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h p://dx.doi.o g/10.1016/j.ij a igue.2012.07.006
h p://hdl.handle.ne /10251/78196
Else ie
Ma ínez Casas, J.; Mazzola, L.; Baeza González, LM.; B uni, S. (2013). Nume ical
Es ima ion o S esses in Railway Axles Using a T ain-T ack In e ac ion Model. In e na ional
Jou nal o Fa igue. 47:18-30. doi:10.1016/j.ij a igue.2012.07.006.
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Nume ical Es ima ion o S esses in Railway Axles Using a T ain-
T ack In e ac ion Model
Au ho s: José Ma ínez-Casas1, Lau a Mazzola2, Luis Baeza1, S e ano B uni2
A ilia ion:
1Cen o de In es igación en Tecnología de Vehículos, Uni e sidad Poli écnica de Valencia, Camino
de Ve a s/n, 46022 Valencia - Spain
2Dipa imen o di Meccanica, Poli ecnico di Milano, Via La Masa 1, 20156 Milano, I aly
E-mail co esponding au ho : s e ano.b [email protected]
Abs ac
The a igue design o ailway axles equi es ha he s esses a ising in he axle in eal se ice a e
accu a ely quan i ied. This pape desc ibes a me hod o compu e he dynamic s esses a ising in
ailway axles as he e ec o ain- ack in e ac ion, based on he nume ical simula ion o he
dynamic in e ac ion be ween a lexible wheelse and a lexible ack. The wheelse is modelled as a
lexible o a ing body using an Eule ian app oach, whe eas ack is ega ded as an in ini e pe iodic
sys em wi h he ail modelled as a Timoshenko beam es ing on disc e e elas ic suppo s,
conside ing he ine ia associa ed wi h he sleepe s.
The pape p esen s an applica ion o he p oposed p ocedu e o he calcula ion o he dynamic
s esses caused in he axle by di e en ypes o geome ic impe ec ion occu ing on he wheel and
ail su aces, conside ing he cases o a single ha monic ail co uga ion, andom ail oughness and
a wheel la .
Keywo ds
Railway ehicles, wheelse design, wheelse axle a igue, dynamic loads, ain- ack in e ac ion, ail
oughness, wheel la
*Manusc ip
Click he e o iew linked Re e ences
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1. In oduc ion
Railway axles du abili y is a key issue in designing and co ec ly main aining ailway ehicles, o
ensu e ha he highes sa e y s anda ds a e me and, a he same ime, o op imise li e-cycle cos s
om a sys em poin o iew, i.e. conside ing no only he ehicle bu also he in e ac ing
in as uc u e. F om he single poin o iew o a igue esis ance, he axle design should end
owa ds inc easing he size o educe s esses, bu his componen also ep esen s a signi ican
con ibu ion o he wheelse un-sp ung mass, which, on he con a y, shall be minimised o educe
he gene a ion o dynamic o ce a wheel- ail con ac and hence damage in he ack and in he
wheels, especially o high-speed ains.
A p esen , wheelse axles a e designed o in ini e a igue li e, howe e a small numbe o axle
ailu es due o a igue con inues o be epo ed, some imes wi h ca as ophic consequences. One o
he easons which ha e been p oposed o explain hese un o eseen ailu es is ha he loads assumed
by he s anda d o he pu pose o a igue design e i ica ion do no ully e lec he ac ual se ice
loads o he ehicle, which a e la gely depending on he ehicle design pa ame e s (e.g. unsp ung
masses, suspension s i ness and damping, …) and on he ehicle se ice p o ile [1, 2]. The p ecise
knowledge o se ice loads is also pi o al o he de ini ion o app op ia e in e als o he non-
des uc i e inspec ion o ailway axles [3, 4], which a e nowadays pa o s anda d main enance
p ac ice o emo e om se ice c acked axles be o e ailu e.
De ailed in o ma ion on he se ice loads is o en de i ed expe imen ally, using ins umen ed
wheelse s [5 - 7]. This app oach howe e p esen s some d awbacks: i s o all, he physical
measu e is no applicable a he design s age o a new ehicle, se ing-up and unning he es s is
expensi e and demanding due o he ha sh measu ing en i onmen and ex ensi e es campaigns a e
equi ed o co e all se ice condi ions (di e en lines and speeds, ope a ion a a e / ull load, new
