic case, only the elastic properties of Mn13 in Table 1 are used. In all calculations, the wheel is modelled with the properties of elastic standard steel.
In this section, results from Hertz, CONTACT, the elastic FE method and the elastic–plastic FE method are compared and discussed. The considered rail cross-section in the crossing nose, see Fig. 2, is selected between section 19 and section 23, see Fig. 1. According to the MBS simulation, the wheel is in one-point contact with the crossing nose at this section, i.e. the wheel has already transferred from the wing rail to the crossing nose. The calculated normal contact force when the wheelset passes the chosen rail cross-section is 111.5 kN. The elastic half-space assumption is violated for this case of contact: The selected cross-section has a sharp curvature at the rail head, and the wheel-rail contact point calculated in GENSYS is positioned close to the corner of the crossing rail and on the transition between tread and flange on the wheel, see Fig. 2. Further, the lateral radii of curvature at the contact point are only about 13 mm on the rail and 18 mm on the wheel. No variation of the contact radius occurs within the contact area. To compare the maximum deformation of the four models, induced by the contact between wheel and crossing, the penetration depth is chosen as representative variable. The penetration depth is defined as the approach of the centres of the two contacting bodies, where in the case of Hertz and CONTACT it is the maximum
Fig. 6. Contact pressure distribution calculated using: (a) Hertz, (b) CONTACT, (c) the elastic FE method and (d) the elastic–plastic FE method.
M. Wiest et al. / Wear 265 (2008) 1439–1445 Table 2 Contact area, maximum contact pressure and maximum deformation (penetration depth) Contact model
Contact area (mm2 )
Max. contact pressure (MPa)
Penetration depth (mm)
Hertz CONTACT Elastic FEM Elastic–plastic FEM
65 67 70 128
2566 2555 2561 1085
0.1447 0.1421 0.1431 0.1710
deformation within the contact patch. For the Hertz contact theory the penetration depth can be taken from Eq. (9). To obtain the penetration depth for the FE method, first the relative displacement of the centres of the contacting spheres is calculated by subtracting the rigid displacement from the total displacement in the contact point for both contacting bodies. These two resulting values are then summed up to get the penetration depth. For the four investigated contact models, the contact area, the maximum contact pressure and the penetration depth are listed in Table 2. The calculated contact areas are plotted in Fig. 5. In Figs. 5 and 6, the x-axis is parallel to the longitudinal direction (direction of travel) and the y-axis is parallel to the transverse direction of the crossing. The results of Hertz, CONTACT and the elastic FE method are very similar both with respect to the shape of the contact area and the pressure distribution, see Figs. 5 and 6. The area calculated by Hertz method is elliptical in shape due to theory. CONTACT delivers a shape of the contact patch that is very close to an ellipse. The contact patch from the elastic FE method is in between an elliptical and a rectangular shape. However, results from the elastic–plastic FE method deviate significantly from the results of the other methods. The contact area is almost twice as large as the areas of the other three methods, and the a and b axes are 25% and 75% longer, respectively. The contact pressure distributions of the four contact models are shown in Fig. 6. CONTACT calculates slightly lower contact pressures compared with the results from Hertz (about −0.5%) and the elastic FE method (about −0.3%). The shape of the contact pressure distribution is ellipsoidal for Hertz and close to an ellipsoid for CONTACT. For the elastic FE method, the shape shows some deviation from the ellipsoidal shape although nominal (unworn) profiles of wheel and rail are used in the calculations. The magnitude of the maximum contact pressure, when using plastic material behaviour in the elastic–plastic FE simulation, is reduced to 42% of the corresponding result from the elastic FE method. Regarding the penetration depth, the value delivered by the elastic FE method is 0.1431 mm, which lies between the values from Hertz and CONTACT. The penetration depth resulting from the elastic–plastic FE method is 20% higher (0.1710 mm). 5. Conclusions Calculations using four different methods for modelling of wheel-rail rolling contact on a crossing nose have been carried out. The contact pressure distributions from the two half- space models agree well with the finite element model for the case when elastic material behaviour is assumed for the wheel and the crossing nose. For the investigated contact point, the minimum radius of the surface curvature of the two contacting bodies is 13 mm, which is very close to the largest semi-axis of the contact patch. It is surprising that methods based on the half-space assumption seem to work well although the assumptions are violated. The same observation was made by Yan and Fischer [2].
