XII Con e ence on T anspo Enginee ing, CIT 2016, 7-9 June 2016, Valencia, Spain
Dynamic Modelling o High Speed Ballas ed Railway T acks:
Analysis o he Beha iou
Inmaculada Gallego Gine
Associa e P o esso , Depa men o Ci il and Building Enginee ing, (UCLM), Spain
Ana Ri as Al a ez
Associa e P o esso , Depa men o Ci il and Building Enginee ing, (UCLM), Spain
San os Sánchez-Camb one o Ga cía-Mo eno
Associa e P o esso , Depa men o Ci il and Building Enginee ing, (UCLM), Spain
Julián Laja a Camacho
Mas e S uden . Ci il Enginee , (UCLM), Spain
Abs ac
The aim o he pape is o p esen a nume ical model o a ballas ed ailway ack ha includes he dynamic e ec o a mo ing
ain load and p edic s he alues o he e ical s i ness o he in as uc u e. This model is he e o e deemed o be a ool o he
e alua ion o he s a e o he ack du ing se ice si ua ions as well as a p edic i e model o he beha iou o he sys em.
Consequen ly, i will be e y use ul when sizing he c oss sec ion o a new ailway line is equi ed.
The main modelling ool is he ini e elemen me hod. In ega d o his, he applica ion o damping elemen s o a oid he elas ic
wa e e lec ion on he bounda ies o he nume ical domain will be s udied. The p oposed dynamic analysis conside he change in
ime o he alue o he ain load, bu no he change in posi ion along he acks.
In he end, a se o sugges ions o he nume ical model wi h mo ing loads will be summa ize aiming o he mi iga ion o he
unusual beha iou o he con ac su ace be ween he ballas and he sleepe s.
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
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1. In oduc ion
P ac icing enginee s equi e a wide ange o skills in oday’s compe i i e wo ld. Al hough hey mus be awa e o
he echnical, en i onmen al and economic con ex in which hei ask a e held, some imes he scale o he p ojec s do
no allow o con ol as many a iables as i should be equi ed. This p esen a ion is an example o how a me hodology
can be implemen ed o help he enginee s in he decision making p ocess when acing design p oblems in high speed
ailways.
In oday’s economy, budge ing and p ojec app aisal has become mo e and mo e impo an du ing he incep ion o
a new in as uc u e and du ing he ende ing p ocess. I is c ucial o p o e ha he money is well expended. Bu when
designe s ha e no he app op ia e ools, sa ing money become a he a hope han a ac . This example i s e y well
he case o he design o High Speed ballas ed acks. In Spain, he me hodology o he design o his kind o ailway
lines ha e elied on he Ca d 719R by he UIC (Union In e na ional de Chemins de Fe , 1994). Acco ding o his, he
hickness o he ballas and sub-ballas laye s depend on he a ic o he line, and he quali y o he ma e ials o he
subg ade. Di e en adminis a ions ha e added hei own ecommenda ions, acco ding o he na ional know-how.
The e ha e no been many discussions abou whe he his me hodology is ou da ed o whe he i can be imp o ed.
The aim o he ollowing documen is o p o ide an idea o he esea ch in his opic, on how nume ical me hods
can be adop ed in o de o gua an ee a cos e ec i e in as uc u e wi h ools ha a e cu en ly a ailable and ha e been
p o ed igh and use ul by he indus y. Fo his ma e , a Fini e Elemen Model has been buil o be desc ibed in his
pape .
When dealing wi h complex p oblems, wi h non-analy ical solu ions o equa ions, o wi h in e ac ions be ween
elemen s we need o ely on he ou pu s p o ided by nume ical analysis. O he wise, he amoun o simpli ica ions may
lead o he s udy o a o ally di e en scena io o e en a w ong one. Wi h his ega ds, he Fini e Elemen Me hod can
be a help ul ool o he ailway designe o e en o he adminis a o s and manage s. Jus imagine he amoun o
money ha would need o be pu in expe imen al es s, o he nuisance o on-si e es ing, causing dis up ions o he
line and a ec ing e en ually o he use s. An ini ial in es men o he calib a ion o a nume ical model can be enough
o cha ac e ise he whole in as uc u e and ha e a be e unde s anding o o eseeable p oblems.
The basic ad an age ha de i es om he use o a model is ha i allows he pa ame isa ion o he basic geome y
and p ope ies. The e o e, hese pa ame e s can be changed and adap ed o he di e en equi emen s o he ack
aking in o accoun spa ial a ia ions, empo al e ec s, a igue o he elemen s, e c.