/ wo n wheel p o iles, …). Fu he mo e, he expe imen al measu e o he se ice load is no mally
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no able o ully explain he causes o ex eme loads, which esul in s ess peaks mos ly a ec ing
he a igue esis ance o he axle and c ack p opaga ion.
Fo his eason, in he las yea a emp s ha e been made o de ine p ocedu es based on he use o
mul i-body models o a ailway ehicle o ain se , o p o ide a nume ical es ima e o he se ice
loads [6 – 8]. This app oach can be used o complemen measu emen s, e.g. add essing se ice
condi ions no co e ed by he es s and o p o ide a be e physical unde s anding o he ac o s
in luencing he se ice spec a, including (among o he s) ehicle design pa ame e s, he condi ion
o he wheel and ail su aces and he se ice scena io. To he Au ho s’ knowledge, hese
app oaches ha e howe e un il now been de eloped conside ing he wheelse as a igid body.
Hence, axle s esses can be co ec ly compu ed only in he low- equency ange (app oxima ely 0-
20 Hz), whe eas hei high- equency componen s, mainly esponsible o he occu ence o s ess
peaks, a e excluded om he analysis [8].
This pape aims he e o e a es ablishing a me hod o de ine nume ically he s esses in a ailway
axle, including he high equency componen s, as unc ion o ehicle- ack pa ame e s and o he
ehicle’s unning condi ion, he eby ex ending he scope and de ail o he exis ing nume ical
app oaches o p edic he wheelse se ice loads. To his end, a model o a lexible wheelse
in e ac ing wi h a lexible ack in angen ack is de i ed and used o simula e ypical se ice
scena ios o a high-speed ailway ehicle. The wheelse is modelled as a lexible o a ing body,
using an Eule ian app oach o ake ad an age om axial symme y and modal syn hesis o educe
he size o he p oblem. The ack is modelled as an in ini e pe iodic sys em, wi h he ail modelled
as a Timoshenko beam es ing on disc e e elas ic suppo s, conside ing he ine ia associa ed wi h
he sleepe s. The nume ical simula ion conside s he e ec o geome ic i egula i ies appea ing on
he ail and wheel su aces, allowing o conside he e ec o e.g. ail co uga ion, ail dips, wheel
polygonalisa ion, wheel la s e c.
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T ain- ack in e ac ion models including a de ailed desc ip ion o he wheelse as an elas ically
de o mable body ha e been p oposed by se e al au ho s. Some models, e.g. [9] a e de ined in he
equency domain and de i e he dynamic o ces a wheel- ail con ac by combining he wheelse
and ack equency esponse unc ions wi h he assumed ail co uga ion. This app oach is
inhe en ly linea and does no allow o conside he ails as disc e ely suppo ed. Modelling
app oaches which inco po a e a disc e e-suppo ack model and he e ec o nonlinea i ies in he
ack and in wheel– ail con ac ha e been de eloped in Re s. [10–14]: in his case, he ain- ack
model is de ined in he ime domain and he wheelse and ack models a e de ined using he ini e
elemen me hod, in oducing a educ ion o he deg ees o eedom using mode supe posi ion.
These app oaches ha e been mos ly applied o he s udy o damage phenomena in he ack and a
wheel- ail in e ace, and he e o e ocus on he de ini ion o wheel- ail con ac o ces a he han
axle s esses. The e o e, a simpli ied model o he wheelse axle (some imes based on beam
elemen s) can be used. Fu he mo e, he e ec o wheelse o a ion is neglec ed o conside ed wi h
some app oxima ion, whe eas his e ec may impo an ly a ec he wheelse esonance condi ions,
which in u n can be a cause o ex eme s ess peaks a ising in he axle: he p esen pape aims
he e o e a de eloping a modelling and simula ion app oach speci ically ailo ed on he p edic ion
o axle s esses
The pape desc ibes he wheelse - ack in e ac ion model de eloped and he p ocedu e o compu e
he dynamic s esses in he axle. Resul s o he nume ical p ocedu e a e p esen ed o di e en
exci a ion cases including ail co uga ion ha ing di e en wa eleng h and a wheel la , and
conside ing he e ec o wheelse speed.