1445
However, significant discrepancies occur when the half-space models are compared to the elastic–plastic FE method. Then, the maximum value of the contact pressure is 42% of the value when using the elastic FE method. The contact pressure distribution in the elastic–plastic case is almost constant in the central part of the contact patch. These results show the importance of using elastic–plastic calculations for studies of contact stresses in the crossing panel. Further investigations on wear, damage and fracture make it necessary to consider results which are not available from the MBS calculation, such as plastic strain and elastic slip. On the other hand, the computer power presently available does not allow for modelling of the whole turnout with finite elements having appropriate meshes. Thus, the combination of the two methods seems to be a promising approach to obtain input data necessary for estimations of component life. The authors are pursuing this approach. Although this in itself is a computationally expensive procedure, there remains the task of including the effects of elastic structure deformations of wheelset and track, which have been shown in [19] to have a considerable influence on the stick/slip behaviour in the wheel-rail contact zone. Acknowledgments ¨ ¨ Financial support by the Osterreichische Forschungsforderungsgesellschaft mbH, the Province of Styria, the Steirische ¨ Wirtschaftsforderungsgesellschaft mbH and the Municipality of Leoben, within research activities of the Materials Center Leoben under the frame of the Austrian Kplus Competence Center Programme, is gratefully acknowledged. Part of this work was performed as an activity within CHARMEC (CHAlmers Railway MEChanics). Mr Erich Scheschy of VAE provided input data on rail cross-sections in the turnout. References [1] J.J. Kalker, Computational Contact Mechanics of the Wheel-Rail System, Rail Quality and Maintenance for Modern Railway Operation, Kluwer Academic Publishers, The Netherlands, 1993, pp. 151–164. [2] W. Yan, F.D. Fischer, Arch. Appl. Mech. 70 (2000) 255–268. [3] T. Telliskivi, U. Olofsson, Proc. Instn. Mech. Eng., Part F. J. Rail Rapid Transit. 215 (2001) 65–72. [4] T. Telliskivi, U. Olofsson, Wear 257 (2004) 1145–1153. [5] M. Pau, F. Aymerich, F. Ginesu, Wear 253 (2002) 265–274. [6] A. Sladkowski, M. Sitarz, Wear 258 (2005) 1217–1223. [7] GENSYS user’s manual, 2004, Release 0403, www.gensys.se. [8] E. Kassa, Simulation of Dynamic Interaction between Train and Turnout, Lic. Thesis, Chalmers University of Technology, Department of Applied Mechanics, Gothenburg, Sweden, 2004, 56 pp. [9] M. Wiest, W. Daves, F.D. Fischer, H. Ossberger, ZEVrail Glas. Ann. 129 (11–12) (2005) 461–467 (in German). [10] E. Kassa, C. Andersson, J.C.O. Nielsen, Veh. Syst. Dyn. 44 (3) (2006) 247–258. [11] Abaqus, ABAQUS User’s Manual, version 6.5, 2005, www.abaqus.com. [12] J.J. Kalker, Veh. Syst. Dyn. 8 (4) (1979) 317–358. [13] M. Wiest, W. Daves, F.D. Fischer, Four different numerical approaches to calculate strains and stresses during impact in wheel-rail rolling contact, in: Proceedings of World Tribology Congress III, Washington, D.C., USA, 2005, pp. 327–328. [14] M. Wiest, W. Daves, F.D. Fischer, H. Ossberger, PAMM 5 (2005) 67–70. [15] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985. ¨ [16] G. Lundberg, H. Sjovall, Stress and Deformation in Elastic Contacts, Chalmers University of Technology, Gothenburg, Sweden, 1958. [17] J.J. Kalker, Three-dimensional Elastic Bodies in Rolling Contact, Kluwer, Dordrecht, The Netherlands, 1990. [18] T.A. Laursen, Computational Contact and Impact Mechanics, Springer, Berlin, Germany, 2002. [19] X. Jin, P. Wu, Z. Wen, Wear 253 (2002) 247–256.