The model p oposed in his pape include a wide ange o pa ame e s, such as he numbe o sleepe s o he model
and he o e all geome y o he subs uc u e and supe s uc u e.
2. S a e-o - he-a in nume ical modelling o ailways
The i s nume ical models aimed a he s udy o he beha iou o ballas ed ailway acks we e de ised in he 70s
and 80s. Those models we e based on a mul ilaye in ini e semi space composed by ho izon al and homogeneous
laye s. The ma e ials we e supposed linea elas ic and he con ac be ween laye s was assumed con inuous.
Howe e , his assump ions we e lacking some consis ency, as he geome y o he laye s play an impo an ole o
he analysis, he ma e ials does no beha e linea ly elas ic and he con ac be ween he laye s is no pe ec .
Fu he mo e, a mul ilaye sys em app oxima es he beha iou o he sys em when he elas ic modulus inc ease in dep h
and he elas ic modulus a e simila . Ac ually his is no e y ep esen a i e, as he elas ic modulus o he s eel o he
ails is e y di e en om he elas ic modulus o he soil o he na u al ill.
Apa om ha , he e is an impo an ques ion o answe : is he ballas well ep esen ed by i s elas ic p ope ies?
The answe is no, and i is mo e accu a e o use a disc e e elemen s model (DEM) o ep esen i han assuming a
con inuous media like he soil. A ini e elemen me hod (FEM) model can es ima e he elas ic beha iou o he ballas
in he o e all s uc u e, bu i canno ep esen accu a ely s esses.
In ailways, some o he p oposed model we e bidimensional, so a s a e o plane de o ma ion was assumed (López
(1977)). This did no wo k as expec ed, u ning he esea che s o analyse 3D models (Sau age and La ible (1982);
P o illidis (1983), (1987); Sahu e al (1999) and he Minis e io de Fomen o (1999)). F om his, he s udies by P o illidis
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
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a he L’École Na ionale des Pon s e Chaussées we e in eg a ed by he D-117 Commi ee o he O ice o Resea ch
and and Expe imen s (ORE) in he UIC 719R File men ioned abo e.
In his cen u y, he model de eloped by Gallego (2006) in his PhD Thesis ook all he expe ise de i ed om
p e ious models and con ibu ed o he pa ame isa ion o he geome y and he soil p ope ies, allowing o a be e
in e p e a ion o esul s and o ca y ou a sensi i i y analysis o wha e e he pa ame e wan ed o be checked, om
he hickness o he laye s o he slope o he shoulde s o he mechanical p ope ies o he ma e ials.
Up o his poin , mos o he models we e s a ic, allowing only o dynamic conside a ion using dynamic
ampli ica ion ac o s on he loads. Fu he mo e, he dynamic s i ness is equal o he s a ic s i ness o he sys em. The
objec i e o modelling he dynamic beha iou o he sys em becomes o pa amoun impo ance when designing a high
speed ailway in as uc u e. This models ha e p oblems which we e no p esen in he s a ic analysis. The main o
hose is he p opaga ion o wa e in elas ic media. A ime dependen load and ime analysis conside he ine ial and
iscous e ec s o he ma e ials.
The e a e 3 ypes o ways o add ess he dynamic p oblem: (i) Comple e ini e elemen s me hod model analysis, as
o example Hall (2003) and Al Shae e al (2008); (ii) ini e elemen s models wi h ini e elemen s applied in he
bounda ies (Cos a e al (2010) and Nguyen (2013)) and (iii) ini e elemen s wi h bounda y elemen s applied in he
bounda ies (O’B ien and Rizos (2005), Chebli e al (2008) and Rome o (2009)).
The di e en pape s ha ha e been ead so a has wo ypes o app oaches. Those ha a e ex emely ma hema ical
and based on equa ions, and hose mainly empi ical based on expe imen al models and da a. This wo k wan o si in
he middle o his wo wo lds. I wan o p o ide an up- o-da e ool using nume ical simula ion and calib a ed
expe imen ally, bu wi hou he need o building complex ma hema ical model. The e o e, i should be a ool accessible
o p o essionals o ailway enginee ing and pe ec o analysis and design in gene al, wi hou a e y speci ic scope.
