Resonance and cancella ion phenomena in wo-span con inuous beams and
i s applica ion o ailway b idges
M.D. Ma ´ınez-Rod igoa,∗, A. Ande ssonb, C. Pacos eb,c, R. Ka oumib
aUni e si a Jaume I, Depa men o Mechanical Enginee ing and Cons uc ion, Cas ell´on, Spain
bDi ision o S uc u al Engiee ing and B idges, KTH Royal Ins i u e o Technology, S ockholm, Sweden
cELU Konsul AB, S ockholm, Sweden
Abs ac
The objec i e o his s udy is o e alua e he ib a o y esponse o wo-span con inuous beams subjec ed
o mo ing loads and, in pa icula , o in es iga e he maximum esonance and cancella ion o esonance
phenomena. The main p ac ical in e es is he e alua ion o he maximum accele a ion esponse in ailway
b idges, which is one o he mos demanding Se iceabili y Limi S a es o a ic sa e y acco ding o cu en
egula ions. Two-span con inuous b idges, in hei simples e sion (i.e. uni o m iden ical spans), p esen
an isymme ic and symme ic modes wi h closely spaced na u al equencies, leading o a mo e in ol ed
dynamic beha iou han ha o simply-suppo ed b idges. Fi s , he ee ib a ion esponse o a Be noulli-
Eule wo-span beam a e he passage o a single load a cons an speed is o mula ed analy ically, and
non-dimensional speeds leading o cancella ion o maximum esponse in ee ib a ion a e ob ained o each
mode. Then, hese condi ions a e equa ed o esonan speeds induced by equidis an load se ies, and span
leng h- o-cha ac e is ic dis ance a ios causing cancelled ou esonances, o ema kably p ominen ones,
a e ob ained. Based on he p e ious de i a ions, a me hodology o de ec ing which could be he mos
agg essi e ains o a pa icula s uc u e based on pu e geome ical conside a ions is discussed. Finally,
he applicabili y o he heo e ical de i a ions is shown h ough he nume ical analysis o wo eal b idges
belonging o he Swedish ailway ne wo k.
Keywo ds: Railway b idges, wo-span con inuous beams, esonance, cancella ion, mo ing loads.
1. In oduc ion
The p og essi e inc ease in ope a ing speeds on ailway lines cons i u es a challenge o adminis a ions,
olling s ock manu ac u e s and enginee s as ib a ion le els ha a e admissible o passenge s, ehicles,
in as uc u es and su ounding s uc u es mus be gua an eed. In his ega d, ailway b idges ha e ecei ed
conside able a en ion du ing he las decades. The pe iodic exci a ion caused by he axle loads c ossing a5
∗Co esponding au ho .
Email add ess: [email p o ec ed] (M.D. Ma ´ınez-Rod igo)
P ep in submi ed o Enginee ing S uc u es July 15, 2020
b idge o a iaduc a cons an speed may induce signi ican le els o e ical oscilla ions on he deck, which
can lead o ad e se consequences such as ballas deconsolida ion, ack misalignmen , passenge discom o
o e en wheel- ail con ac loss and associa ed isks [1],[2],[3].
Among he longi udinal ypologies o ailway iaduc s, bo h b idges wi h con inuous decks es ing on
mul iple suppo s and b idges composed o simply-suppo ed (SS) spans coexis . The o me , common10
in coun ies such as Spain, Sweden o F ance, a e s uc u ally mo e e icien and able o ansmi he
ho izon al b eak and accele a ion o ces o he g ound wi h he collabo a i e ac ion o he subs uc u e
elemen s. The la e , equen ly ound in coun ies like Ge many o China, may be cons uc ed in a a he
sys ema ic way, allow possible p e ab ica ion, pa ial eplacemen o he SS decks and acili a e con inuous
ails. Ne e heless, he highe numbe o join s and suppo ing de ices inc eases he main enance cos s and15
hese s uc u es a e usually app op ia e only when pie s ha e a limi ed heigh [4],[5]. Simply-suppo ed
b idges ha e ecei ed conside ably mo e a en ion han con inuous s uc u es. This wo k is de o ed o wo-
span con inuous b idges, as he simul aneous con ibu ion o he ans e se ib a ions o an isymme ic and
symme ic modes cons i u es a mo e complex p oblem and hese s uc u es may s ill expe ience impo an
ampli ica ions unde ailway a ic [6],[7].20
The basic phenomenon go e ning he le el o ib a ions induced in a b idge by a ailway con oy is
he ampli ude o he ee ib a ions ha each axle lea es on he s uc u e a e i s passage, as hese ee
ib a ion wa es accumula e and may add in phase o ce ain speeds. Depending on he a io be ween he
a elling ime o he load and he na u al pe iod o he s uc u e, he ampli ude o he ee ib a ions in
ha pa icula mode may be maximum o may be cancelled ou , implying ha he s uc u e will emain25
a es unde ce ain ideal condi ions (i.e. in he absence o s uc u al damping). This basic p oblem has
been analysed in de ail by au ho s such as Yang e al. [8],[9], Sa in [10], Muse os e al. [11] and Kuma
e al. [12] o simply- and elas ically-suppo ed beams. The in e es o knowing hese condi ions a p io i
is ha when esonance, caused by se ies o loads, akes place a a maximum ee ib a ion eloci y o , on
he con a y, close o a cancella ion condi ion, he ain will induce ei he a e y p ominen esponse o an30
almos impe cep ible one, espec i ely.
