CERTIFICATION OF NUMERICAL DESIGN AND
STRENGTH CALCULATIONS OF RAIL VEHICLES
ELEMENTS
Sitarz M., Chruzik K.
Katedra Transportu Szynowego Politechniki Śląskiej Europejskie Centrum Doskonałości TRANSMEC
Abstract: Advantages arising from standardisation and certification in economic activity and in
particular in manufacture are plain. It is difficult to imagine large-scale production without any standards. This is also true in case of construction, both in the design and prototype testing stage. Manufacture of rail elements in common European market necessitates unification of design processes, numerical methods and software. These demands are related to minimisation of design time and costs and, first of all, safety of people and cargo.
1. INTRODUCTION
Certification of design process is closely related to standardisation of railway transport in European Union. Standardisation means introducing standards used in a given process – in this case, rail transport. On one hand, it leads to unification, and on the other, minimum acceptable levels are set. Unification of different elements of design and production should result in allowing for comparison of effects of different processes, e.g. numerical analyses. This is a huge challenge due to progressing globalisation, in particular globalisation of rail transport. In order to unify, it is necessary to adopt some common reference level – and this is what we call a standard, related to elements being investigated.
Advantages arising from standardisation and certification in economic activity and in particular in manufacture are plain. It is difficult to imagine large-scale production without any standards. This is also true in case of construction, both in the design and prototype testing stage. The number of manufacturers present at European market brings about the need for creation of unified numerical design methods.
Keeping in mind that 25 years ago the experimental tests of rail vehicles constituted c. 95 percent and numerical tests c. 5 % of total testing, and anticipating that in 10 years
Classification of information – more and less important
General theory - - mechanics FEM theorem User – selection of parts Choice of appropriate computer software Results Selection of significant parameters Idealisation I -physical model Idealisation II Useful parts Object information
simulation and strength testing using numerical methods will probably constitute 90% of total testing, the need for certification of numerical methods becomes obvious. The several issues here are certification of software and R&D centres.
2. ACCURACY OF TESTING AND NUMERICAL METHODS
Since numerical methods are more and more widely used in strength calculations for railway wheelsets (BONATRANS, LUCCHINI, KLW WHEELCO, GRIFFIN, VALDUNES) and since results of these calculations are the bases for different technical and economical decisions, the credibility of these methods is a very important issue. In particular, the errors and accuracy of these methods and how to get them down to acceptable levels are a big question.
The sources of errors in computer computations are diverse, for instance database set-up or programming. These kinds of errors are relatively easy to find and remove. However, errors resulting from badly defined procedures in the setup of computation model, the so-called discretisation errors, cannot be entirely removed. Complex research has been carried out, with the view of working out methods of evaluating discretisation errors, ways of pointing them out and limiting them [4].
While realising FEM possibilities and usefulness, it must be noted that using this method must be very carefully considered. It is an approximate method. The results are not related to real-life devices, but to their models. That is why there exists a significant difference between real-life problem and results of calculations based on the model (Fig.1). It is clear that in this last stage of modelling the software used is crucial to end results. The authors of presented work have analysed the railway wheelset wheels strength using three different computer programs (ANSYS, NASTRAN, COSMOS), with identical input data (geometry, material, boundary conditions, finite elements definition). The comparison of these three programs will in future make possible the selection of most appropriate program for discussed types of calculations.
The finite element method is an approximate method. If we assume that the program is free of hidden errors and that computations are done correctly, then there remains a question of discretisation errors. These are specific to the method and cannot be entirely removed. During computer analyses additional errors due to rounding of numbers also arise, depending on processor type.
Fig. 1. Graduation – different stages and simplifications on the way from
reality to calculation results Discretisation error depends mostly on the number of node parameters, used to describe the model [4]. Number of these parameters, denoted by N and called number of degrees of freedom (LSS), depends on the number of elements (LE), and if the surface under consideration is limited, then the error value will depend on the size of greatest element hmax, number of nodes (LW) and number of degrees of freedom in the node (LSSW). Both LW and LSSW depend on interpolating functions, e.g. with polynomials the degree of polynomial p is crucial. It can be stated that N=f(h,p). Since discretisation error should approach zero with N approaching infinity, then with N=f(h,p) this may take place in two cases:
lim e = 0 przy hmax 0 (1)
or
lim e = 0 przy pmax . (2)
In practice, N is increased not to infinity, but to a finite big number.