Table 1: Mechanical p ope ies o he elemen s o he ack
Ma e ial
E (N/m2)
S eel ( ails)
2.1x1011
Basepla es
2.952x108
Sleepe s
5.02x1010
Ballas
1.3x108
Sub-ballas
1.2x108
Fill
1.25x107- 3.0x109
The e a e some speci ic issues ega ding he use o he FEM o model ailway acks:
 The e is a big di e ence be ween he mechanical p ope ies o he elemen s. Table 1 shows some
examples.
 The e is a big di e ence in he size o he elemen . While he ails a e jus a ew cen ime e s, he heigh
subg ade may be a ound se e al me e s. Again, we ind ha d o wo k wi h di e en o de s o magni ude.
 Al hough he media is assumed con inuous, and his assump ion wo ks well in mos cases, he s esses in
he ballas and sub-ballas canno be es ima ed co ec ly using con inuum mechanics, as he size
dis ibu ion o he g ains is es ic i e.
Ne e heless, a FEM model becomes use ul o de e mining he o e all beha iou o he s uc u e in e ms o
de lec ion, i makes he in e ac ion be ween elemen s and laye s easy o ea and unde s and and makes possible he
applica ion o di e en bounda y condi ions, as well as he in oduc ion o di e en geome ies wi hin he same model.
3. Desc ip ion o model nume ic p oposed
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
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The model de eloped in Gallego (2006) and ex ended in Gallego e al (2009), (2011), (2012), (2013) and (2015) is
he base o he analysis he eby p esen ed. The model has been used a e ha in o he s udies o cha ac e ise he
sensi i i y o di e en pa ame e s such as he quali y and hickness o he ill and he na u al ill laye s in e ms o
de lec ions. This model was based on he p oposed by he Spanish Minis y o Public Wo ks wi h he ollowing
di e ences (apa om he pa ame isa ion o he geome y):
 The slope o he shoulde s o he subs uc u e a e conside ed in he analysis, al hough no bounda y
condi ions apply in his a ea.
 The geome y is simpli ied by symme y wi h espec o he axis o he alignmen .
 The load is applied in s ages. In he i s s age only he sel -weigh is p esen . In he second s age he loads
a e applied. This loads a e di ided in di e en load scena ios o simula e a ain passing o e he ails. The
load ac s o e 4 sleepe s assuming di e en coe icien s simula ing he in luence o he load in he es o
he sleepe s.
The model was implemen ed in ANSYS ©. ANSYS © Mechanical is a ini e elemen analysis ool o s uc u al
analysis, including linea , nonlinea and dynamic s udies. ANSYS LS-DYNA combines he LS-DYNA explici ini e
elemen p og am wi h he powe ul p e and pos p ocesso o ANSYS © so wa e. In o de o use wisely he capabili ies
o he so wa e, he LS-DYNA and he s anda d compu a ion we e used o he quasi-s a ic si ua ion and he s a ic
espec i ely.
A ansien analysis has been used, in oducing di e en loading si ua ions. This in ol e he use o an implici
analysis and an explici analysis wi h he LS-DYNA. In he implici analysis, he s i ness ma ix o he whole s uc u e
has o be in e ed in o de o sol e o he accele a ion ield i e a i ely. This has a huge compu a ional cos when he
model becomes g ea e in size o o he use o con ac elemen s. The explici me hod is much mo e app op ia e,
because i is as e and i does no need he in e sion o he s i ness ma ix.
3.1. S a ic / Dynamic
The me hodology o he g oup has ollowed a na u al e olu ion o he model. Fi s , a s a ic analysis was ca ied
ou , accoun ing o dynamic e ec s using a dynamic ampli ica ion ac o o he s a ic loading condi ions.
A e ha , a quasi-s a ic model in ol es he disc e isa ion o a ime dependen load. In his case, o each ime s ep
he so wa e needs o sol e a di e en load case acco ding o he a ia ion o he load wi hin ime. Bu ha load has
no been conceded as a mo ing load. Tha is he nex s ep o pe o m. A ully dynamic model in which he load a ies
wi hin ime bu also i is able o mo e along he ails. This will comple e he cycle o simula ions in o de o ake in o
conside a ion all he e ec s o a eal case si ua ion. The e is also many mo e issues o add ess. Fo example, he
in luence o he dynamic e ical s i ness in he o e all beha iou o he s uc u e. Ano he s ep would be he
inco po a ion o an elas oplas ic law o he soils while main aining he dynamic analysis.
3.2. Mesh o he model
Al hough in he s a ic model he size o he model does no play an impo an ole, in he dynamic si ua ion he
wa e p opaga ion will a ec he size o he elemen s, he size o he model and he ime s ep conside ed in he analysis.