The p oblem o a con inuous uni o m beam wi h wo equal spans a e sed by a concen a ed o ce
mo ing a cons an speed was i s sol ed by Ay e e al. [13], who desc ibed he mo ing load as a se ies o
pulsa ing o ces. The ea ly wo ks on he p oblem o mo ing loads ac ing on simple s uc u es is desc ibed in
de ail in he classic e e ence by F ´yba [14]. In he 1990s, Zheng e al. [15] and Cheung e al. [16] analysed35
he ib a ions o mul i-span beams subjec ed o mo ing o ces and oscilla o s, espec i ely, using modi ied
beam ib a ion unc ions as he assumed modes. The au ho s showed as con e gence o he me hod wi h
a small numbe o unknowns when compa ed o Fini e Elemen (FE) solu ions. Yau [7] in es iga ed he
e ec o he numbe o spans in he dynamic esponse o mul i-span uni o m beams subjec ed o ain loads.
The au ho concluded ha he inc ease in he numbe o spans esul s in he appea ance o mo e esonan 40
2
peaks as he speed inc eases, bu also in a educ ion in he impac esponse. This is a ibu ed o he
ansmission o he ib a ion ene gy o he neighbou ing spans and o he highe es aining e ec o he
suppo s on he ans e se displacemen . Dugush and Eisenbe ge [17] ob ained he exac solu ion o he
mo ing load p oblem on mul i-span non-uni o m beams o any polynomial a ia ion in he c oss-sec ion
p ope ies using he exac elemen me hod. Johansson e al. [18] de i ed a closed- o m exac solu ion o 45
e alua ing he dynamic beha iou o a gene al mul i-span Be noulli-Eule (BE) beam unde cons an mo ing
loads, conside ing s epped sec ions and elas ic bounda y condi ions.
Publica ions de o ed o esonance o i s cancella ion in wo-span beams o b idges a e a he sca ce and
ecen . Yau [7] in es iga ed he impac esponse o con inuous beams and, conside ing mo ing loads, de ec ed
mul iple esonan peaks due o he coincidence o he exci a ion equency wi h he beam equencies. Kwa k50
e al. [19] p esen ed one o he i s expe imen al s udies on he dynamic beha iou o a wo-span con inuous
b idge unde he ci cula ion o Ko ean High-Speed ains along wi h FE p edic ions including ehicle-b idge
in e ac ion. Resonances o only he undamen al an isymme ic mode we e cap u ed o he pa icula design
speed. The au ho s ocused on he e ec o damping and ehicle-s uc u e in e ac ion on he ans e se
displacemen esponse. Wang e al. [20] analysed he esonan esponse o wo-span b idges modelled as BE55
uni o m beams subjec ed o ains o mo ing oscilla o s. The au ho s ocused on he e ical esponse in
e ms o displacemen s, and on he appea ance o wo c i ical speeds causing esonance associa ed o he i s
an isymme ic and i s symme ic modes o ib a ion. Wang e al. [6] nume ically in es iga ed he e ical
accele a ion o wo-span con inuous b idges wi h long spans (40 and 45 m) and uni o m c oss-sec ions unde
he ac ion o High-Speed ains, modelled as equally spaced 2-deg ees-o - eedom mass-sp ing-dampe uni s.60
This is one o he ew publica ions o ocus on he accele a ion esponse o he b idge a he han on he
displacemen . The au ho s concluded ha he esonan accele a ion esponse in b idge and ehicle may be
ampli ied o a ai ly high deg ee, especially o he i s wo esonan speeds ha may all wi hin he ange
o ope a ing speeds o High-Speed ains. Mo eo e , due o he p esence o sub- esonan speeds o highe
modes, he maximum accele a ion may occu a sec ions o he han mid-span.65