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2. THE VEHICLE-TRACK INTERACTION MODEL
The ehicle – ack in e ac ion model, see Figu e 1, is de ined adop ing a sub-s uc u ing echnique
[15, 16], acco ding o which he whole sys em is di ided in o subs uc u es: he ehicle, he ails
and he ail suppo s. Fo each subs uc u e, he equa ions o mo ion a e w i en sepa a ely, and
in e ac ion e ec s be ween ehicle and ack a e ep esen ed by he wheel- ail con ac o ces and by
he o ces gene a ed a he ailpads [15]. The simula ion app oach is de eloped conside ing he
mo ion o he wheelse in angen ack.
Gi en he ange o equencies add essed, he ehicle model is con ined o one wheelse wi h
p ima y suspension, see sec ion 2.1. The ib a ion o he lexible wheelse is exp essed using
Eule ian modal coo dina es, see Sec ion 3, aking ad an age o m he axial symme y o he body.
Fu he mo e he modal app oach is chosen o desc ibe he wheelse mo ion in o de o educe he
compu a ional e o equi ed by he simula ion.
The ack is modelled by means o a cyclic app oach, which p o ides some bene i wi h espec o
classical ack modelling o he ini e one. The sub-s uc u ing echnique is s ill adop ed he e o
simpli y he sys em modelling; a de ailed desc ip ion o he ack is p o ided in sec ion 2.2.
2.1 The Vehicle Model
The dynamic s esses a ising in he wheelse axle a e mos ly ela ed wi h ain- ack in e ac ion
e ec s in he equency ange abo e 20 Hz, which a e exci ed by sho wa eleng h geome ic
impe ec ions in he wheel and ail p o iles and by singula i ies such as ail dips and wheel la s.
In his equency ange he dynamics o he sp ung masses (bogie ame and ca body) a e
e ec i ely isola ed om he mo ion o he un-sp ung masses (wheelse s and axle boxes) on accoun
o he mechanical il e in oduced by he suspensions. The e o e, he ehicle model used in his
pape conside s one single wheelse , modelled as an elas ically lexible body, and he p ima y
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suspension, ep esen ed using isco-elas ic lumped pa ame e elemen s. The s a ic load associa ed
wi h he g a i a ional o ces ac ing on he bogie and on he ca body masses is ep esen ed by wo
s a ic o ces applied h ough he p ima y suspension on he wo sides o he wheelse .
2.2 The T ack Model
The ack is modelled by means o a cyclic app oach, whe e a ini e sec ion o he ack is de ined,
in oducing cyclic bounda y condi ions a he ends o he model, hence he ack, see Figu e 2, can
be in e p e ed as an in ini e ack, nego ia ed by an in ini e se o iden ical ehicles, uni o mly
dis ibu ed in such a way ha each ehicle is se a a cons an dis ance L apa om he adjacen
ones. Due o he pe iodici y o he s uc u e and o he loading condi ions, he s udy is educed o a
single sec ion ha ing ini e leng h L, whose alue is se la ge enough o a oid in e ac ion be ween
he ehicles.
The model adop ed o he di e en ack componen s is illus a ed in Figu e 3: he ails a e
modelled as Timoshenko beams, including bending de o ma ions in e ical/la e al di ec ions, as
well as o sional de o ma ions. Rail ib a ion is in oduced in e ms o modal supe posi ion o he
uncons ained ail wi h cyclic bounda y condi ions, hence esul ing in o a se o de-coupled 1-d.o. .
equa ions. Mo eo e due o he symme y o he sub-sys em wi h espec o he ack cen eline,
only one ail is modelled in he p esen s udy.
The disc e e ail suppo s a e in oduced in he o m o lumped pa ame e sys ems, see Figu e 3
igh . The ail pads a e modelled as lumped isco-elas ic elemen s gene a ing he in e ac ion o ces
be ween he ails and he sleepe s, ep esen ed as lumped masses. Ballas dynamics is neglec ed
he e, being no ele an o he s ess analysis on he wheelse , bu he equi alen ballas s i ness
and damping a e accoun ed o by means o lumped sp ing and dashpo elemen s.
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2.3 The Model o Wheel-Rail Con ac Fo ces
The ack model is coupled wi h he lexible wheelse ia he wheel- ail con ac o ces, which a e
exp essed as unc ion o he ela i e wheel- ail displacemen and eloci y in he con ac pa ch. In
he nume ical model p esen ed he e, he heo y o He z is adop ed o de ine he no mal con ac
o ce componen and FASTSIM [17] is used o de ine he angen ial con ac o ces as unc ion o
he no mal con ac o ce and o he c eepage componen s.
A each ime s ep, he displacemen and eloci y o he wheel in he con ac poin a e ob ained using
Eq. (1) e alua ed a he con ac poin posi ion.
3. THE FLEXIBLE WHEELSET MODEL
In o de o model he kinema ics o he lexible wheelse , wo con igu a ions (unde o med and
de o med) a e de ined. The unde o med con igu a ion is associa ed wi h he spinning eloci y o he
wheelse (a cons an angula eloci y

