SIST-TP CEN/TR 17469:2020
(Main)Railway applications - Axle design method
Railway applications - Axle design method
This document presents the stage of knowledge resulting from the Euraxles project about the design of the axle, and further steps to be taken.
It is the support:
- to define the loads to be taken into account;
- to describe the stress calculation method using finite elements and the validation processes associated;
- to specify the maximum permissible stresses to be assumed in calculations and the safety factors to be used.
This technical report is applicable for:
- wheelset Axles defined in EN 13261 as "pure wheelset";
- other axle designs such as those encountered in particular rolling stocks e.g. with independent wheels, variable gauges, urban rail...
This document has not for aim to replace EN 13103-1 and CEN/TS 13103-2 but to present a complementary method to the existing ones.
Bahnanwendungen - Konstruktionsverfahren von Radsatzwellen
Applications ferroviaires - Méthode de conception des essieux
Le présent document décrit l'état des connaissances acquises dans le cadre du projet EURAXLES en ce qui concerne la conception des essieux-axes, ainsi que les autres éléments à prendre en compte.
Il a pour objet de :
- définir les charges à prendre en compte ;
- décrire la méthode de calcul des contraintes selon la méthode des éléments finis, ainsi que les processus de validation associés ;
- spécifier les contraintes maximales admissibles à utiliser pour les calculs, ainsi que les coefficients de sécurité à appliquer.
Le présent Rapport technique est applicable aux :
- essieux montés définis dans l'EN 13261 ;
- autres conceptions d'essieux-axes rencontrées sur les matériels roulants particuliers, par exemple les essieux-axes équipés de roues indépendantes, à écartement variable ou empruntant le réseau ferroviaire urbain, etc.
Le présent document n'a pas pour objet de remplacer l'EN 13103 1 et la CEN/TS 13103-2, mais de décrire une méthode complémentaire aux méthodes actuelles.
Železniške naprave - Metoda načrtovanja osi
General Information
Standards Content (Sample)
SLOVENSKI STANDARD
01-junij-2020
Železniške naprave - Metoda načrtovanja osi
Railway applications - Axle design method
Bahnanwendungen - Konstruktionsverfahren von Radsatzwellen
Applications ferroviaires - Méthode de conception des essieux
Ta slovenski standard je istoveten z: CEN/TR 17469:2020
ICS:
45.040 Materiali in deli za železniško Materials and components
tehniko for railway engineering
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
CEN/TR 17469
TECHNICAL REPORT
RAPPORT TECHNIQUE
March 2020
TECHNISCHER BERICHT
ICS 45.040
English Version
Railway applications - Axle design method
Applications ferroviaires - Méthode de conception des Bahnanwendungen - Konstruktionsverfahren von
essieux Radsatzwellen
This Technical Report was approved by CEN on 24 February 2020. It has been drawn up by the Technical Committee CEN/TC
256.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,
Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and
United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2020 CEN All rights of exploitation in any form and by any means reserved Ref. No. CEN/TR 17469:2020 E
worldwide for CEN national Members.