In he au ho s’ opinion, wha i is p esen ed he ein is use ul, no el and con ibu es o he knowledge on
he dynamic pe o mance o ailway b idges as (i) i p o ides a comp ehensi e s udy on he cancella ion
and maximisa ion o esonance in wo-span con inuous b idges based o he ee ib a ion esponse o he
s uc u es; (ii) p e ious publica ions gene ally add ess speci ic s uc u es and ains and, in he opinion o
he au ho s, he p oblem should be o mula ed non-dimensionally, in o de o each gene al conclusions; (iii)70
mos o he p e ious con ibu ions ocus on he displacemen esponse o he b idge, gene ally go e ned by
only a ew modes [21]-[22], and no on he accele a ion, which is a a mo e es ic i e Se iceabili y Limi
S a e o hese s uc u es acco ding o cu en egula ions [23]; and (i ) being able o p edic eloci ies
leading o maximum ee ib a ion o cancella ion is o p ac ical in e es as no only he mos and leas
agg essi e ains may be de ec ed o a pa icula s uc u e bu , in addi ion, his in o ma ion could also be75
3
use ul when planning expe imen al campaigns on b idges wi h he aim o iden i ying ampli ude-dependen
magni udes (e.g. modal damping).
The objec i es o his s udy a e o (i) in es iga e analy ically he p oblem o ee ib a ions in wo-span
con inuous beams; (ii) e i y whe he maximum ee ib a ion and cancella ion condi ions ake place and, i
so, o de e mine hei alue o any longi udinal bending mode and o any s uc u e; (iii) ob ain geome ical80
a ios leading o maximum esonance and cancella ion o i o symme ic and an isymme ic modes and o
p o e hei applicabili y when ideal condi ions a e no me ; and (i ) apply he o me heo e ical de i a ions
o he applica ion o wo-span b idges unde High-Speed a ic by p oposing a me hodology o de ec which
could be he mos and leas agg essi e ains o a pa icula design speed o he line, and wha kind o
esonance (o de and mode) is esponsible o i .85
The con en s o he manusc ip a e o ganised as ollows. In sec ion 2 he ee ib a ion esponse o a
BE wo-span con inuous beam unde a single mo ing load is o mula ed analy ically, and maximum ee
ib a ion and cancella ion non-dimensional speeds a e p esen ed and ob ained o each mode. In sec ion
3 he exci a ion caused by ains o equidis an loads is conside ed and leng h- o-cha ac e is ic dis ance
a ios a e de i ed leading o maximum o cancelled di e en o de esonances o any mode o he beam.90
In sec ion 4 a me hodology is p oposed so as o be able o compa e he e ec o di e en ains on he
maximum accele a ion esponse o he b idge admi ing a maximum ope a ional speed o he line. Finally,
wo b idges om he Swedish ailway ne wo k a e e alua ed in o de o show he applicabili y o he o me
heo e ical de i a ions. Conclusions a e p esen ed in sec ion 5.
The de i a ions and conclusions p esen ed he ein a e limi ed o he e ec o he geome y o he ains95
on he b idge esponse. Addi ional phenomena con o ming he ain-induced ib a ion p oblem may a ec
he maximum accele a ion esponse, such as ehicle- ack-b idge ([24],[25],[26]) o soil-s uc u e in e ac ion
([27],[28]). The in luence o hese e ec s on he maximum esponse o he b idge may o may no be
ele an depending on he le el o coupling be ween he subsys ems. This s udy p e ends o cons i u e a
i m heo e ical base on which addi ional in e ac ion e ec s, as hose p e iously men ioned which en ail an100
impo an le el o unce ain y, may be e alua ed in subsequen in es iga ions.