). The de o med con igu a ion conside s he lexibili y and
small igid solid displacemen s. The displacemen ield ela es he de o med con igu a ion wi h he
unde o med con igu a ion as i will be shown in Eq. (1).
The coo dina es ha a e implemen ed in he wheelse model do no ollow he ma e ial poin s o he
solid which is he commones p ocedu e in Mechanics, ne e heless hey a e associa ed wi h spa ial
poin s (Eule ian app oach). Le
u
an Eule ian ec o coo dina e in a ixed coo dina e ame. Any
p ope y o he solid
),( u

co esponds o he ma e ial poin o he solid whose unde o med
con igu a ion is in he spa ial poin
u
a ins an
. Following his c i e ion, he displacemen ield is
de ined by means o he ollowing o mula:
),( uwu 
, (1)
whe e
is he inal posi ion o he pa icle, and
w
is he displacemen s associa ed wi h lexibili y
and small igid body displacemen s.
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The coo dina e ame is chosen so ha he spin o a ion is in he second axle. The ollowing
ma ices a e de ined as ollows:












001
000
100
J
;











100
000
001
E
. (2)
The angula eloci y enso e i ies:
JΩ
















00
000
00
~
; and
EΩΩ 2
~~


. (3)
The eloci y due o he igid body spinning is:
 
uuJuΩ ~
~
T
321 ΩΩ 
, (4)
whe e
 
T
13 0
~uu u
. The eloci y o he pa icle is compu ed h ough he ma e ial de i a i e
o
, and ha is
 








ii
i
ii
i
ii
iu
uΩΩ
u
u
w
wuJ
w
w
uwu ~
D
D
D
D
D
D
. (5)
In o de o calcula e he kine ic ene gy o he solid, he squa e o he pa icle eloci y Eq. (5) has o
be ob ained. I has he ollowing o mula:
.
~
2
~
22
~~
D
D
D
D
T
T
2T
T
2TT2
T













































ii
i
ii
i
ii
i
ii
i
u
uΩ
u
uΩΩ
u
u
u
uΩΩ
w
wuJ
w
uJw
ww
wwuEu


(6)
The nex modal app oach is adop ed:
)()(),( quΦuw 
, (7)
being
)(uΦ
he mode shape unc ions ma ix o he ee-bounda y wheelse . The small igid
displacemen s o he solid a e conside ed in his app oach h ough he igid modes o he wheelse .
I mus be poin ed ou ha he mode shape unc ions do no depend on ime since he o a ion o he
solid does no change he mode shapes unc ions in ixed coo dina es, because he axial symme y
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ampli ude o he dynamic o ce luc ua ion is app oxima ely 81 kN, and is he e o e conside ably
lowe han o he 60 mm wa eleng h (c . Fig. 6), on accoun o he ail showing a g ea e mobili y
in his esonance condi ion. Also in his case, he peak-peak o ce ampli ude is nea ly p opo ional
o he co uga ion ampli ude.
The lowe sub igu e shows he e ical con ac o ce plo ed s. he a elled dis ance o a
co uga ion wa eleng h exci ing he i s o wa d bending mode o he wheelse . In his case, he
peak-peak ampli ude o he con ac o ce o he la ge co uga ion ampli ude is 82.2 kN, again
signi ican ly lowe han o he 60 mm wa eleng h. In his second case howe e he inc ease o he
peak-peak o ce ampli ude wi h he co uga ion ampli ude is signi ican ly less han p opo ional,
and also he wa e o m o he con ac o ce is di e en o he wo co uga ion ampli udes: his shall
be asc ibed o he di e en impo ance o he sleepe -passing e ec in he wo cases, a he han o
he e ec o non-linea i ies.
In Figu e 9, he y-axis no mal s ess in he axle is shown o he same cases as in Fig. 8. Fo he
co uga ion wa eleng h exci ing he pinned-pinned esonance o he ack, no signi ican di e ence
is obse ed wi h espec o he cases epo ed in Figu e 7 o he same speed (300 km/h), and again
an inc ease o he co uga ion ampli ude lea es una ec ed he ampli ude o he axial s ess.
Howe e , when he co uga ion wa eleng h exci ing he wheelse esonance and he la ge
co uga ion ampli ude is conside ed, he s ess ampli ude becomes 33% highe han in Figu e 7
(despi e he con ac o ce being lowe in his case han o he 60 mm wa eleng h) and de ia es
signi ican ly o m he sinusoidal wa e o m. This esul is jus i ied by he e ec o wheelse
lexibili y playing a pa icula ly impo an ole in he case conside ed, on accoun o one mode o
ib a ion being exci ed in esonance.

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A use ul pa ame e o quan i y he ele ance o dynamic e ec s a ec ing he wheel- ail con ac
o ces and he s esses in he axle is he dynamic ac o , de ined as he a io be ween he maximum
dynamic alue o he quan i y conside ed and he co esponding s a ic alue. Fo he con ac o ce,
he dynamic ac o
Q
k
is de ined acco ding o he ollowing equa ion:
s
max,d
QQ
Q
k
(20)
wi h
max,d
Q
he maximum alue o he con ac o ce and
s
Q
he s a ic wheel load. Fo he s ess in
he axle he dynamic ac o

k
is de ined as:
max,s
max,d
k




(21)
wi h
max,d

maximum dynamic s ess in he ma e ial poin conside ed and
max,s

he
co esponding maximum s ess unde he ac ion o o a ing bending p oduced by he s a ic loads
ac ing on he wheelse .
In Figu e 10 he dynamic ac o s
Q
k
and