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Assessment of methods for calculating contact pressure in wheel-rail/switch contact M. Wiest a,b , E. Kassa c,∗ , W. Daves a,b , J.C.O. Nielsen c , H. Ossberger d a
Materials Center Leoben Forschung GmbH, Franz-Josef-Straße 13, A-8700 Leoben, Austria Institute of Mechanics, Montanuniversit¨ at Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria c CHARMEC/Department of Applied Mechanics, Chalmers University of Technology, SE-41296 Gothenburg, Sweden d VAE GmbH, Alpinestraße 1, A-8740 Zeltweg, Austria b
a r t i c l e
i n f o
Article history: Accepted 14 February 2008 Available online 27 May 2008 Keywords: Wheel-rail contact Crossing Contact pressure and contact forces Finite element modelling Multibody dynamics
a b s t r a c t Two models for wheel-rail rolling contact that are based on the half-space assumption are compared in this paper: Hertz and the non-Hertzian method implemented in the computer program CONTACT. These two models are further restricted by the assumption of linear-elastic material behaviour. Moreover, one elastic and one elastic–plastic finite element model of the contact are investigated with the commercial code ABAQUS. The finite element method is not limited by the half-space assumption or applicable to a linear-elastic material model only. The objective is to assess the four methods based on calculated contact pressure, contact patch size and penetration depth. Contact loads and contact locations, used as input data in the analysis, are taken from a vehicle dynamics simulation in the software GENSYS. The comparison is performed at a given cross-section in the crossing panel of a selected turnout design. To fulfil the requirements of the half-space assumption, the dimensions of the contact area must be small compared to the radii of curvature of the bodies in contact. On the selected cross-section, however, the half-space assumption does not hold since the smallest radius of rail curvature at the contact point is 13 mm, which is comparable to the largest semi-axis of the contact area. Nevertheless, it is found that the contact pressure distributions calculated using Hertz and CONTACT correlate well with those results obtained from the finite element method as long as no plastification of the material occurs. © 2008 E. Kassa. Published by Elsevier B.V. All rights reserved.
1. Introduction Several numerical tools are available to study rolling contact problems. The wheel-rail rolling contact problem includes elastic and plastic contact of two bodies with similar material properties. A survey of computational methods for studies of contact mechanics, including elastic contact problems and plastic rolling contact theory, is performed by Kalker [1]. In the literature, a couple of attempts exist to analyse discrepancies and conformities between the different methods. As various authors arrive at different conclusions regarding the agreement or disagreement of results between the several available calculation methods, the authors of the present paper perform calculations with four different methods to receive a conclusive result. Yan and Fischer studied the applicability of Hertz contact theory to wheel-rail contact problems for a standard rail, a crane rail and a crossing nose in [2]. They found that for elastic material prop-
∗ Corresponding author. Tel.: +46 31 772 1499; fax: +46 31 772 3827. E-mail address: [email protected] (E. Kassa).
erties, and if the surface curvature does not change within the contact area, the contact pressure distributions from their threedimensional finite element (FE) calculations agreed well with those from Hertz contact theory. Their results are especially interesting as the agreement between the elastic FE method and Hertz theory remained for cases where the half-space assumption was violated. In addition, Telliskivi and Olofson [3,4] used the FE method to investigate the contact conditions for measured wheel-rail profiles of an ordinary track. Two methods based on the half-space assumption, Hertz analytical method and Kalker’s program CONTACT, were compared with the FE method for two test cases. For the case where the contact condition satisfied the half-space assumption, the difference in maximum contact pressure between the FE method and Hertz/CONTACT was negligible, but large discrepancies were observed for the case where the radius of one of the profiles was small compared to the dimensions of the contact area. Pau et al. [5] investigated the distribution of contact pressure in the wheelrail contact area by an ultrasonic technique. Substantial agreement between their results and Hertz theory was found, although significant differences were observed in terms of regularity of the shape of the contact pressure ellipsoid. Good conformity of contact stresses
0043-1648/$ – see front matter © 2008 E. Kassa. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2008.02.039
1440
M. Wiest et al. / Wear 265 (2008) 1439–1445
Fig. 1. Example of sampled rail cross-sections in the crossing panel. Dimensions are in millimeters.