Contents Page
European foreword . 4
Introduction . 5
1 Scope . 9
2 Normative references . 9
3 Terms, definitions, symbols and abbreviations . 9
3.1 Terms and definitions . 9
3.2 Symbols and abbreviations . 9
4 Loads . 11
4.1 Reliability analysis based on the Stress Strength Interference Analysis method . 11
4.2 Fatigue load analysis method . 13
4.2.1 General . 13
4.2.2 Load signals processing and Fatigue-Equivalent-Load . 13
4.2.3 Method to generate the distribution of in-service load severities . 19
4.3 Fatigue reliability assessment of a railway passenger coach axle . 23
4.3.1 Load measurements . 23
4.3.2 Load spectra classification and generation and distribution of load severity. 27
4.3.3 Estimation of the probability of a crack initiation . 31
5 Modelling . 33
5.1 General . 33
5.2 Stress concentration factors . 33
5.3 Length of the transition . 36
5.4 Numerical modelling of axles . 38
5.4.1 Development of numerical models and validation . 38
5.4.2 Analysis of mounted components . 42
5.4.3 Modelling recommendations . 43
5.5 Axle calculation method . 44
6 Fatigue limits . 45
6.1 Testing method principals . 45
6.1.1 F1 tests . 45
6.1.2 F4 tests . 46
6.1.3 Fatigue limit estimation . 46
6.2 Test plan . 47
6.3 Axle body fatigue limit results . 51
6.3.1 F1 standard surface – transitions and groves (EA4T axles) . 51
6.3.2 F1 Blasted surface – transitions (EA4T axles) . 52
6.3.3 F1 Standard surface – transitions (EA1N axles) . 53
6.3.4 F1 Corroded surfaces – transitions of unpainted axles . 54
6.4 Axle press-fit seat fatigue limits (F4) . 55
6.4.1 Diameter ratio = 1,12 (EA4T axles) . 55
6.4.2 Diameter ratio = 1,08 (EA4T) . 56
7 Safety factors. 57
7.1 Aims and problem statement. 57
7.2 Probabilistic fatigue assessment . 60
7.2.1 Failure probability under constant amplitude stress . 60
7.2.2 Fatigue damage under VA loading . 60
7.2.3 Bignonnet method . 61
7.3 Input data for probabilistic fatigue assessment of railway axles . 62
7.3.1 Definitions of reference S-N diagrams . 62
7.3.2 Miner Index at failure. 63
7.3.3 Target reliability and failure rate for railway axles . 66
7.4 Probabilistic fatigue damage calculations for railway axles . 66
7.4.1 Format for the calculations . 66
7.4.2 Montecarlo simulations . 67
7.4.3 Stress spectra . 67
7.5 Results . 68
7.5.1 General . 68
7.5.2 Safety factor and reliability under constant amplitude stress . 69
7.5.3 Safety factor for damage calculations . 70
8 Conclusions of Euraxles Project . 71
9 Recommendations of CEN TC256/SC2/WG11 . 75
Annex A (informative) Application example of the axle calculation method . 76
A.1 General . 76
A.2 General descriptions . 76
A.3 Load distribution . 77
A.4 Results according to EN 13103-1 . 78
A.5 Design of EURAXLES method . 80
A.6 Comparison of results . 82
Bibliography . 83
European foreword
This document (CEN/TR 17469:2020) has been prepared by Technical Committee CEN/TC 256 “Railway
applications”, the secretariat of which is held by DIN.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
Introduction
The first railway accident due to the fatigue failure of an axle occurred on 1842, May 8th, in France, near
Meudon, on the Versailles-Paris line.
In those days, the fatigue phenomenon was unknown. This failure initiated numerous studies including
German Engineer August WOHLER works on wheelset failures at the end of XIX century.
In the middle of XX century, M. KAMMERER, an engineer working for French railways, established the
bases for the calculation of wheelset axles.
At international level, the report ORE B136 RP11 « Calculation of fret wagon and passenger coaches’
wheelset axles » was edited in April 1979, using in particular the French approach.
This document allowed editing on 1994, July 1st of UIC leaflet 515-3 «Railway rolling stock – Bogie –
Running gears – Axle calculation method».
The first edition of the European Standards about design of axles occurs on April 2001 (EN 13103 for
non-powered axles for powered axles).
The ongoing European standardization has allowed the merging of EN 13103 in only one standard
(EN 13103-1 Railway applications – Wheelsets and bogies – Part 1: Design method for axles with external
journals) and the creation of a new Technical Specification about internal journal (CEN/TS 13103-2
Railway applications – Wheelsets and bogies – Part 2: Design method for axles with internal journals).
All these documents, including M. KAMMERER’s work up to EN 13103-1 and CEN/TS 13103-2, use the
beam theory calculation method. The stresses taken into account are then the nominal stresses. The
fatigue limit is determined from full scale tests in which nominal stresses are taken into account.