k
a e epo ed as unc ion o he ehicle speed o he
di e en ypes o ail co uga ion p esen ed abo e, conside ing a co uga ion ampli ude en imes
g ea e han he ISO 3095 limi ; in his case, also he esul s o a co uga ion wa eleng h exci ing
he second o wa d bending mode o he wheelse a e p esen ed. Fo he con ac o ce dynamic
ac o
Q
k
, in he en i e speed ange conside ed, he la ges alues a e ob ained o he co uga ion
wa eleng h exci ing he i s o wa d bending mode o he wheelse , wi h a maximum alue close o
1.8. A local maximum appea s o mos o he co uga ion cases conside ed o speeds in he 125-
135 km/h ange, his is due o a esonance e ec associa ed wi h he sleepe passing equency ha
exci es P2 equency (see P1 and P2 equencies in Re . [15])..
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As a as he s ess dynamic ac o

k
is conce ned, e y simila ends and alues a e ob ained o
he case wi h no oughness, o oughness wa eleng h exci ing he pinned-pinned equency,
leading o low alues o s ess ampli ica ion (max. alue below 1.2). Only O he wise, in he case o
he o co uga ion wa eleng h exci ing one o he wheelse he bending esonances o he wheelse
he s ess dynamic ac o is signi ican ly highe and can be eaches a maximum alue close up o
abou 1.8 o he case o co uga ion wa eleng h exci ing , wi h he esonance o he i s o wa d
and up o 1.4 app oxima ely o co uga ion wa eleng h exci ing bending mode leading o la ge
dynamic s esses han o he second o wa d bending mode. The ends wi h speed a e close o
mono onically inc easing, and he e ec o he esonance associa ed wi h he sleepe passing is less
e iden han in he end o he
Q
k
dynamic ac o .
These esul s sugges ha he bending s esses in he axle a e highly a ec ed by he wa eleng h o
ail co uga ion, and ha combina ions o ain speed and co uga ion wa eleng h leading o a
esonance o a bending mode may be especially ele an in iew o axle esis ance o a igue.
Pa icula ly signi ican in iew o eal applica ions is he case o ail co uga ion, a o m o i egula
wea o he ail o en appea ing in ailway sys ems and cha ac e ised quasi-ha monic wea pa e ns
de eloping on he ail head in longi udinal di ec ion [18]. Co uga ion wa eleng h can ange om
50 mm o less up o mo e han 1m in he case o hea y haul co uga ion, and ypical wea dep h
alues a e in he ange o some en hs o millime e.
To quan i y he ele ance o a igue o ail co uga ion, we no e ha he design me hods p esc ibed
o ailway axles by he Eu opean S anda ds EN13103 and EN13104 [19, 20] assume a 1.25
dynamic ampli ica ion ac o on he loads gene a ed a he p ima y suspension. Since he calcula ion
o he bending s esses is hen based on s a ic equilib ium, also he bending s esses a e magni ied
by 25% wi h espec o he s a ic case, whe eas he esul s in Figu e 10 sugges ha la ge dynamic
ac o s migh apply when a pa icula combina ion o co uga ion wa eleng h and ain speed
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exci es one bending mode. Fo ins ance, a sinusoidal ail co uga ion ha ing app oxima ely 350 mm
wa eleng h and 0.15 mm dep h (i.e. en imes he ISO3095 limi o he conside ed wa eleng h)
would p oduce a dynamic s ess ampli ica ion close o 40%. I shall be poin ed ou howe e ha he
equi alen s a ic loads p esc ibed by EN13103/104 also include la e al wheel- ail con ac o ces due
o cu ing, which a e no included in he analysis p esen ed he e.
4.2 Resul s o andom ail co uga ion and o a wheel la
The simple exci a ion cases conside ed in Sub-sec ion 4.1 allow o poin ou he in luence o
di e en dynamic e ec s on ain- ack in e ac ion and on he s esses in he wheelse , bu do no
ep esen ealis ic exci a ion cases. In his Sub-sec ion, wo sou ces o exci a ion ypically occu ing
in se iced ains a e conside ed: andom ail oughness and a wheel la . Random oughness akes
place on he ail head on accoun o di e en causes: geome ic impe ec ions associa ed wi h he
manu ac u ing and ins alla ion o he ail, i egula wea caused by ain passage and, o la ge
wa eleng hs, non-uni o m ack se lemen due o pe manen de o ma ions in he soil and ballas (in
his case, he e m ― ack i egula i ies‖ is used ins ead o co uga ion). S udies ha e shown ha ail
oughness and ack i egula i y ake he o m o s a iona y andom p ocesses cha ac e ised by hei
powe spec al densi y, which can be de ined based on he quali y o ack main enance [2181]. On
he o he hand, wheel la s a e localised de ec s occu ing on he wheel su ace as he esul o ull
slip o he wheel, ypically caused by poo ly adjus ed o aul y b akes [1922].
Figu e 11 shows he e ical con ac o ce plo ed s. he a elled dis ance o he ain speeds o
100 and 300 km/h, conside ing he exci a ion p oduced by andom ail co uga ion (ampli ude
co esponding o he ISO 3095 limi ). The esul s ob ained conside ing a igid and lexible wheelse
model a e compa ed. Impo an dynamic luc ua ions o he con ac o ce a e e idenced, wi h he
maximum dynamic alue co esponding o app oxima ely 1.6 imes he s a ic load. The di e ences
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be ween he esul s ob ained o he igid and lexible wheelse a e ela i ely small in his case, and
do no a ec signi ican ly he maximum con ac o ce alue.
Figu e 12 p esen s he e ical con ac o ce s. a elled dis ance o a ehicle a ec ed by a
wheel la . The calcula ion has been ca ied ou o 50 and 300 km/h by means o he igid and
enhanced wheelse models. A se e e dynamic e ec is obse ed, consis ing o he comple e loss o
con ac be ween he wheel and he ail, ollowed by an impac leading o a maximum alue o he
con ac o ce which is be ween 3 and 4 imes he s a ic load and inally by a ansien ib a ion. The
di e ences be ween he esul s o he igid and lexible wheelse model a e small in e ms o
du a ion o he con ac loss and o maximum o e loading, bu he ansien ollowing he impac is
a ec ed qui e ema kably by wheelse lexibili y, as demons a ed by he di e en equency
con en s o he con ac o ce signal, see pa icula ly he zoomed iew on he igh side o he igu e.
The y-axis s esses o he andom ail co uga ion and wheel la exci a ion cases a e shown in
Figu e 13. In he andom co uga ion case, he dynamic luc ua ions o he s ess componen a e
ela i ely low and lead o a maximum peak-peak s ess ampli ude a ound 91.2 MPa, co esponding
o a dynamic ampli ica ion wi h espec o he maximum s ess p oduced by he s a ic load