between finite element simulations and a quasi-Hertz approach was found by Sladkowski and Sitarz for elastic calculations in [6]. In this paper, vehicle motion, magnitudes of the contact forces and locations of the contact patches are predicted for a vehicle that runs through a crossing using the multibody system (MBS) software GENSYS [7]. Then, the contact loads and contact locations are used as input data in the contact pressure analysis. A comparison of four contact models is performed for a rail cross-section in the crossing panel of a turnout, see Fig. 1. The crossing panel is the part of the turnout which allows two tracks to cross each other at the same level. To enable wheel flanges to pass the intersecting rails, the continuous rails are interrupted at the crossing point. Crossings are designed so that the wheel should be continuously supported throughout the crossing, giving a smooth transition. However, this is only true for ideal conditions and with a new wheel running over the crossing. In reality, when a wheelset with worn wheels passes the crossing, high impact loads may arise [8,9]. Consequences are damage, fracture and wear of both the crossing and the wheels leading to high maintenance costs. Therefore, investigations of dynamics and damage in a crossing are of great interest. The basis for the prediction of component life is an adequate determination of stresses and strains. Even with the computer power available today, this still remains a challenging task. The present paper is starting a promising approach in this direction. Two contact models based on the half-space assumption are compared: Hertz and the non-Hertzian method implemented in the computer program CONTACT. Moreover, one elastic and one elastic–plastic finite element model of the contact are investigated. The objective is to assess the four methods based on the results of calculated contact pressures, contact patch sizes and penetration depths for a selected cross-section where the half-space assumption is violated. The future goal of the authors work is to develop an interface between the MBS model and the FE model to be able
to use results from the MBS simulation as input to the FE analysis. By performing the present comparison for a selected cross-section, we further succeeded in figuring out the reasons for possible problems which can arise when driving a finite element model using results from a multibody system simulation, e.g. different contact point locations resulting from the two algorithms. During the passage of the wheel over the crossing, situations with several simultaneous contact points between wheel and double rail arrangements, e.g. wing rail with through rail and stock rail with check rail, are possible. Two contact points on the wheel are common at the crossing, with one contact point with the wing rail and the other with the crossing nose [8,10]. At some stage, two contact points may occur on the crossing nose. The contact point locations in crossings are extremely sensitive to the displacements and rotations of the two bodies in contact. The FE method and the MBS method are based on quite different algorithms for locating the contact points. In contrast to the FE method, in most MBS analyses, deformations of the wheel and the rail are not taken into account. Therefore dissimilar results may arise when comparing the MBS and the FE method. A correct interpretation of the discrepancies is necessary when the results are used for damage, wear and fracture investigations. The described problem is most significant for applications in the switch and crossing panels, and it does not seem to appear when the method of driving a FE analysis with results from a MBS simulation is applied for a normal track where there are two standard rails on each side of the track centreline. 2. Vehicle-track interaction Simulation of dynamic interaction between train and railway turnout involves modelling of vehicle, track and wheel-rail contact. A freight vehicle travelling along the standard UIC60-760-1:15 turnout geometry (curve radius 760 m, turnout angle 1:15) in the
M. Wiest et al. / Wear 265 (2008) 1439–1445
facing move is considered. Rail cross-sections are sampled at several locations along the turnout and are used as input, see Fig. 1. The numerical simulation is performed using the commercial MBS software GENSYS. The specific features of a turnout, such as rails with no inclination, check rail and variations in rail cross-section, are considered in the simulation [8,10]. To reduce differences in the GENSYS and FE models, the track model in GENSYS is taken as almost rigid with a track stiffness of 600 MN/m both in vertical and lateral directions. The corresponding track damping coefficients amount to 250 kNs/m. The track model is rigid in the longitudinal direction. This enables a fair comparison of calculated contact pressures from Hertz, CONTACT and the FE models. The contact geometry problem is solved in advance. The precalculated contact point functions are collected in tables and used in the subsequent time integration analysis. The elastic contact problem is solved based on Hertz theory for the normal contact and Kalker’s simplified theory (FASTSIM) for the tangential contact problem. In these solutions, several simplifications are necessary to arrange fast calculations, and plastic deformations in and near the contact zone are not considered. Despite the fact that assumptions made in Hertz theory are often violated, this approach is normally adopted in vehicle-track interaction simulations because it is simple and fast. The magnitude and orientation of the normal contact force depends on the curvature difference of the two contacting surfaces and the contact angle at the contact point. Often a small shift in the location of the contact point is significant since this could lead to major differences in the contact geometry. To determine the elastic deformation in the contact, the contact geometry and the relative displacement between the wheel and the rail are used. The deformation ın normal to the contact plane is computed at the contact point. Then, the normal force can be calculated based on the linear Hertzian contact stiffness CH . The relation between the normal force Fn and the normal deformation ın is given by
Fn =
CH ın , if ın > 0 . 0, else
1441
Fig. 2. Contact point location on the sampled rail cross-section and wheel crosssection.