Concentration factors are defined from tests to consider the local geometry and to increase the nominal
stress locally. The method is quite simple, with no need of sophisticated calculations or dedicated
software.
On another hand, in the middle of XX century, the need in mechanics to have a tool to calculate
complicated parts lead to the development of the finite element method.
Along with the theoretical study of this method, the use of new mathematical objects and the growth of
calculation capacities of computers, the finite element method raised to a large and common use in
design.
The stresses then calculated are local stresses, and not anymore nominal stresses, and the fatigue limit
to be applied with this methodology are based on local stresses.
In the Euraxles project, the objective was to propose the use of a new assessment method based on load
measurements, finite element method, experimental fatigue limit and new safety concept for the design
of axles in particular for axle designs requiring more complex geometries. This design procedure is
different from today’s proven methods given by the EN standards and not in a status to substitute them.
Nevertheless, it was considered interesting to gather the Euraxles project results inside this Technical
Report. The content should be considered as partial and only for informative uses at this stage. For
example, the reliability of the input data, the variability of parameters, boundary conditions and the
confidence in the partial results should be assessed at full extent.
Where relevant, CEN TC256/SC2/WG11 comments and responses to preliminary enquiry inside the
community were inserted for additional use for the reader, as Observations of CEN/TC 256/SC 2/WG 11.
Besides, a general recommendation of use has been drafted by WG11 members in chapter 9.
This new method is described in this Technical Report in order to allow the possibility for wheelset
designers to apply it and to collect return of experience for further improvement.
Work Program summary
Clause 4 deals with the definition of a new fatigue design method which enables to assess the in-service
reliability of axles with regards to fatigue failure. The proposed approach, based on the “Stress Strength
Interference Analysis” (SSIA) and the “Fatigue-Equivalent-Load” (FEL) methods, aims at estimating the
probability of axles’ fatigue failure by characterizing the variability of in-service loads and the scatter of
the axles fatigue strength.
First of all, the main lines of the SSIA method are recalled. This method aims at evaluating the in-service
reliability of components for their design or their homologation. In the second part, the fatigue load
analysis method that is proposed for railway axles is described. It starts with a post-processing of an axle
load measurement: from a time signal of forces applied to both wheels fitted on the axle, fatigue cycles of
bending moment applied to the axle are identified and transformed into a cyclic equivalent load, Meq,
which is a measurement of the severity of the initial variable load. Then, virtual but realistic load spectra
are generated, thanks to a classification operation followed by a random draw of elementary load data
that considers the operation and maintenance conditions of the axle. All the spectra are then analysed
with the FEL method in order to build the distribution of in-service load severities. This distribution gives
a picture of the stress to which the axles are submitted. In the third and last part, the methods are applied
to real data of SNCF. Sensitivity analyses are performed in order to quantify the effect on Meq of variations
of parameters and to verify the convergence and robustness of the process. Finally, results obtained for
a passenger coach are given. The comparison between the distribution of load severities and the
normative load, defined as according to standards EN 13103-1, shows that, for the studied axle, the
normative load is very conservative. Finally, using the axles fatigue limits identified on full-scale tests, a
Stress Strength Interference Analysis is performed to calculate the probability of failure of the axle.
Figure 1 — Flowchart for load analysis and reliability assessment
Clause 5 concerns the mechanical modelling of an axle and defines a procedure to obtain local stresses
from the applied loads.
The characteristics of the finite element models to be applied to railway axles are analysed in terms of
element definition, convergence analysis, boundary conditions. A parametric analysis was performed to
assess the applicability of the models. The numerical models generated were validated through the
comparison with experimental results coming from full scale fatigue tests. Finally, a methodology to
design axles using modelling tools as a complement to current European norms is proposed looking for
a compromise between the computational effort and the results obtained.
Figure 2 — Flowchart for modelling
The main scope of Clause 6 is to provide the fatigue limits for standard steel grades considering also the
effect of surface conditions that may be different from the normal newly machined axles, like surface
corrosion that can appear during the service or surface blasting as a method to improve paint adhesion.