2. F ee ib a ions o a wo-span con inuous beam unde a single mo ing load
2.1. Ampli ude o ee ib a ions in undamped case
The pa ial di e en ial equa ion go e ning he undamped ans e se ib a ions o a BE beam, neglec ing
shea de o ma ion and o a o y ine ia, a e sed by a cons an - alued load Pmo ing a cons an speed105
(see Fig. 1(a)) is gi en by
ρA(x)∂2w(x, )
∂ 2+∂2
∂x2EI(x)∂2w(x, )
∂x2=−Pδ (x−V )H( )−H −2L
V (1)
4
whe e w(x) is he ans e se displacemen o a gene ic sec ion xa ime ,ρA(x) is he mass pe uni o
leng h o he beam and EI(x) ep esen s he c oss-sec ion bending s i ness. In Eq. 1, δand Hs and o
Di ac Del a and Hea iside uni unc ions, espec i ely. The solu ion o Eq. 1 may be exp essed as a linea
combina ion o he beam no mal modes o ib a ion φi(x) as pe 110
w(x, ) =
∞
X
i=1
ξi( )φi(x) (2)
Fo he simples case o a wo-equal-span uni o m beam wi h o al leng h 2L(see Fig. 1(a)), applying
app op ia e bounda y condi ions and pe o ming a ee ib a ion analysis [29], he analy ical no mal modes
a e ob ained o an isymme ic and symme ic modes, φa
iand φs
i, espec i ely, and can be exp essed as
φa
i(x) = sin λa
ix
L0≤x≤2L λa
i=i+ 1
2π i = 1,3,5, ... (3a)
φs
i(x) = 






sin λs
ix
L−sinh λs
ix
L
sin λs
i
sinh λs
i
0≤x≤L
sin λs
ix0
L−sinh λs
ix0
L
sin λs
i
sinh λs
i
0≤x0≤L
λs
i=i+ 0.5
2π i = 2,4,6, ... (3b)
whe e x0= 2L−x. In Eqs. 3a-3b and in wha ollows, supe sc ip s aand sa e used o di e en ia e
an isymme ic and symme ic modes when conside ed necessa y. λa
iand λs
ia e he oo s o he equency115
equa ion (Eq. 4a) and a e ela ed o he ci cula equencies as pe Eq. 4b
sin (λ) (cos (λ)−co h (λ) sin (λ)) = 0 ⇒λa
i, λs
i(4a)
ωa
i=λa
i
L2sEI
ρA ωs
i=λs
i
L2sEI
ρA (4b)
In Figs. 1(c)-(d) he i s h ee an isymme ic and i s h ee symme ic modes a e ep esen ed o equal
maximum displacemen no malisa ion. An isymme ic and symme ic modes al e na e wi h he equency
numbe , and each modal equency inc eases in ela ion o he undamen al one acco ding o he ac o s
i= a
1·{1,1.56,4,5.06,9,10.56,16,18.06, ...}(5)
whe e bold numbe s co espond o an isymme ic modes equencies.120
By subs i u ion o Eqs. 3a, 3b and 2 in o 1, mul iplica ion by he n- h mode, in eg a ion along he beam
leng h, and in i ue o he o hogonali y condi ion o he modes, he uncoupled equa ion go e ning he
n- h modal ampli ude is ob ained:
¨
ξn( ) + ω2
nξn( ) = −P
Mn
φn(x=V )Mn=Z2L
0
ρAφn(x)2dx =ρAL (6)
5

L
x,u x’x’x,u
z,w
P
(a)
(c)
(b)
(d)
V
dd d
P
k
P
k
P
k+1
P
k+1
P
k+2
P
k+2P
k-1
1
s
l 1.25p=
2
s
l 2.25p=
3
s
l 3.25p=
2
a
l 2p=
1
a
l p=
3
a
l 3p=
z,w
LL L
Figu e 1: Two-span uni o m beam unde mo ing loads a elling a cons an speed (a)-(b). Fi s h ee (c) an isymme ic and
(d) symme ic no mal modes.