faces and that no plastic deformation occurs in the contact patch. However, for the general problem, including situations of conformal contact and plasticity, the FE method is the only available tool. FE models are able to describe the contact behaviour of two bodies in greater detail, including their real geometry without the need of making any assumptions on constant curvature, for example. Other major strengths of the FE method are the ability to describe accurately the inelastic deformation of the material and to calculate residual stresses and strains [2,9,13]. Multiple point contacts are also accounted for. Routines for investigation of damage can be included, such as demonstrated, e.g. in [14]. Further, monitoring of the material behaviour due to cyclic loading is possible. 3.1. Hertz contact theory
(1)
The coefficient CH depends on the contact geometry of the contact point and on the material properties of the two contacting bodies. A look-up table based on Hertzian formulae is used to determine this coefficient. Vehicle motion, magnitudes of forces and creepages in the wheel-rail contacts, and dimensions and locations of the contact patches are predicted when the vehicle reaches the turnout crossing. The resulting displacements of the leading wheelset of the leading bogie for the time during which the wheel passes the crossing nose are used as input to the subsequent FE analysis in the commercial code ABAQUS [11]. Based on contact forces and contact positions from GENSYS, a detailed finite element investigation of the wheel-rail/switch contact is performed. Here, a comparison of contact mechanics methods is carried out for a rail profile located in the crossing panel of the turnout at the time when the wheel is passing it, see Fig. 2. 3. Wheel-rail contact models Based on the elastic half-space assumption, a number of theories have been derived for the normal wheel-rail contact problem [12]. Wheel and rail cross-sections are profiled, and the normal and tangential contact stresses are highly concentrated on the contact region. Hence, contact stresses can be approximated by considering the wheel and the rail as semi-infinite elastic solids, i.e. elastic half-spaces. This means that the dimensions of the contact patch are small compared to the radii of curvature of the contacting sur-
In Hertz contact theory, due to the elastic deformation of the two profiled bodies, the common surface will attain an elliptic boundary [15]. Hertz contact theory is based on a number of assumptions. The two bodies in contact are regarded as half-spaces, where the contact area is small compared to the minimum dimensions of the bodies in contact. In addition, no plastic deformation in the contact patch is assumed, and the radii of curvature of wheel and rail profiles in the contact patch are assumed to be constant. It is clear that these are rough assumptions in cases of conformal contact and two-point contacts, which are common in turnout applications. For example, for contacts near the gauge corner of the crossing rail, there is a significant variation of the wheel and rail profile curvatures within the contact patch and the dimensions of the contact patch may not be small compared to the radii of curvature of the contacting surfaces. In some cases, the half-space assumption is not valid when contact occurs close to the gauge corner. In this case, a non-elliptical contact area may ensue. According to Hertz theory, the normal pressure is distributed as an ellipsoid over the elliptic contact area with semi-axes a and b. The ellipsoidal contact pressure distribution p(x,y) is expressed by
3Fn p(x, y) = 2ab
1−
x 2 a
−
y 2 b
,
(2)
where Fn is the dynamic normal contact force obtained from the GENSYS simulation, while x and y are local coordinates with the origin at the centre of the contact ellipse. The x coordinates are bounded by the semi-axis a, |x| ≤ a, and the y coordinates are bounded by the semi-axis b, |y| ≤ b. The maximum pressure is at
1442
M. Wiest et al. / Wear 265 (2008) 1439–1445
the centre of the contact ellipse. The semi-axes of the contact area are calculated as 1/3
a = ah Fn b=
,
(3)
1/3 bh Fn ,
(4)
where ah and bh are parameters that depend on the contact geometry and on the material properties of the two contacting surfaces. The two bodies that are in contact are assumed to have identical material properties with Young’s modulus 200 GPa and Poisson’s ratio 0.3. ¨ According to the method in Lundberg and Sjovall [16], the contact geometry is defined by a parameter . This parameter is given by
1
2
1 1 − + = r rx1 ry1 1 1 1 1 1 + + + = r rx1 rx2 ry1 ry2
1 1 − rx2 ry2
2
+2
1 1 − rx1 ry1
(A2 + 1)E − 2K , (A2 − 1)E
(6)
where E and K are the complete elliptic integrals with the modulus k2 = 1 − A−2 . The semi-axes of the contact area, a and b, are then given as
a=
3A2 E(1 − ) G
b=
1/3
· Fn
,
(7)
,
(8)
1/3
3E(1 − ) AG
1/3
1/r
1/3
1/r
· Fn
where G is the shear modulus and is Poisson’s ratio. The elastic deformation (penetration depth) at the centre of the contact patch is computed by ı=
K
1/r
2A2 E
1/3
3(1 − ) 2G
2/3
2/3
· Fn
.