The areas of the axles considered were the free body transitions or groves and the wheel seats where at
high bending rates relative micro slips take place generating the so called fretting fatigue phenomena.
The paper provides in the conclusions a comparison with the fatigue limits that are today included in the
European Standards.
Another aspect that is treated in this work is the stress concentration effect that takes place along the
transitions where the body fatigue limit is verified. These parameters were measured by strain gauges
during each test and used inside the Euraxle project to validate their estimation through FE model
calculation.
Figure 3 — Full-scale and small-scale fatigue tests
1 Scope
This document presents the stage of knowledge resulting from the Euraxles project about the design of
the axle, and further steps to be taken.
It is the support:
- to define the loads to be taken into account;
- to describe the stress calculation method using finite elements and the validation processes associated;
- to specify the maximum permissible stresses to be assumed in calculations and the safety factors to be
used.
This technical report is applicable for:
- wheelset Axles defined in EN 13261 as “pure wheelset”;
- other axle designs such as those encountered in particular rolling stocks e.g. with independent wheels,
variable gauges, urban rail…
This document has not for aim to replace EN 13103-1 and CEN/TS 13103-2 but to present a
complementary method to the existing ones.
2 Normative references
There are no normative references in this document.
3 Terms, definitions, symbols and abbreviations
3.1 Terms and definitions
No terms and definitions are listed in this document.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— IEC Electropedia: available at http://www.electropedia.org/
— ISO Online browsing platform: available at https://www.iso.org/obp/ui
3.2 Symbols and abbreviations
FEL fatigue equivalent load method
SSIA Stress Strength Interference Analysis
KMR Consequent Miner Rule
FEM Finite Element Method
Nomenclature is given in Table 1.
Table 1 — Nomenclature
Y (t), Y (t) lateral force applied to both wheels
1 2
Q (t), Q (t) vertical force applied to both wheels
1 2
P1, P2 Vertical loads applied on the journals
C Minimum transition length
min
F Fatigue load
F Fatigue equivalent load
eq
F Severe representative load that can be defined for test and simulation validation
n
x, y, z Longitudinal, axial, vertical direction of wheelset reference axis
Bending moment applied to the y-section of the axle in the x direction (train
M (y)
x
circulation direction)
M Bending moment applied to the most critical section of the axle in the x direction
x
Equivalent bending moment applied to the most critical section of the axle in the
M
x,eq
x direction
M Normative bending moment
x,EN
MR Resultant bending moment
P Probability
P Probability of failure
f
P Probability of having a more severe load than F
n n
E Young modulus
K Stress correction factor
Kt Stress concentration factor
K Fatigue stress concentration factor
f
K Stress concentration factor based on strain measurements
t,s
D Total fatigue damage
d Partial damage generated by the ith class of a load spectrum
i
n Occurrence of the ith class of a load spectrum
i
N Number of cycles for a crack initiation for the ith class of a load spectrum
i
CV Coefficient of variation for the X variable
X
m Slope of the S-N diagram when using a one single slope curve
k Slope of the S-N diagram for S > S
D
k’ Slope of the S-N diagram for S < S
D
N number of cycle for the knee of the S-N diagram
D
S Stress load
S stress amplitude for the knee of the S-N diagram
D
K total mileage of an axle
ref
σ Allowable fatigue stress
f
σ Calculated dynamic stress
d
σ Von Mises stress
VM
σ Hydrostatic stress
h
τ Shear stress
σ Principal stress
ε Principal strain
σn Nominal stress
D/d Diameter ratio (diameter of wheel seat divided diameter of nearby body)
D Outer diameter of hub
N
d Diameter of the axle shaft
Maximum value of the radius of the transition r
r
max
F1 Full scale axle body fatigue limit
F4 Full scale axle seat fatigue limit
4 Loads
4.1 Reliability analysis based on the Stress Strength Interference Analysis method
Fatigue is known to be a damage phenomenon which is very dispersive. The sources of variability are
linked to the material properties that depend on its composition but also on the manufacturing process,
the geometry of the structure, loads, usages, environment, etc. To ensure safety, margins applied to the
specified loads and the prescribed fatigue limits, associated to a stress calculation method were defined
in standards EN 13103-1 [21] for the design and validation of railway axles. They were established in the
past decades, based on experience of railway experts and experimental and modelling works. Today, they
enable to guaranty a high level of safety for the European railway sector, as feedback from operation
shows. But, to gain competitiveness, it can be very useful to measure the available margins in order to
ensure that when a new design or a new technology is introduced, the level of safety is maintained.