In mechanics, the functions describing data fields should be determined (e.g. stress, displacement). In FEM the parameters values are determined for a single node, and on that basis the appropriate functions within the element surface are determined. Therefore FEM calculation results relate to functions. Hence, function calculations errors are important in FEM method. Error measures must be determined here.
Let us consider function of a single variable. If u(x) is a precise descriptor and uh(x) is an approximate descriptor, then the error is related to different function values and may be defined as the difference u(x)-uh(x) or the distance between those two functions:
x u
x x
a,h
u max u u e h h (3)This type of error measure is called submetric and is the simplest of all measures – usually more complex measures are used.
In mechanics different data fields are considered, namely displacements, deformations and stress. If the components of these fields are denoted as u, e, , respectively, then the measure in relation to several fields may be defined as:
2 1 T 2 1 T 2 1 Tud , e e ed , d u u
(4) This error measure is rather inconvenient to use. It is easier to interpret relative error:U U U e h w (5)
which may be also expressed as a percentage error:
100 e
ew,p w (6)
Most often the energy error described via stress is calculated [4].
In our case the basis for error assessment will be a comparison of test results with numerical analysis results.
Nowadays issues related to accuracy of results obtained by numerical investigation and their precise defining are based on:
analysis of the phenomenon by experimental and numerical methods comparison of experimental and numerical analysis results
analysis of differences between results of experimental and numerical analysis drawing conclusions from the above analyses and modification of computation
systems.
3. SELECTED EXAMPLES OF MISTAKES COMMITED DURING
RAILWAY WHEELSET DESIGN PROCESS
Design of railway wheelsets using numerical methods must be considered not only by taking into account the software technicalities, but first of all practical knowledge and know-how of designers. At present big companies manufacturing railway wheelsets use different programs (ANSYS, NASTRAN, COSMOS etc.) with different features [1÷3]. The differences are in particular noticeable in case of strength calculation of wheels determining thermal and fatigue stresses.
It may be noted that strength issues related to railway wheelsets cover a wide range of problems. It must be emphasised that the recent ten years are characterised by accelerated research into the railway wheelset issues in the different phases of its service. The resultant progress is due to implementing of numerical computational methods, which are used in the strength analyses more often.
The advancement in the computers computational speed and the elaboration of complex software based on finite element method (FEM) and devoted to the railway industry demands, results in running calculations and simulations, which have not been previously possible.
There are many issues, which so far have been only experimentally/analytically investigated. The use of numerical methods limits or wholly eliminates the need for some tests or calculations.
The investigation time is therefore decreased, the complex test stands can be replaced with suitable software; hence, financial advantages are gained.
On the basis of the above analysis related to state-of-the-art in the railway wheelsets subject matter, we may conclude that computation by numerical methods might play significant role in each of the above issues.
It is certain that numerical calculations should be the starting point in the design, assembly and service stress analysis and thermal analysis of a material.
However, the numerical methods are saddled with errors due to the imperfect transformation of the real model into the virtual model. Still, if the investigation methods are used jointly, i.e. experimental tests are backed by numerical analysis, the results obtained may be close to reality.
Basing on the references, the present state-of-the-art of the numerical calculations of the railway wheelset wheels can be summarised as follows:
lack of universally accepted computational algorithm of railway wheels,
inadequate experimental confirmation of the correctness of computational procedures used at present,
absence of comparison of numerical calculations of railway wheels done with the help of different software,
discrepancies in set boundary conditions,
absence of precise algorithm for creating FEM model for railway wheels (type, distribution and size of elements),
complexity of stress calculation method, when the stress is due to assembly-time interference,
lack of UIC certification for calculations of thermal stress due to braking. The computer software used by leading R&D centres has been compared – Table 1.