3.3. Finite element model Detailed investigations on stresses and strains in the contacting bodies require a very dense mesh when using the finite element method. Thus, because of limited computational power, only sections of the wheel and the crossing are modelled, see Fig. 3. The length of the sections of both wheel and crossing is 250 mm, while
1 1 − rx2 ry2
where rx1 and ry1 are the principal radii of curvature in the x- and y-directions of the first body, and rx2 and ry2 are the principal radii of curvature in the x- and y-directions of the second body. The angle between the principal planes is denoted ϕ. The ratio of the semi-axes, A = a/b, depends on the curvature of the two bodies in contact and on the angle between the principal axes. To determine the semi-axes of the contact area, first the ratio of the semi-axes A is determined by the following equation =
However, like the Hertz contact solution, the CONTACT algorithm is based on the half-space assumption. Wheel and rail are considered as elastic half-spaces. In addition, CONTACT is restricted to elastic problems, which means that plastic deformation in the contact patch is not accounted for.
(9)
⎫ ⎪ ⎪ cos 2ϕ ⎬ ⎪ ⎪ ⎭
,
the depth of the sections at the contact zone is 35 mm for the crossing and 30 mm for the wheel. Both bodies are modelled with a very fine mesh in the contact zone with an element size on the surface of 0.5 mm × 0.5 mm. Altogether the model consists of 110,000 eightnoded linear hexahedral solid elements with reduced integration. Each element has 24 degrees of freedom. The total number of nodes is 130,000. “Tied contact”-constraints are used for coarsening of the mesh of the wheel and the crossing in regions with sufficient distance from the contact zone, see Fig. 4. In the contact region care was given for the elements to be of good quality, while less effort was spent outside the contact region and distorted elements were accepted. Various algorithms are available to study the contact situation. In ABAQUS/Standard, contact between the two bodies is defined using a strict “master–slave” algorithm. The two surfaces that may interact are called a “contact pair” which consists of a “master” surface and a “slave” surface. For each node of the “slave” surface the closest point on the “master” surface is calculated where the normal to the “master” surface passes through the node on the “slave” surface. Between this point on the “master” surface and the corresponding “slave” node the interaction is then discretized. The “slave” nodes are constrained so they cannot penetrate the “master” surface. As the nodes of the “master” surface are in principle able to penetrate the “slave” surface, special attention has to be paid to the order in which the two surfaces are specified. To enforce the contact constraints, the Lagrange multiplier method is used [18]. Any arbitrary motion of the surfaces is allowed for by the contact
3.2. Contact In the case of conformal or two-point contact, the Hertzian contact model may constitute a source of error. Unlike Hertz solution, Kalker’s programme CONTACT is able to deal with arbitrary surface geometries of the two contacting bodies, which results in a non-elliptical contact patch. The program CONTACT is based on the boundary element (BE) method [17]. Rough surfaces can be considered since an arbitrary discretization of the potential contact region is possible. The potential contact region is divided into a number of rectangular elements of equal size. The exact position of the origin of the contact does not need to be known in advance. Here, the potential contact region is discretized with rectangular elements with 63 elements in the x-direction and 59 elements in the y-direction. The element size is 0.5 mm × 0.2 mm. The magnitude of the normal pressure is positive in the contact area, and it is equal to zero outside of it.
(5)
Fig. 3. Finite element mesh and geometry of the whole analysed model.
M. Wiest et al. / Wear 265 (2008) 1439–1445
Fig. 4. Contact region of the finite element mesh at the studied section of the crossing nose.
1443
formulation. Frictionless contact is assumed for the FE model to simplify comparisons with Hertz and CONTACT. Only the normal contact response is compared. To restrain and move the two parts of the system, the crossing and the wheel are each connected to a steering point by use of rigid connections. Boundary conditions are restricting the movement of the crossing part in vertical and longitudinal directions and by a horizontal spring with spring stiffness 600 MN/m in the lateral direction, see Fig. 3. The vertical track spring included in the MBS model is omitted in the FE model as GENSYS does not consider vertical displacements of the rail when contact point locations are determined. Further, the wheel axle is oriented horizontally in the FE model as GENSYS does not include the roll angle of the wheel, i.e. the angle between the wheel axle and the horizontal plane, when finding the contact point location. The yaw angle of the wheel, which is defined as the angle between the wheel axle and the plane normal to the track centreline in the actual position of the wheelset, is neglected because its influence on the selected cross-section is not significant. The steering point of the wheel is located at the centre of the wheel axle. Loads, displacements and velocities, received from the MBS software, can be applied to this point to guide the wheel.
Fig. 5. Contact area computed using: (a) Hertz, (b) CONTACT, (c) the elastic FE method and (d) the elastic–plastic FE method.