For that reason, it would be beneficial to switch little by little from conservative approaches towards
reliability approaches. Maximalist approaches ensure safe designs by defining safety factors that make
the load specifications more severe and underestimate the allowable fatigue limits. The consequence is
that optimized solutions can’t be found. Moreover, when a significant change occurs in the system, it is
difficult to evaluate its impact on reliability. In reliable approaches, the aim is to have a “just necessary”
design associated to a target probability of failure. For safety critical components, the probability of
−5 −8
failure during the lifetime generally vary from 10 to 10 . In the example given in [4] on an automotive
engine part, the target probability is 10-6. For railway safety applications, if one considers that the
−6 −7
number of accidents due to mechanical failures is rather small, a target between 10 and 10 sounds
reasonable.
−6 −7
Observations of CEN/TC 256/SC 2/WG 11: The target value quoted (10 to 10 ) is a failure rate per
−9
axle during its whole life. It is approximately in line with the 10 failure rate per operational hour defined
in the CSM (EU regulation 402/2013) for technical systems for which a functional failure with immediate
disastrous consequences is assumed.
Reliability approaches have been developed for many years in various industries. The methods always
consist in characterizing the variabilities of the system to be designed and in calculating its probability of
failure by propagating the uncertainties in a deterministic modelling, as shown in Figure 4.
Figure 4 — Reliability approaches
The SSIA (Stress Strength Interference Analysis) is one of the reliability methods. It is widely used in
industries, due to its simplicity. Within this approach, only two global parameters are used to characterize
the variability of the problem: the load severity - referred to as STRESS - and the component strength -
referred to as STRENGTH. These two parameters are chosen so as to be comparable (for instance, local
stresses vs fatigue limits). Once the distributions of the Stress and the Strength are identified and
modelled, a probability of failure can be easily calculated, by means of analytical expressions or numerical
resolutions, as explained for example in [1][2][3][4]. The SSIA is therefore often used in the validation
process of a new component. But it is even more powerful in the design process: from a pre-defined target
probability of failure Pf, for a given distribution of Stress, the Strength distribution (mean value and
scatter) can be specified. Let’s notice that there is not a unique solution to this problem, as shown in
Figure 5.
a) SSIA for the validation process b) SSIA for the design process
Figure 5 — SSIA approach
To apply the SSIA approach, it is necessary to characterize both distributions. The distribution of Strength
can be identified thanks to laboratory fatigue tests. To have a good estimation of its distribution and not
only its mean value, a sufficient number of tests is necessary. In the Euraxles project, new tests were
performed for steel grades EA1N and EA4T. They are described in [17] and in Clause 4 of the report. For
the Stress distribution, measurements of loads shall be carried out in various operating conditions in
order to capture the variability of the behaviour of the component, as described in 4.2.3. Using
complementary information on the variability of usages from an axle to another, a distribution of in-
service load severity S can be generated. Accordingly, a representative and severe load can be defined as
a specification to be used in the numerical and experimental design and validation process. This severe
load is defined by its probability of occurrence P(S > S ) = P as shown in 2. This probability is a
severe severe
parameter that needs to be chosen by the industry and depends on the application. The idea is to define
a severe load to accelerate the physical fatigue tests, but not to define a solicitation that would induce a
damage process that is not the one that is studied. Typical values vary from 1/100 to 1/50000.