In o de o a oid he nume ic dispe sion o he esul s, he ollowing expe imen al ela ion ha e been ollowed:
1
5<∆𝑠
𝜆<1
10 (1)
Whe e ∆s is he dis ance be ween consecu i e nodes and λ is he wa e leng h. To a oid an excessi e numbe o
nodes in he model, he less es ic i e alue was chosen (1/5). I we ela e he wa e leng h wi h a ain o loads, we
ind ha he sepa a ion be ween nodes will be in luenced by he cele i y o he wa e in he soil and by he equency
o he load ( his is he ain passing o e ):
∆𝑠 = 𝜆
5=𝑣𝑇
5=𝑣
5𝑓 (2)
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
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Whe e 𝑣 is he cele i y o he wa e and is he equency gene a ed by a ain. I we wan o ind he maximum
allowable dis ance be ween nodes we should choose he minimum cele i y o p opaga ion (co esponding o he weake
ma e ial) wi h he maximum equency ( he as e he ain can pass). The wa e cele i y can be easily de i ed, and is
unc ion o he mechanical p ope ies o he ma e ial (Young’s modulus and Poisson’s a io). F om he wo ypes o
wa es (P ima y and Seconda y), he mos es ic i e a e he Seconda y wa es. The pe iod o he ain will allow us o
de i e he equency o ac ua ion o he load (T = 1/ ). Assuming a load pe axis o he ain, and being able o know
he dis ance be ween bogies (d) and he eloci y o he ain ( ), he equency can be ob ained om:
1
𝑓=𝑑
𝑣
(3)
This allow us o ind he maximum allowable ∆s be ween nodes.
I we measu e now a dis ance D (see igu e 1) in pa allel o he main di ec ions o ou model om he poin o
applica ion o he load, we would be able o ind he dimensions o he model i we ha e a es ic ion ega ding he
wa e p opaga ion and mos impo an , he wa e e lec ion. The e o e, we mus gua an ee ha he model is big enough
o allow a wa e o a el bu no o e lec in he bounda ies and dis o he ou pu s o he model.
Figu e 1: The p oposed 3D nume ical model
This condi ion is sa is ied o any D > max/2 min. The maximum eloci y is ound in he ails, as he cele i y o
he P ima y wa es in s eel is abou 6140 m/s, and he minimum equency co esponds wi h he minimum allowable
eloci y o he ain. An easy calcula ion will gi e us a alue o D = 276 m. This alues makes he model un easible
unless we use damping bounda y elemen s able o abso b he wa e e lec ion.
The elemen s used by ANSYS © mus change be ween s a ic and dynamic analysis. Fo he s a ic one, SOLID95
p o ides he bes solu ion, whe eas o dynamic analysis wi h con ac s algo i hm is manda o y o use he SOLID164.
3.3. The con ac s be ween he sleepe s and he ballas
When dealing wi h mul ilaye s, he con ac be ween su aces need a special cha ac e isa ion. In his model, he
ocus has o be pu on he con ac su ace be ween he sleepe s and he ballas . Because o he loads ac ing on he ails,
he sleepe s a e subjec ed o high s esses, and hose need o be ansmi ed o he ballas . I is he e o e ine i able ha ,
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
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when wo e y di e en ma e ial a e in con ac , hey beha e di e en ly. So he ques ion is now how o accommoda e
he displacemen s and s esses o he sleepe s wi h he displacemen s and s esses o he ballas . Bu , again, he ballas
is p oblema ic, because we a e modelling i as a con inuum soil. So, i he de lec ion o he sleepe is such ha c ea es
ension in he media below, ha si ua ion mus be co ec ed, because i is a physical nonsense.
The nume ical ea men o he in e aces o he model equi e a e y speci ic ea men . The c i ical su ace we e
his codes ha e been applied a e he con ac su ace be ween he sleepe s and he ballas , which a e c i ical o he
analysis.