In o de o de e mine he ampli ude o he ee ib a ions once he load Plea es he s uc u e, he
p e ious equa ion and i s i s de i a i e a e sol ed by con olu ion o he pa icula ins an = 2L/V125
ξn( =2L
V) = −P
ωnMnZ2L
V
0
φn(V τ) sin [ωn( −τ)] dτ ˙
ξn =2L
V=dξn( )
d  =2L
V
(7)
Finally, he ampli ude o he ee ib a ions in each mode is ob ained and non-dimensionalised by he
modal s a ic displacemen as pe Eq. 8. By doing so, he so-called no malised ampli ude o he ee
ib a ions, Rn, can be ob ained o an isymme ic and symme ic modes in e ms o a single non-dimensional
eloci y, Kn. This speed pa ame e has also been used by p e ious au ho s [9]. In Eq. 9 he pa icula closed
o m exp ession o he an isymme ic modes is p o ided. Fo he symme ic case he analy ical exp ession130
is a he in ol ed and i is no included o he sake o conciseness. None heless, Rnmay be compu ed o
any mode ei he analy ically o by nume ical e alua ion o Eq. 7. No ice ha he ampli ude o he ee
ib a ions does no decay, as damping is no conside ed in his sec ion.
Rn=ω2
nMn
−P
u
u
˙
ξ2
n =2L
V
ω2
n
+ξ2
n =2L
V⇒Rn= Kn=λnV
ωnL(8)
Rn=√2Kn
1−K2
ns1−cos (1 + n)π
Knn= 1,3,5, ... (9)
6
In Figu es 2(a) and 2(b), he e olu ion o Rnis p esen ed e sus he speed pa ame e co esponding
o he undamen al mode, K1, o he i s an isymme ic (n= 1) and he i s symme ic (n= 2) modes
o he wo-span uni o m beam, espec i ely. Fig. 2(c) also ep esen s Rn e sus K1bu o he i s h ee
an isymme ic modes (i.e. n= 1,3,5). As bo h Rnand Kna e non-dimensional, hese ep esen a ions
and he conclusions de i ed he ea e a e applicable o any wo-equal-span uni o m beam. Rep esen ing he
esponse wi h espec o he same speed pa ame e (K1ins ead o Kn) allows di ec compa ison in e ms
o he speed V o he di e en modal esponses. The ela ion be ween any modal speed pa ame e Knand
he one e e ed o he undamen al mode can easily be deduced o an isymme ic and symme ic modes
as
Kn=




2
n+ 1K1n= 1,3,5, ...
2
n+ 0.5K1n= 2,4,6, ...
(10)
F om he analysis o Fig. 2 he ollowing can be concluded:
•Depending on he a elling speed, he ampli ude o he ee ib a ions ha a pa icula beam unde -135
goes in a ce ain mode once he load lea es he s uc u e can be maximum o cancelled ou , aking
in o accoun ha no damping is p esen in he sys em. The speed pa ame e s o hese cancella ion
and maximum ee ib a ion condi ions, Kci
1and Kmi
1, can be ob ained analy ically and a e poin ed
ou in Figu es 2(a)-(b) o he i s an isymme ic (n= 1) and he i s symme ic (n= 2) modes,
espec i ely. No ice ha index i e e s o a pa icula cancella ion o maximum ee ib a ion e en ,140
and ha i= 1 co esponds o he e en aking place a he highes eloci y.
•The eloci ies cancelling ou he ee ib a ions o he undamen al mode also cancel ou he esponse
o he emaining an isymme ic modes (see Fig. 2(c)). This does no occu among he symme ic
modes o among hese and he undamen al mode.
•When damping is p esen and o mode a e le els o i , like hose usually iden i ied in ailway b idges145
[1], Rnp esen s a simila e olu ion in e ms o he speed pa ame e Kn. The main di e ences in
he damped case a e ha (i) ampli udes a local maxima a e lowe , and ha (ii) he esponse is no
comple ely cancelled ou a cancella ion speeds, al hough he esponse is ema kably low. The e ec
o damping will be accoun ed o in sec ions 3 and 4.
Finally, i is o in e es o no e ha , due o he selec ed no malisa ion and o he ela ion be ween
displacemen and accele a ion ampli udes in he undamped case, he dimensional e ical displacemen and
accele a ion ampli udes in a pa icula sec ion o he wo-span beam in he ee ib a ion phase induced by
a load Pis ela ed o Rnacco ding o
w ee
n(x) = Rn·P
ω2
nρAL ·φn(x)a ee
n(x) = Rn·P
ρAL ·φn(x) (11)
7
n
(b)
(a)
(c)
1
K
1
c3
K
1
c3
K
1
m3
K
1
m3
K
1
c2
K
1
c2
K
1
m2
K
1
m2
K
1
c1
K
1
c1
K
1
m1
K
1
m1
K
2
R1
R
R
n= 1
n= 1
n= 3
n= 5
n= 2
1 ( =1)
s cancella ion i
1 ( =1)
s cancella ion i
2 ( =2)
nd cancella ion i
2 ( =2)
nd cancella ion i
3 d cancella ion i( =3)
3 d cancella ion i( =3)
1 ( =1)
s local maximum i
1 ( =1)
s local maximum i
2nd local maximum i( =2)
2nd local maximum i( =2)
3 d local maximum i( =3)
3 d local maximum i( =3)
Figu e 2: Rn o (a) i s an isymme ic mode, (b) i s symme ic mode, and (b) i s h ee an isymme ic modes s. K1.