4.2 Fatigue load analysis method
4.2.1 General
The method for the axle fatigue load analysis is described in this part. The first step is the load signal post-
processing: from a time signal of forces applied to the both wheels, elementary fatigue cycles applied to
the axle are identified. Then, by calculating the total damage induced by all these elementary load cycles,
a cyclic equivalent load, F , which is as damaging as the initial variable load, is calculated. Its amplitude
eq
is taken as a measurement of the variable load’s severity. The second step of the method is the generation
of virtual load spectra which are realistic and representative of real usages of axles. To do so, information
from operation and maintenance are used (total average number of kilometres before the axles reach the
acceptable wear ratio limit, the way the axles are changed in a whole train during maintenance, the
characteristics of the tracks on which the axles circulate, etc.). A strategy to generate those load spectra
is described. The third and last step is the processing of all these virtual spectra with the FEL method
which enables to build the distribution of in-service loads severities (“Stress” in the SSIA) to be compared
to the standard load.
4.2.2 Load signals processing and Fatigue-Equivalent-Load
In mechanical high-cycle fatigue, Wöhler works showed that the main parameters that affect the lifetime
of a material under a cyclic one-dimensional load are the amplitude and, to a less extend, the mean value
of the load. To estimate the total damage induced on a component by a load with variable amplitude, it is
necessary to identify all the elementary cycles included in the load signal and finally cumulate the partial
damages induced by all these cycles. When defined by their amplitudes and their mean values, the
elementary cycles can be identified thanks to a Rainflow counting method, as explained in [6]. When the
mean values variations of the elementary cycles are small, one can consider that only the amplitudes of
the cycles have to be considered in the fatigue analysis. Hence, a simple cumulative damage law, as the
linear Miner rule [7], can be used to calculate the total damage. Finally, a cyclic equivalent load, called the
F , producing the same total damage, can be easily calculated, using Wöhler curves. Its amplitude is taken
eq
as the parameter of load severity. The whole process is explained in details in [1] and is presented in
Figure 6, where F(t) is the initial load time signal and F is the amplitude of the final cyclic equivalent
eq
load.
Figure 6 — The FEL method
An adaptation of this FEL approach to the railway axle is proposed and presented in the next paragraphs.
During operation, axles are subjected to vertical loads due to the train mass on both axle journals, the
wheel/rail contact forces and the braking load. Let’s call Y , Q , Y , Q the lateral and vertical forces applied
1 1 2 2
to both wheels and P1 and P2 the train loads, as shown in Figure 7. These forces generate bending
moments in each cross section of the axle, defined by the coordinate y. It is assumed that the bending
moment in the train circulation direction, Mx(y), sizes the axle to the first order and that the bending
moments in the other directions and the other forces (torques) applied to axles can be neglected in the
fatigue analysis since their occurrences are low.
Observations of CEN/TC 256/SC 2/WG 11: The assumption that only the bending moments in the
train circulation direction are needed to design the axle should be demonstrated for a full extent
assessment of an axle. The actual design philosophy of an axle in EN 13103 series standard takes into
account all bending moment directions and torsion moments. Thus, in a cumulative damage approach,
not only the number of occurrences is to be taken into account but also the inherent level of the damages.
Figure 7 — Forces applied to wheels and axle
When submitted to a rotating bending moment M (y), any material point of the axle, defined by a section
x
y and a vertical location z, undergoes a stress cycle σ y,,zt during each revolution of the axle. Its
( )
amplitude ∆σ yz, is proportional to M and can be calculated following Formula (1):
( ) x
r yz,
( )
∆σσ yz, =2×max( yz,,t= M y (1)
( ) ( ) ( )
t x
Iy
( )
To these stress variations, mean stresses can be added (mainly due to the press-fitting of wheels and
pinions on the axle; residual stresses due to manufacturing are not considered as they are included in the
fatigue tests made to determine the fatigue limits of full scale axles) but they can be assumed to be
constant and to depend only on the y coordinate.