k
o
1.18 app oxima ely. When he wheel la de ec is conside ed, he esul s a e highly a ec ed by he
ehicle speed, wi h la ge dynamic e ec s aking place a low speed (wi h a dynamic ac o

k
o
1.34 app oxima ely) on accoun o he epea ed loss o con ac o he wheel o he ail, and wi h a
educ ion o he dynamic s esses a highe speed.
Finally, in Figu e 14 he dynamic ac o s
Q
k
and

k
a e epo ed o he andom co uga ion and
wheel la as unc ion o he ehicle speed. The con ac o ce dynamic ac o
Q
k
is nea ly
mono onically inc easing up o a maximum alue close o 1.8 o andom co uga ion exci a ion,
whe eas in case o wheel la exci a ion he dynamic ac o is much highe , in he ange o 2.8-4.2,
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wi h la ge alues occu ing a lowe speeds: his is because a low speed he du a ion o he con ac
loss caused by he wheel la is longe and hence he ollowing impac is la ge . The s ess dynamic
ac o

k
o he andom co uga ion exci a ion case is almos mono onically inc easing wi h speed,
wi h a maximum alue close o 1.3. Fo he wheel la exci a ion case, a comple ely di e en end is
obse ed, he s ess dynamic ac o

k
being ini ially dec easing wi h he speed and hen inc easing
abo e 200 km/h. The maximum alue o he s ess dynamic ac o o his exci a ion case is
ob ained a he lowes speed conside ed in he analysis and is sligh ly below 1.5.
As in he case o exci a ion gene a ed by a single-ha monic ail co uga ion, i is in e es ing o
obse e ha in some cases he dynamic ac o

k
exceeds he 1.25 alue assumed in EN13103/104.
Fo wheel la exci a ion, his happens a low speeds (below 75 km/h) which a e ypical e.g. o
eigh applica ion whe eas o andom ail oughness he 1.25 alue is exceeded only a e y high
speeds abo e 300 km/h, which a e only ele an o e y high speed ains. I shall be s essed
howe e ha he esul s shown in Figu e 14 depend on he ampli ude o he de ec s being
conside ed and, in case o mo e se e e i egula i ies, la ge dynamic s esses shall be expec ed.
I is also in e es ing o poin ou ha by using a s a ic calcula ion o de i e he bending s esses,
S anda ds EN13103/104 inhe en ly imply a p opo ionali y be ween he con ac o ces and he
s esses, whe eas he esul s in igu es 10 and 14 show ha he s ess dynamic ac o

k
is
gene ally lowe (some imes much lowe ) han he dynamic ac o o he e ical con ac o ce
Q
k
.
This ci cums ance is due o he ine ia o ces gene a ed in he wheels and o he di e en
magni ica ion o wheelse lexible modes p oduced when esonance condi ions occu and sugges s
ha a me hod based on a s a ic calcula ion could be no ully sui ed o es ima e se ice s esses in
he axle. Dynamic ain- ack in e ac ion models such as he one p oposed in his pape could be
en isaged as a means o de i e a mo e ealis ic es ima e o se ice s esses in he axle, bu his