F om Eq. 11 i is seen ha he ela i e di e ences be ween Rn alues in Fig. 2 a e p opo ional o150
he ela i e di e ences be ween he accele a ion ampli udes in ee ib a ion in a ce ain mode and in a
pa icula sec ion.
2.2. Cancella ion and maximum ee ib a ion speeds
F om he solu ion o Rn= (Kn) (Eq. 8) cancella ion and maximum ee ib a ion non-dimensional
speeds may be calcula ed as
Rn(Kn)=0⇒Kci
n
∂Rn(Kn)
∂Kn
= 0 ⇒Kmi
ni= 1,2,3, ... (12)
F om now on, Kci
nand Kmi
ns and o he speed pa ame e s leading o he i- h cancella ion and i- h maximum
ee ib a ion condi ions in mode n, espec i ely. In Table 1 he alues o he i s ou cancella ion and155
8
maximum ee ib a ion speeds a e included o he i s wo an isymme ic and he i s wo symme ic
modes. As should be expec ed, he alues o he undamen al mode coincide wi h hose o he SS beam o
he second bending mode i Lis he span leng h [11].
nKci
nKmi
n
i= 1 i= 2 i= 3 i= 4 i= 1 i= 2 i= 3 i= 4
1 0.5000 0.3333 0.2500 0.200 0.8883 0.4094 0.2886 0.2235
3 0.6667 0.5000 0.4000 0.3333 0.9653 0.5812 0.4478 0.3652
2 0.4835 0.3624 0.2758 0.2282 0.7312 0.4202 0.3157 0.2509
4 0.6201 0.5107 0.4044 0.3488 0.8409 0.5625 0.4542 0.3758
Table 1: Values o he i s ou cancella ion and maximum ee ib a ion speed pa ame e s o he i s wo an isymme ic
(n= 1,3) and i s wo symme ic (n= 2,4) modes.
As he main p ac ical in e es o his esea ch is he e alua ion o ailway-induced ib a ions in wo-span
b idges, i is wo h men ioning ha a ealis ic uppe limi o K1, which will always p esen he highes alue160
o Knacco ding o Eq. 10, can be es ima ed. Admi ing an a e age b idge undamen al equency in e ms
o he span leng h as pe [30], conside ing a maximum ope a ional speed o 500 km/h and a minimum span
leng h o 15 m, a maximum alue K1≃0.5 is ob ained. The e o e, he o e all i s maximum co esponding
o i= 1 will ne e be eached in a ealis ic si ua ion.
3. Fo ced ib a ions unde equidis an loads: maximum esonance and cancella ion165
3.1. Span leng h- o-cha ac e is ic dis ance a ios o maximum esonance and cancella ion o esonance
In his sec ion he dynamic esponse o he wo-span beam is in es iga ed unde ains o equidis an
loads a elling a cons an speed V(see Fig. 1(b)). When he ime in e al be ween he passage o wo
consecu i e loads is a mul iple o one o he beam na u al pe iods, esonance is induced. A esonance, he
ee ib a ions le by e e y single load add in phase. The e o e, depending on he ampli ude o he ee170
ib a ions s udied in sec ion 2, esonance could esul in a ema kably ampli ied esponse (i he esonan
speed coincides wi h a maximum ee ib a ion speed) o may no e en be pe cep ible (i i coincides wi h
o is close o a cancella ion condi ion). Ideally, a ain o equidis an loads wi h cha ac e is ic dis ance d
induces a j- h o de esonance o he n- h mode when a elling a V
nj, as pe Eq. 13, [9]. Mo eo e , he
esonan speed may be exp essed non-dimensionally, acco ding o he speed pa ame e de ini ion in Eq. 8,175
as
V
nj =d n
j⇒K
nj =λn
ωnL
d n
j(13)
9
and, he e o e, does no ake place ei he . I is impo an o s ess ha he esponse is calcula ed including
he con ibu ion o six modes and in he p esence o damping. Fo hese wo easons, he supp ession o he
esponse is no comple e.