During load measurements, time signals of lateral and vertical forces Y , Q , Y , Q at the wheel/rail
1 1 2 2
contact applied to both wheels are recorded. To identify all the elementary fatigue cycles included in the
time signals Y (t), Q (t), Y (t), Q (t), it is therefore necessary to identify all the axle revolutions and to
1 1 2 2
determine the bending moment value M , assumed to be constant during an axle revolution. To be
x
conservative, it is suggested to use the maximum value of M encountered during each revolution. Finally,
x
the post-processing of a test campaign gives a bending moment spectrum (M (i), n) where n is the
x i i
occurrence of the M (i) cycles for any point of the axle.
x
To measure the severity of this bending moment spectrum, the FEL method is used. It consists in
calculating the cyclic force that would reproduce the same damage as the load spectrum, when repeated
millions of times. In the specific case of the railway axle, the amplitude of the stresses cycle at each
material point depends on the value of the bending moment M , while the mean value mainly depends on
x
the location of the point on the axle and the value of the press fitting of the wheels on the axle. For this
reason, only the value of the bending needs to be recorded in the load analysis; there is no need to take
into account the mean value of the cycles.
Generally, to calculate the equivalent bending moment M , the traditional and simple linear cumulative
eq
damage Miner rule is used: the total damage produced by the whole load spectrum is given by
Formula (2):
n
i
Dd (2)
∑ i ∑
N
ii
i
where
N is the number of cycles at failure for the load M (i).
i x
The cyclic load M which, repeated N times, produces the same damage D is calculated with the same
eq eq
damage law. It is expressed with Formula (3):
m
m
n 11nM.
mm m
i ii
(3)
∀i N .M=N M D== ⇒=M nM.
i i eq eq ∑∑ eq ∑ i i
m
NN M N
ii i
i eq eq eq
Figure 8 — Calculation of M for a bending moment spectrum
eq
In the special case of railway axles, the number of cycles supported by a railway axle is generally more
than 10 and experimental tests show that the Wöhler curves of the axle steel grades exhibit a change of
slope after 10 cycles. Hence, instead of using the traditional Basquin law with one slope, a Basquin law
with two slopes following the rule proposed by Haibach (k for N < N and k’ = 2k-1 for N > N ) can be
D D
used [8][9].
Another alternative method for the calculation of the cumulated damage generated by a load spectrum is
the “consequent miner-rule method” that is explained in FKM guidelines for structural assessment [10]
or in [11]. The consequent-Miner-rule assumes that a totally new component has a Wöhler curve with an
asymptotic tail S , which can be identified by constant amplitude tests. When submitted to a random
D
variable amplitude spectrum, the application of cycles above the fatigue limit progressively increases the
damage D and reduces little by little the current endurance limit of the component S . As a
D,D≠0
consequence, more and more cycles of the spectrum contribute to the damage (even those who were
underneath the initial fatigue limit). The process to calculate the total damage and the associated lifetime
induced by a load spectrum is represented in Figure 9. The initial load spectrum is in black plain line and
==
the initial and current damage laws are represented by the two red curves S and S . . The KMR
D,D = 0 D,D≠0
calculation gives the lifetime N(D) associated to a damage D as the load cycles are progressively imposed.
Figure 9 — Consequent Miner Rule lifetime calculation
The formulas to calculate the total number of cycles N(D) in terms of the total damage D within the KMR
are the following:
k
S = 0
D,0D=
DA −+1.D 1. .N
( ) ( )
kon D,0D=
S
a,1
k−1
j
S
ZZ
a,1
A . +
kon ∑
S NN
D,0D= 1 v=m 2
kk−−11 k
m−1
S S S
h
D,0D= a,m ai,
i
ZN=−= .