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equi es he s udy o be ex ended o conside he e ec o cu ing, which is en isaged as a nex
de elopmen o he wo k desc ibed he e.
.
5. CONCLUSIONS
Dynamic e ec s may be ex emely impo an in de e mining he a igue esis ance o ailway axles,
and need o be p ope ly conside ed in he axle design p ocess. This pape has p esen ed a me hod
o he nume ical es ima ion o he dynamic s esses in ailway axles, based on he simula ion o he
in e ac ion be ween a lexible wheelse and a lexible ack. The me hod is p esen ly limi ed o he
case o angen ack unning o he wheelse , bu he ex ension o he cu ing condi ion is
en isaged as a nex s ep o he esea ch, o conside he addi ional axle loading due o wheel- ail
con ac o ces in a cu e.
Resul s o he nume ical p ocedu es we e p esen ed in he pape , conside ing he dynamic exci a ion
caused by ail co uga ion and by a wheel la o di e en wheelse speed alues. The esul s clea ly
show ha dynamic e ec s may lead o a signi ican dynamic ampli ica ion o he s esses in he
axle, which is in some cases close o 70%. Howe e , he ac ual ele ance o dynamic e ec s
a ec ing axle s esses is s ongly depending upon he ype o exci a ion and he ehicle speed.
When single ha monic co uga ion is conside ed, he dynamic ampli ica ion o axle s esses is
ela i ely low excep in he case when he co uga ion wa eleng h exci es he bending modes o he
wheelse . In he case o andom mul i-ha monic ail co uga ion, he dynamic ampli ica ion ac o s
a e ela i ely low because he exci a ion is sp ead o e a wide ange o equencies, hus educing
he impo ance o esonance e ec s. In all ail co uga ion cases conside ed, ega dless he
wa e o m o he exci a ion, he dynamic s ess ampli ica ion ends o inc ease wi h he ehicle
speed, so ha he mos c i ical case is ob ained o high-speed applica ions.
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Wheel la exci a ion also appea s o be c i ical in iew o he axle a igue li e, because high
dynamic ampli ica ion ac o s up o 45-50% a e ob ained, bu on accoun o he localized na u e and
small size o his de ec in his case he mos se e e dynamic exci a ion occu s a low speeds, a
which he equency o he exci a ion gene a ed by he la exci es he bending esonances o he
axle. Fo he wheel la case, he e o e, he mos c i ical case appea s o be ha o eigh ca s,
which may be o en a elling a low speed.
O e all, he analyses p esen ed in his pape show ha dynamic e ec s a e essen ial o co ec ly
es ima e he wheelse ’s a igue li e. Mo e esea ch and alida ion is needed o ans e he indings
p esen ed he e in o axle design p ac ices. Hope ully howe e he pape has highligh ed an impo an
a ea o u u e esea ch, in iew o u he imp o ing he eliabili y o ailway anspo .
Acknowledgemen s
The au ho s g a e ully acknowledge he suppo o his wo k p o ided by he P ojec TRA2010-
15669 (Minis e io de Ciencia e Inno ación) and TRA2007-67167 (Minis e io de Educación y
Ciencia-FEDER).
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ack esponse de i ed om ield es s and nume ical simula ions, P oc. Ins n. Mech. Eng s.
Figu e 3: De ail o he ack model. Le : model o some sleepe bays. Righ :sleepe and
ailpad model.
Figu e 3

Figu e 4: Fini e elemen model o he lexible wheelse .
Figu e 4
Figu e 5: S udied sec ion o wheelse and spa ial poin whe e he s esses ha e been
calcula ed.
y
z
z
x
Figu e 5
Figu e 6: Wheel- ail con ac o ce when he ehicle ci cula es on a co uga ed ack
wi h co uga ion wa eleng h 60 mm. Abo e: e ec o speed and o wheelse lexibili y
o co uga ion ampli ude co esponding o he ISO 3095 limi . Below: e ec o
co uga ion ampli ude o speed 300 km/h and lexible wheelse .
0 0.2 0.4 0.6 0.8 1 1.2
52
54
56
58
60
62
64
66
68
70
Rail oughness
Dis ance x [m]
Ve ical con ac o ce [kN]
Flexible wheelse V = 100 km/h Flexible wheelse V = 300 km/h Rigid wheelse V = 300 km/h
0 0.2 0.4 0.6 0.8 1 1.2
10
20
30
40
50
60
70
80
90
100
110
Rail oughness
Dis ance x [m]
Ve ical con ac o ce [kN]
ISO 3095 10 x ISO 3095
Figu e 6
Figu e 7: y-axis no mal s ess in he s udied sec ion o he axle (c . Fig. 5) when he
ehicle ci cula es on a co uga ed ack wi h co uga ion wa eleng h 60 mm. Le : wo
ehicle speeds, co uga ion ampli ude co esponding o he ISO 3095 limi . Righ : wo
co uga ion ampli udes ( he esul s a e undis inguishable).
  