The so wa e ANSYS © allows wo ypes o solu ions: con ac elemen s and node coupling:
 Con ac Elemen s: i equi es he duplica ion o he nodes o he con ac su ace. I in oduces a ic i ious
laye (null hickness) be ween he elemen s. This laye has no mal s i ness and a shea s i ness. Basically,
i allows he sepa a ion o he nodes o he ballas and he sleepe s, i.e., no physical con ac . This sol es
he p oblem o inding ension s esses in he ballas . On he o he hand, i a comp essi e s ess is ac ing,
he angen shea moduli allow o he de elopmen o shea s esses. This ic ion is ele an in he
p oblem, no only in e ms o s esses, bu also in e ms o ene gy dissipa ion when dealing wi h dynamic
analysis. Howe e , his algo i hm inc eases conside ably he compu a ional cos o he p oblem and
inco po a e new pa ame e s o he model ( he ic ional law o he in e aces), which would equi e special
cha ac e isa ion and a special ea men .
 Coupling Nodes: i equi es he duplica ion o he nodes in con ac , one o he nodes is pa o he uppe
laye and he o he one is pa o he bo om laye . Be ween opposi e nodes he e is kinema ic cons ain .
This condi ion is applied o he “mas e ” node, and he o he , he “sla e” node aces he condi ion imposed
by he mas e . The no mal mo emen wi h espec o he con ac su ace is ee whe eas he ela i e
displacemen be ween su aces is es ained. This algo i hm p o ides a solu ion o he discon inui y o
s esses and displacemen s be ween sleepe and ballas wi hou inc easing he compu a ional ime.
Howe e , i does no ake in o accoun he ic ion be ween he elemen s and he associa ed dissipa ion o
ene gy and i does no allow he physical sepa a ion be ween sleepe and ballas .
Wha is he bes solu ion o modelling in e aces? Acco ding o he li e a u e and he expe ience h oughou he
yea s, he sepa a ion be ween sleepe and ballas does occu (see Figu e 2), and he ic ion be ween g anula and no
g anula ma e ials is ele an in he analysis, so he con ac elemen s a e needed o model co ec ly he ack. Howe e ,
he inc ease in compu a ional ime may become es ic i e in some cases, and he de e mina ion o a ic ional law
be ween he conc e e and he ballas is no ob ious. As a i s app oach, he ic ion can be es ima ed using geo echnical
c i e ia assuming pe ec in e ac ion be ween conc e e and g anula ma e ials. This is assumed as 2/3 o he ic ion
angle o he ballas .
3.4. Bounda y elemen s
The p oblem o he e lec ion o he wa es in he bounda ies can be o e come using damping elemen s. When
modelling geomechanical sys ems, and in ini e domain is o en equi ed o ep esen he g ound o o he la ge bodies.
Fo his ype o si ua ions, using non- e lec ing elemen s become use ul. This condi ions, which he so wa e applies
au oma ically o he nodes, will p e en he a i icial wa e e lec ions gene a ed a he bounda y om een e ing he
model and con amina ing he esul s. When his condi ions a e included, he algo i hm compu es an impedance
ma ching unc ions o all he bounda y segmen s based on an assump ion o linea ma e ial beha iou . This p oblem
can also be add essed using nume ical abso bing bounda y condi ions based on a damped wa e equa ions.
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
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Figu e 2: De o ma ion o he ail-pla e-sleepe assembly e sus he unde o med con igu a ion
3.5. Cons i u i e laws
Ini ially, we can ind wo g oups o ma e ial: ails, basepla es and sleepe s plus he g anula ma e ials. The
beha iou o he ails, basepla es and sleepe s can be assumed as linea iso opic elas ic. The es o he ma e ials a e
conside ed o be soils ( om a modelling pe spec i e iew). Those a e assumed o be con olled by a pe ec
elas oplas ic law wi h a D ucke -P age yielding su ace. This is only alid o s a ic simula ions. The dynamic analysis
equi e a linea cons i u i e law. Unde se ice loads, howe e , a plas ic analysis is no equi ed, as he esponse o
he ail oad can be pe ec ly app oxima ed using non-linea elas ic cons i u i e laws.
3.6. Loading condi ions
The way he load is applied need a special ea men in he p oblem, because depending on he way he load is inpu
in he model we would need o change he geome y. The ul ima e objec i e is o simula e he pass o a ain o e he
ails, and his can be achie ed in wo ways:
 T iangula load: The iangula load is he easies example o a ain load. I is applied di ec ly o e he
nodes o he ails, and i ep esen s a simple axis o he bogie. The pa ame e s o his a e he maximum
load and he ime i akes o he load o each ha maximum. By he momen , only a ime dependen load
has been conside ( u u e model will include space a iabili y as well).