1444
M. Wiest et al. / Wear 265 (2008) 1439–1445
Table 1 Material properties
4. Results and discussion
Property
Manganese steel Mn13 (crossing)
Standard steel (wheel)
Young’s modulus (GPa) Poisson’s ratio Density (kg/m3 ) Yield stress (MPa)
190 0.3 7800 360
210 0.3 7800 –
In the present investigation the wheel is not rolled along the crossing. Instead, the wheel is positioned at one given contact point location calculated by GENSYS for the chosen cross-section and loaded with the normal contact force received from GENSYS for this contact point. The final centre of the contact patch that is calculated by the FE simulation can deviate slightly from the given initial contact point location. Table 1 summarizes the properties for the materials included in the FE model. Manganese steel Mn13 is used for the crossing. Typical for this kind of steel is the very low yield stress and an extremely good hardening behaviour. The elastic–plastic material behaviour of Mn13 is realised by using the von Mises plasticity model. For the elastic case, only the elastic properties of Mn13 in Table 1 are used. In all calculations, the wheel is modelled with the properties of elastic standard steel.
In this section, results from Hertz, CONTACT, the elastic FE method and the elastic–plastic FE method are compared and discussed. The considered rail cross-section in the crossing nose, see Fig. 2, is selected between section 19 and section 23, see Fig. 1. According to the MBS simulation, the wheel is in one-point contact with the crossing nose at this section, i.e. the wheel has already transferred from the wing rail to the crossing nose. The calculated normal contact force when the wheelset passes the chosen rail cross-section is 111.5 kN. The elastic half-space assumption is violated for this case of contact: The selected cross-section has a sharp curvature at the rail head, and the wheel-rail contact point calculated in GENSYS is positioned close to the corner of the crossing rail and on the transition between tread and flange on the wheel, see Fig. 2. Further, the lateral radii of curvature at the contact point are only about 13 mm on the rail and 18 mm on the wheel. No variation of the contact radius occurs within the contact area. To compare the maximum deformation of the four models, induced by the contact between wheel and crossing, the penetration depth is chosen as representative variable. The penetration depth is defined as the approach of the centres of the two contacting bodies, where in the case of Hertz and CONTACT it is the maximum
Fig. 6. Contact pressure distribution calculated using: (a) Hertz, (b) CONTACT, (c) the elastic FE method and (d) the elastic–plastic FE method.
M. Wiest et al. / Wear 265 (2008) 1439–1445 Table 2 Contact area, maximum contact pressure and maximum deformation (penetration depth) Contact model
Contact area (mm2 )
Max. contact pressure (MPa)
Penetration depth (mm)
Hertz CONTACT Elastic FEM Elastic–plastic FEM
65 67 70 128
2566 2555 2561 1085
0.1447 0.1421 0.1431 0.1710
deformation within the contact patch. For the Hertz contact theory the penetration depth can be taken from Eq. (9). To obtain the penetration depth for the FE method, first the relative displacement of the centres of the contacting spheres is calculated by subtracting the rigid displacement from the total displacement in the contact point for both contacting bodies. These two resulting values are then summed up to get the penetration depth. For the four investigated contact models, the contact area, the maximum contact pressure and the penetration depth are listed in Table 2. The calculated contact areas are plotted in Fig. 5. In Figs. 5 and 6, the x-axis is parallel to the longitudinal direction (direction of travel) and the y-axis is parallel to the transverse direction of the crossing. The results of Hertz, CONTACT and the elastic FE method are very similar both with respect to the shape of the contact area and the pressure distribution, see Figs. 5 and 6. The area calculated by Hertz method is elliptical in shape due to theory. CONTACT delivers a shape of the contact patch that is very close to an ellipse. The contact patch from the elastic FE method is in between an elliptical and a rectangular shape. However, results from the elastic–plastic FE method deviate significantly from the results of the other methods. The contact area is almost twice as large as the areas of the other three methods, and the a and b axes are 25% and 75% longer, respectively. The contact pressure distributions of the four contact models are shown in Fig. 6. CONTACT calculates slightly lower contact pressures compared with the results from Hertz (about −0.5%) and the elastic FE method (about −0.3%). The shape of the contact pressure distribution is ellipsoidal for Hertz and close to an ellipsoid for CONTACT. For the elastic FE method, the shape shows some deviation from the ellipsoidal shape although nominal (unworn) profiles of wheel and rail are used in the calculations. The magnitude of the maximum contact pressure, when using plastic material behaviour in the elastic–plastic FE simulation, is reduced to 42% of the corresponding result from the elastic FE method. Regarding the penetration depth, the value delivered by the elastic FE method is 0.1431 mm, which lies between the values from Hertz and CONTACT. The penetration depth resulting from the elastic–plastic FE method is 20% higher (0.1710 mm). 5. Conclusions Calculations using four different methods for modelling of wheel-rail rolling contact on a crossing nose have been carried out. The contact pressure distributions from the two half- space models agree well with the finite element model for the case when elastic material behaviour is assumed for the wheel and the crossing nose. For the investigated contact point, the minimum radius of the surface curvature of the two contacting bodies is 13 mm, which is very close to the largest semi-axis of the contact patch. It is surprising that methods based on the half-space assumption seem to work well although the assumptions are violated. The same observation was made by Yan and Fischer [2].