Table 1. Comparison of software used in strain analysis of railway wheelsets
Parameter Method
NASTRAN ANSYS COSMOS 1. Facility of data preparation
creation of mesh input of material data
time-consumption
dependence on temperature
setting boundary conditions in displacements setting loads +/-+ +/-+ + +/-+ + +/-+ + +/-+ +/-+ +
+/-2. Rime consumption of: data preparation computation results visualisation +/-+ + +/-+ +/-+ 3. File import + +/- +/-4. Contact issues + + + 5. Static issues + + + 6. Dynamic issues + + +
7. Nonstationary heat flow - + +
8. Thermal stresses for a given temperature field + + +
9. Thermal stress issues due to braking time consumption + +/-+ +/-+
+/-10. Facility of accounting for assembly stresses
(interference) - -
-11. Facility of accounting for centrifugal forces + + +
12. Simple wear analysis - - +
13. Superposition of stresses and displacements + + +
where:
+ is present, - is not present, +/-is present in a limited degree.
Creation of calculation algorithm for railway wheels can be divided into two phases: – definition of correct numerical model of a railway wheel,
– working out of numerical analysis algorithms of different loads present during operation and their experimental verification.
3.1. Wheel disc stress
Analysis of the wheel disc model [1] shows that the minimum number of elements along the disc cross-section should be equal to 6. In order to prove that the proposed mesh density is correct, three different densities were examined: 2,6, and 12 elements per disc cross-section – Fig.2. PN920/185_s wheel was used in the analysis.
The analysis has shown that with the rough mesh (2 elements) the difference of results related to model mesh (6 elements) was equal at most to 20 MPa. Doubling mesh density
did not affect results greatly (max 7 Mpa discrepancy). However, calculation time was 9 times greater than in case of model mesh. The results are set out in Table 2 and Figs. 3,4,5.
a) b) c)
Fig. 2. Discrete models of wheels: a) rough mesh – 2 elements, b) model mesh – 6 elements, c) fine mesh – 12 elements in disc
Table 2. Analysis of the mesh density impact on calculation time and accuracy
Type of load
Wheel PN920/185_s Ø 920
2 elements in disc 6 elements in disc 12 elements in disc Stress red. [MPa] Analysis time [s] Stress red. [MPa] Analysis
time [s] Stress red.[MPa]] Analysistime [s] Max. axle load
22 500 kg/axle 207 15 227 450 234 5 400 Assembly interferential fit 0,21 mm 216 15 225 375 228 7 200 Centrifugal forces 160 km/h 10 15 11 330 11 5 400 Loads due to braking 2700 s, 30 kW 419 12 410 10 500 408 28 800
a) b) c)
Fig. 3. Analysis of the mesh density impact of calculation time and accuracy – carriage loaded with maximum load: a) rough mesh – 2 elements, b) model mesh – 6 elements, c) fine mesh – 12 elements in disc
a) b) c)
Fig. 4. Impact of mesh density on calculation time and accuracy – assembly interferential fit: a) rough mesh – 2 elements, b) model mesh – 6 elements, c) fine mesh – 12 elements in disc
a) b) c)
Fig. 5. Impact of mesh density on calculation time and accuracy – temperatures due to braking process: a) rough mesh – 2 elements, b) model mesh – 6 elements, c) fine mesh – 12 elements in disc
Fig. 6. Impact of mesh density on calculations accuracy
Fig. 7. Impact of mesh density on calculations time 0 50 100 150 200 250 300 350 400 450
Mass of the carriage Pressing on of wheels Centrifugal Applying of the brakes
-temperature
Applying of the brakes -stress Type of Loads R e s u lt s o f n u m e ri c a l c a lc u la ti o n s [ M P a ], [ C ] 2 elements in disc 6 elements in disc 12 elements in disc 0,1 1 10 100 1000 10000 100000
Mas s of the c arriage Pressing on of wheels Centrifugal Applying of the brakes -temperature
Apply ing of the brak es -stres s Type of Loads T im e o f c a lc u la ti o n s [ m in ] 2 elements in disc 6 elements in disc 12 elements in disc
Diagrams in Figs. 6 and 7 show comparison of results in respect to calculation time and accuracy.