 M load: This ype o load simula es he e ec o a bogie wi h wo axis ollowing he app oach de ined in
Sau age (1983) (see Figu e 3). I has an M shape wi h wo peaks, co esponding o he maximum alues
o he load. This load is di ec ly applied o he sleepe s, so he e is no need o model he ail. This may
become handy, because wi hou he ail, he e is no in luence o he wa e p opaga ion on he s eel (which
was he mos limi ing ma e ial), he e o e i has many ad an ages. Among some, a smalles model. On he
con a y, he unc ion o he M load depends on wo non-dimensional empi ical a iables (X and Y) ha
a e ela ed wi h he elas ic modulus o he soil and wi h he e ical s i ness o he ack.
Rega ding he M load, se e al issues ha e been de ec ed. One o he mos impo an is ha he lack o ails
dis o he eal beha iou o he acks. The ails ac e en ually as a ie o he sleepe s, making hem
connec ed wi h one ano he . When he M load is applied di ec ly o he sleepe s, he e is no elemen
connec ing he sleepe s, so he e is no in luence o s ess dis ibu ion be ween hem. The ail is also
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esponsible o main aining he ela i e posi ion o he sleepe s. This implies ha we ha e o ely on he
load unc ion o do his wo k spa ially. This issues a ec no ably he model o he s uc u e.
Figu e 3: M load simula ing he e ec o a bogie wi h wo axis (Sau age G. (1993))
The ime s ep is also an impo an pa ame e when deciding how o ep esen he load. This also ep esen s he
equency he so wa e is ge ing da a o pe o m he calcula ions. The e a e many ecommenda ions o he ime s ep
in he li e a u e, bu he wo mos ecommended a e he one p oposed by Chop a (∆ /T < 0.1) whe e T is he pe iod
o he highes equency and he one ela ed o he wa e cele i y, which can be de i e om:
∆𝑡 < 𝑉
𝐴∙𝑐
(4)
Whe e V is he olume o a ini e elemen is, A is he a ea o he la ges side o he ini e elemen and c is he highes
wa e cele i y in he di e en ma e ials.
Apa om his, only e ical load ha e been conside ed. In ailway enginee ing he e is an impo an ho izon al
load due o he hun ing mo emen o he ain, bu his ou o he scope o his documen .
3.7. Compu a ional cos
The compu a ional cos o he model depends on wo ac o s:
 Size o he mesh: in p e ious sec ions we ha e ound a ela ion o de i e he spacing be ween nodes. I is
ecommended o he dynamic model no o ha e symme y planes (because he wa e e lec ion would be
in luenced by ha ac ). The e o e, he g a e he model becomes, he numbe o nodes inc ease cubically.
Bu i he model is no big enough, he in luence o he wa e e lec ion would cause a signi ican dis o ion
o he esul s p o iding useless ou pu s.
 Time s ep: Due o he load disc e isa ion and o gua an ee he con e gence c i e ia, he ime s eps need o
be much educed. This may lead o simula ions o a ew seconds wi h mo e han hund eds in e media e
s eps and calcula ions o he whole model.
This makes necessa y o use supe compu e s o sol e he model, e en wi h educed geome ies and simpli ica ions.
4. Conclusions
In spi e o he p oblems exposed in his pape , a FEM model becomes use ul o de e mining he o e all beha iou
o he s uc u e in e ms o de lec ion, i makes he in e ac ion be ween elemen s and laye s easy o ea and unde s and
and makes possible he applica ion o di e en bounda y condi ions, as well as he in oduc ion o di e en geome ies
wi hin he same model. Howe e , i is necessa y o conside he ollowing ecommenda ions:
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 I we a e dealing wi h a dynamic analysis, hen he cons i u i e law o ma e ials should be linea . One s ep
would be he inco po a ion o an elas oplas ic law o he soils while main aining he dynamic analysis.
 Al hough in he s a ic model he size o he model does no play an impo an ole, in he dynamic si ua ion
he wa e p opaga ion will a ec he size o he elemen s, he size o he model and he ime s ep conside ed
in he analysis.
 Acco ding o he li e a u e and he expe ience h oughou he yea s, he sepa a ion be ween sleepe and
ballas does occu , and he ic ion be ween g anula and no g anula ma e ials is ele an in he analysis,
so he con ac elemen s a e needed o model co ec ly he ack.
 The p oblem o he e lec ion o he wa es in he bounda ies can be o e come using damping elemen s,
o non- e lec ing elemen s, o one model big enough o allow a wa e o a el bu no o e lec in he
bounda ies and dis o he ou pu s o he model.
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