Ro a ed angle [ ad]

y [MPa]
V = 100 km/h V = 300 km/h
  











Ro a ed angle [ ad]

y [MPa]
ISO 3095 10 x ISO 3095
Figu e 7
Figu e 8: Wheel- ail con ac o ce when he ehicle ci cula es on a co uga ed ack
wi h di e en co uga ion wa eleng hs. Two co uga ion ampli udes a e conside ed,
co esponding o he ISO 3095 limi and 10 imes he ISO 3095 limi . Abo e: he
co uga ion wa eleng h exci es he pinned-pinned mode o he ail. Below: he
co uga ion wa eleng h exci es he i s o wa d bending mode o he wheelse .
0 0.2 0.4 0.6 0.8 1 1.2
10
20
30
40
50
60
70
80
90
100
110
Rail oughness
Dis ance x [m]
Ve ical con ac o ce [kN]
ISO 3095 10 x ISO 3095
0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
10
20
30
40
50
60
70
80
90
100
110
Rail oughness
Dis ance x [m]
Ve ical con ac o ce [kN]
ISO 3095 10 x ISO 3095
Figu e 8

Figu e 9: y-axis no mal s ess in he s udied sec ion o he axle (c . Fig. 5) when he
ehicle ci cula es on a co uga ed ack a 300 km/h speed. Le : he co uga ion
wa eleng h exci es he pinned-pinned mode o he ail. Righ : he co uga ion
wa eleng h exci es he i s o wa d bending mode o he wheelse .
  











Ro a ed angle [ ad]

y [MPa]
ISO 3095 10 x ISO 3095
  







Ro a ed angle [ ad]

y [MPa]
ISO 3095 10 x ISO 3095
Figu e 9
Figu e 10: Dynamic ac o s
Q
k
( e ical con ac o ce, uppe sub igu e) and

k
(y-axis
s ess, lowe sub igu e) o di e en exci a ion cases and ehicle eloci ies. The
oughness ampli ude is en imes highe he ISO 3095 limi .
050 100 150 200 250 300 350
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Vehicle eloci y [km/h]
Dynamic/s a ic con ac o ce a io
No oughness
Co uga ion exci es pinned-pinned equency
Co uga ion exci es 1s o wa d mode
Co uga ion exci es 2nd o wa d mode
050 100 150 200 250 300 350
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Vehicle eloci y [km/h]
Dynamic/s a ic s ess a io
No oughness
Co uga ion exci es pinned-pinned equency
Co uga ion exci es 1s o wa d mode
Co uga ion exci es 2nd o wa d mode
Figu e 10
Figu e 11: Wheel- ail con ac o ce when he ehicle ci cula es a 100 and 300 km/h
speeds on a andomly co uga ed ack.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
10
20
30
40
50
60
70
80
90
100
Rail oughness
Dis ance x [m]
Ve ical con ac o ce [kN]
Flexible wheelse V = 100 km/h Flexible wheelse V = 300 km/h Rigid wheelse V = 300 km/h
Figu e 11
Figu e 12: Wheel- ail con ac o ce when he ehicle ci cula es a 50 (abo e) and 300
km/h (below) speeds on a pe ec ly e en ail in p esence o a 50 mm wheel la .
0 0.5 1 1.5
0
50
100
150
200
250
Dis ance x [m]
Ve ical con ac o ce [kN]
View A
Flexible wheelse V = 50 km/h Rigid wheelse V = 50 km/h
0.1 0.2 0.3 0.4 0.5
0
50
100
150
200
250
Dis ance x [m]
Ve ical con ac o ce [kN]
Zoomed iew A
0 0.5 1 1.5
0
20
40
60
80
100
120
140
160
180
Dis ance x [m]
Ve ical con ac o ce [kN]
View A
Flexible wheelse V = 300 km/h Rigid wheelse V = 300 km/h
0.1 0.2 0.3 0.4 0.5 0.6
0
20
40
60
80
100
120
140
160
180
Dis ance x [m]
Ve ical con ac o ce [kN]
Zoomed iew A
Figu e 12