(L
/d)n j i m/c E en Fig. 6
0.563 1 1 1 m 1s maximum o 1s mode 1s esonance (a)-(b)
0.855 2 1 1 m 1s maximum o 2nd mode 1s esonance (c)-(d)
1.000 1 1 1 c 1s cancella ion o 1s mode 1s esonance (e)-( )
1.725 2 1 2 c 2nd cancella ion o 2nd mode 1s esonance (g)-(h)
2.000 1 1 3 c 3nd cancella ion o 1s mode 1s esonance (i)-(j)
Table 4: L/d a ios and associa ed e en s ep esen ed in Fig. 6.
F om he p e ious analysis i can be concluded ha when esonance is induced on a wo-span con inuous
beam o b idge by a ain o equidis an loads, i s ampli ica ion will depend on he le el o ee ib a ions265
associa ed wi h he pa icula eloci y. Resonances o ei he he i s (an isymme ic) o second (symme ic)
mode a e p one o be esponsible o he o e all maximum e ical accele a ion o he b idge. The same
ain will equi e a highe speed in o de o induce he same esonance o he second mode (when compa ed
o ha o he undamen al), wi h a highe le el o ee ib a ions le by each axle load. The e o e, he
second mode can be he one esponsible o he maximum esponse as long as he ain speed is su icien ly270
high. In o he wo ds, he mode causing he maximum o e all esponse o he b idge will depend on he
maximum ain speed.
Depending on he a io be ween he leng h and he ain cha ac e is ic dis ance, he esponse a esonance
may be a he p ominen o almos impe cep ible. Fu he mo e, he analy ical p edic ions o hese L/d a ios
leading o maximum esonance o cancella ion o i , which we e ob ained in closed o m in he absence o 275
damping and admi ing sepa a e modal con ibu ions, show hemsel es o be good es ima es o he eal
alues. This is due o he mode a e damping alues in ailway b idges and also o he ac ha , a esonance,
he con ibu ion o modes o he han he one unde going esonance is e y limi ed.
4. Case s udies
In he p e ious sec ion, he condi ions o maximum esonance and i s cancella ion ha e been analysed280
in he ideal case o pe ec ly equidis an ains o loads and unlimi ed ain speeds in o de o in es iga e he
e olu ion o he esonan ampli udes in he comple e domain o non-dimensional speeds and L/d a ios. In
his sec ion, he analysis o he dynamic pe o mance o wo eal b idges is p esen ed unde he ci cula ion
o a icula ed load ains conside ing ealis ic uppe -limi design speeds.
16

The b idges o in e es , desc ibed in subsec ions 4.1 and 4.2, belong o he Swedish ailway ne wo k, in285
pa icula , o he Bo hnia and o he Wes Coas lines, espec i ely. The maximum ain speed a bo h
si es is a p esen 200 km/h. The possibili y o inc easing he ope a ing speed on hese lines wi h a a ge
o 250 km/h is cu en ly unde s udy, and he e alua ion o he pe o mance o he b idges i hese lines a e
upg aded is a opic o majo in e es o he Swedish ailway adminis a ion [32], [33].
The s uc u es unde s udy a e single- ack and, in a i s app oach, he con ibu ion o modes o he han290
he longi udinal bending ones (e.g. o sion, ans e se bending) is dis ega ded. The esponse o he b idges
unde he ci cula ion o he High Speed Load Model-A (HSLM-A) om Eu ocode [30] (see Appendix A o
ain model de ini ion) is ob ained by ime in eg a ion, admi ing ha he o al esponse can be exp essed
as a combina ion o he analy ical modes o he wo-span con inuous BE beam as desc ibed in sec ion 3
(i.e. he na u al equencies and mode shapes a e ob ained applying Eqs. 3a, 3b and 4b). The design speed295
conside ed o bo h b idges is 300 km/h (i.e. 1.2 imes he a ge ope a ing speed).