1445
However, significant discrepancies occur when the half-space models are compared to the elastic–plastic FE method. Then, the maximum value of the contact pressure is 42% of the value when using the elastic FE method. The contact pressure distribution in the elastic–plastic case is almost constant in the central part of the contact patch. These results show the importance of using elastic–plastic calculations for studies of contact stresses in the crossing panel. Further investigations on wear, damage and fracture make it necessary to consider results which are not available from the MBS calculation, such as plastic strain and elastic slip. On the other hand, the computer power presently available does not allow for modelling of the whole turnout with finite elements having appropriate meshes. Thus, the combination of the two methods seems to be a promising approach to obtain input data necessary for estimations of component life. The authors are pursuing this approach. Although this in itself is a computationally expensive procedure, there remains the task of including the effects of elastic structure deformations of wheelset and track, which have been shown in [19] to have a considerable influence on the stick/slip behaviour in the wheel-rail contact zone. Acknowledgments ¨ ¨ Financial support by the Osterreichische Forschungsforderungsgesellschaft mbH, the Province of Styria, the Steirische ¨ Wirtschaftsforderungsgesellschaft mbH and the Municipality of Leoben, within research activities of the Materials Center Leoben under the frame of the Austrian Kplus Competence Center Programme, is gratefully acknowledged. Part of this work was performed as an activity within CHARMEC (CHAlmers Railway MEChanics). Mr Erich Scheschy of VAE provided input data on rail cross-sections in the turnout. References [1] J.J. Kalker, Computational Contact Mechanics of the Wheel-Rail System, Rail Quality and Maintenance for Modern Railway Operation, Kluwer Academic Publishers, The Netherlands, 1993, pp. 151–164. [2] W. Yan, F.D. Fischer, Arch. Appl. Mech. 70 (2000) 255–268. [3] T. Telliskivi, U. Olofsson, Proc. Instn. Mech. Eng., Part F. J. Rail Rapid Transit. 215 (2001) 65–72. [4] T. Telliskivi, U. Olofsson, Wear 257 (2004) 1145–1153. [5] M. Pau, F. Aymerich, F. Ginesu, Wear 253 (2002) 265–274. [6] A. Sladkowski, M. Sitarz, Wear 258 (2005) 1217–1223. [7] GENSYS user’s manual, 2004, Release 0403, www.gensys.se. [8] E. Kassa, Simulation of Dynamic Interaction between Train and Turnout, Lic. Thesis, Chalmers University of Technology, Department of Applied Mechanics, Gothenburg, Sweden, 2004, 56 pp. [9] M. Wiest, W. Daves, F.D. Fischer, H. Ossberger, ZEVrail Glas. Ann. 129 (11–12) (2005) 461–467 (in German). [10] E. Kassa, C. Andersson, J.C.O. Nielsen, Veh. Syst. Dyn. 44 (3) (2006) 247–258. [11] Abaqus, ABAQUS User’s Manual, version 6.5, 2005, www.abaqus.com. [12] J.J. Kalker, Veh. Syst. Dyn. 8 (4) (1979) 317–358. [13] M. Wiest, W. Daves, F.D. Fischer, Four different numerical approaches to calculate strains and stresses during impact in wheel-rail rolling contact, in: Proceedings of World Tribology Congress III, Washington, D.C., USA, 2005, pp. 327–328. [14] M. Wiest, W. Daves, F.D. Fischer, H. Ossberger, PAMM 5 (2005) 67–70. [15] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985. ¨ [16] G. Lundberg, H. Sjovall, Stress and Deformation in Elastic Contacts, Chalmers University of Technology, Gothenburg, Sweden, 1958. [17] J.J. Kalker, Three-dimensional Elastic Bodies in Rolling Contact, Kluwer, Dordrecht, The Netherlands, 1990. [18] T.A. Laursen, Computational Contact and Impact Mechanics, Springer, Berlin, Germany, 2002. [19] X. Jin, P. Wu, Z. Wen, Wear 253 (2002) 247–256.