These investigations have made possible generation of most advantageous mesh in respect to calculations time and accuracy.
3.2. Stress due to thermal loads
Assembly loads and service loads of the railway wheelset wheel have been determined and shown in Fig.8. These are static, dynamic and thermal loads. The static loads are due to interferential fit during the assembly and to carriage weight. The dynamic loads are related to the vehicle run over the track and centrifugal forces resulting from run speed. The thermal loads occur in the wheel during braking.
The numerical analysis has been conducted for ER7 material, which is used for railway wheels in Europe. The material properties have been selected in accordance with UIC Report 169.1 and research.
Fig. 8. Loading of railway wheelset wheel
The UIC reports quoted above are only recommendations for design process. The unequivocal definition of numerical and load models causes great discrepancy in results, up to 50 per cent (if quadrilateral and hex meshes are used) [1]. Additionally, some divergences occur during numerical analysis conducted with different software packages. A good instance may be analysis of loads during braking (brake shoe applied to wheel). In order to check the correctness of algorithm used to calculate temperature field, the comparison of numerical calculations with experimental test results has been run. The tests were conducted according to UIC 510-5 braking cycle – see Fig.9. Numerical thermal analysis has been done in accordance with UIC Report [5], wheel with maximum wear of rolling profile. The thermal flux has been determined for different braking stages according to Eq. 7. ] [W/m Ø d 7 , 0 000 30 HX 2 (7)
During simulated braking cycles of 2700 s duration each, cooling has been applied (14400 s) after each braking. The nodes temperatures have been noted after every cycle (nodes corresponded to thermocouples positions) – Fig.10. The result of the analysis was to determine average temperature in separate nodes for several braking cycles.
Fig. 9. Braking cycle according to UIC 510-5
Fig. 10. Node corresponding to temperature analysis location – marked in yellow In order to enhance visualisation of results they are shown in the form of a diagram – Fig.11.
Comparison of braking cycles test results and numerical analysis results leads to following conclusions:
- the least difference in temperatures has been obtained in Package 1 (up to 5 per cent); - the difference for Packages 1 and 2 has been as great as 22 per cent.
Explanation of existing differences calls for separate work and co-operation with software-producing companies.
Fig. 11. Node temperatures (nodes correspond to thermocouples positions)
4. FINAL REMARKS AND CONCLUSIONS
4.1 R&D centres doing research use different FEM wheel models, different load models and different software – this leads to differences in computation procedures, making impossible comparison of numerical tests results.
4.2 Two given examples of FEM strength analysis of railway wheelsets wheels show that significant differences in calculation results may occur in design practice, which might affect safety of people and cargoes
4.3 The computer simulations in rail transport are becoming widespread. Hence it seems necessary to specify the rules for certification of software and research centres using this software. Some activity aimed at defining these rules is currently undertaken in Department of Railway Engineering of Silesian University of Technology, in the framework of European grant EURNEX.
5. REFERENCES
[1]. M. Sitarz, A. Sładkowski, K. Chruzik: Metody Numeryczne w projektowaniu kół kolejowych zestawów kołowych. Monografia, Wydawnictwo Politechniki Śląskiej Nr 60, Gliwice 2003, s.128
[2]. M. Sitarz, A. Sładkowski, K. Bizoń, K. Chruzik: Design and investigation of railway wheelsets. Rozdział w Monografii Railway Wheelsets, Wydawnictwo Politechniki Śląskiej Nr 59, Gliwice 2003, s. 21-61
[3]. M. Sitarz, K. Bizon, K. Churzik: Numerical – Experimental Strength Analysis of Wheels of Railway Wheelsets. W: 14 th International Wheelset Congress, 17 – 21 Październik 2004, CD
[4]. Rakowski G., Kacprzyk Z.: Metoda elementów skończonych w mechanice konstrukcji. Warszawa: Oficyna Wydawnicza Politechniki Warszawskiej, 1993 [5]. Raport ERRI B 169. Termische grenzen der raden und bremsklotze. MTEL P 98005