11 ∑
S S HS
i=1
aa,1 ,1 a,1
k−11k− k
v
SS S
h
av, av,1+ a,i
i
ZN=−= . (4)
22 ∑
S S HS
i=1
a,1 a,1 a,1
=
=
The parameters in the formulas correspond to:
k slope of the Wöhler-line
S initial endurance limit
D,D = 0
N number of cycles at knee-point
D,D = 0
D miner sum
i index of step in the spectrum (beginning with 1 for the maximum value)
h number of cycles in step i of the spectrum
i
x relative load amplitude (related to the maximum load) of step i of the
i
spectrum
S absolute value of the load amplitude of step i of the spectrum
a,i
S absolute value of the maximum load amplitude
a,1
j number of steps in the spectrum
m number of the step smaller or less than the endurance limit
N permissible number of cycles
Z1, N1, Z2, N and A are for purposes of the calculation
2 kon
γ parameter describing the influence of the damage on S and N set
D,D > 0 D,D > 0
equal k-1 (see [11]).
A new method to define a Fatigue-equivalent load which is consistent with the KMR (Consequent Miner
Rule) is necessary. Since only stress ratios are necessary to calculate N, and since the amplitudes of
stresses in the axles can be assumed to be proportional to the bending moment, the formulas remain
correct when replacing stresses ratios by bending moments ratios. The Consequent Miner damage law
applied to bending moments can therefore be directly used for the Fatigue Equivalent Bending Moment
calculation: the bending moment corresponding to the lifetime N calculated for a damage D = 1 is taken
D
as the Fatigue-Equivalent-Bending-Moment.
The methodology described above is applied to an SNCF axle made of steel EA1N and put on a passenger
coach running up to V = 160km/h on a French classical line, for which load data are available (111 km of
load acquisition). This small sample is simply repeated many times to simulate a much higher total
mileage K .
ref
In Figure 3. 7, the SNCF load spectrum can be seen, with, in red, the value of the partial damage generated
by each level of bending calculated with the Basquin law with one slope. It can be seen that with this
damage law, the small cycles do not damage the axle. The Fatigue-Equivalent-loads obtained with the
Basquin laws with one or two slopes (with k = 8; k’ = 15; change of slope for N = 1,00E+07 cycles) or with
D
the Consequent Miner Rule are given in the table in Figure 10. The total circulated distance (defining the
total number of cycles) used to define the fatigue-equivalent-load is K = 7,10 km, being the average
ref
mileage observed by SNCF for such an axle. Minimum and maximum typical encountered values during
6 7
operation are 4,10 km and 10 km. These values are also tested. The normative loads defined as
according to standard EN 13103-1 are also given.
Figure 10 — SNCF data analysis - Fatigue-Equivalent-Bending-Moments calculated with various
damage laws and comparison with the standardized load
These calculations show that:
• The standardized load does not vary with the target lifetime of the axle, while the Fatigue Equivalent
Load takes into account the effect on the severity of the load of the target lifetime of the axle, which
makes physical sense.
• The fatigue equivalent bending moments obtained with the KMR are significantly lower than those
obtained with the Miner rule. This is consistent with the initial motivation of the KMR development:
the classical Miner rule used with a Basquin law with an asymptotic tail is known to be too severe,
especially when most of the loads included in the spectrum are lower than the initial fatigue limit of
the component.
• The Fatigue equivalent Loads obtained with the KMR are always significantly lower than the
normative loads, which is reassuring! The distance between the normative load and the equivalent-
load calculated with KMR seem reasonable. This means that, for this example, the standardized loads
are on the safe side. But one shall keep in mind that the load sample which is used is quite short. If
the measurements were longer, higher load amplitudes could be encountered and the final
equivalent load could be higher. Therefore, the conclusions shall be taken with caution because the
studied loads are not various and representative enough.
4.2.3 Method to generate the distribution of in-service load severities
The philosophy of the work that is led is to compare the normative load not to the fatigue-equivalent-
load of a “representative load spectrum” but with a distribution of fatigue-equivalent-loads. The idea is
to take into account, in the final overall reliability approach, the fact that railway axles have various lives.
The loads undergone by an axle that is on a trailer position during its whole life and that circulates on a
track with lots of straight lines are definitively different from the loads applied to an axle that is put ahead
of a wagon and circulates on a line with many tight curves. Since it is impossible to put sensors on all
axles that circulate on the Europea
...
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