4.1. Case 1. B idge o e Ri e L¨ogde in V¨as e bo en, Sweden
The i s s uc u e unde s udy is a b idge c ossing he Ri e L¨ogde on he Bo hnia ailway line, be ween
he ci ies o ¨
O nsk¨olds ik and Gimon¨as. I is a con inuous b idge wi h wo 43 m iden ical spans and a
uni o m s eel-conc e e composi e deck, as shown in Fig. 7. The deck accommoda es a single ballas ed ack.300
The main p ope ies o he beam model o his b idge a e hose also used in [33], and a e summa ised in
Table 8. A modal damping a io o 0.5% is admi ed as ecommended in [30] o composi e b idges o he
a o emen ioned span leng h.
3.735 m 3.735 m
Figu e 7: B idge o e he L¨ogde i e . Ele a ion iew and c oss-sec ion.
The i s wo bending equencies o he b idge a e 2.34 and 3.66 Hz. Table 6 p esen s he c i ical speeds
o he en HSLM-A ains’ cha ac e is ic dis ances o he lowes esonance o de a ainable gi en he design305
speed, and o he i s wo modes, which will be he ones ha con ibu e mos o he accele a ion esponse,
as shown la e on. F om now on, dks ands o he cha ac e is ic dis ance o he k- h ain. The dimensional
esonan speeds a e compu ed applying Eq. 13, and he non-dimensional ones e e o he undamen al mode
o bo h esonan eloci ies o he i s and he second mode, o con enience. All he ains induce i s
17
L¨ogde b idge F¨o sl¨o b idge
L(m) 43.00 23.50
EI Nm21.05 ·1011 7.14 ·1010
ρA (kg/m) 13816 23010
ζn(%) 0.50 1.00
1, 2(Hz) 2.34,3.66 5.01,7.83
Table 5: P ope ies o he L¨ogde and F¨o sl¨o b idges.
esonance o he undamen al mode unde 300 km/h. None heless, only he i s i e ains, wi h smalle 310
cha ac e is ic dis ances, a e able o do so in he case o he second mode, due o i s highe equency. The
lowes esonance o de a ainable o his second mode o ains A6 o A10 is hen second o de (j= 2).
T dk(m) L/dk
n= 1 n= 2
j V
1j(km
h)K
1jR1FP k R1FP k/ω2
1j V
2j(km
h)K
1jR2FP k R2FP k/ω2
2
A1 18 2.39 1 151.8 0.209 0.28 1.30 ·10−31 237.1 0.327 0.39 7.38 ·10−4
A2 19 2.26 1 160.2 0.221 0.54 2.52 ·10−31 250.3 0.345 0.01 2.00 ·10−5
A3 20 2.15 1 168.6 0.233 0.42 1.95 ·10−31 263.5 0.363 0.48 9.05 ·10−4
A4 21 2.05 1 177.1 0.244 0.17 7.90 ·10−41 276.7 0.382 0.83 1.56 ·10−3
A5 22 1.95 1 185.5 0.256 0.15 7.12 ·10−41 289.8 0.400 0.79 1.50 ·10−3
A6 23 1.87 1 193.9 0.267 0.45 2.06 ·10−32 151.5 0.209 0.01 2.20 ·10−5
A7 24 1.79 1 202.4 0.279 0.65 3.02 ·10−32 158.1 0.218 0.34 6.45 ·10−4
A8 25 1.72 1 210.8 0.291 0.70 3.22 ·10−32 164.7 0.227 0.38 7.22 ·10−4
A9 26 1.65 1 219.2 0.302 0.68 3.12 ·10−32 171.3 0.236 0.15 2.89 ·10−4
A10 27 1.59 1 227.6 0.314 0.47 2.18 ·10−32 177.9 0.245 0.23 4.37 ·10−4
Table 6: L¨ogde b idge highes a ainable esonan speeds and ee ib a ion ampli udes o n= 1,2 unde HSLM-A ains.
In o de o compa e he le el o ee ib a ions associa ed o each ain a he esonan speeds, he
alues o K
1ja e supe imposed o he no malised ampli ude o he ee ib a ions o he i s wo modes,
R1and R2in Fig. 8. The e ical black lines s and o he esonan speeds o he undamen al mode and315
he g ey e ical ones o hose o he second mode. The in e sec ion o each e ical line wi h ei he R1(K1)
o R2(K1) (only in e sec ions o aces o he same colou should be conside ed) p o ides an es ima ion o
he le el o accele a ion expe ienced by he b idge due o he accumula ion o ee ib a ions in a ce ain
mode a esonance. Mo eo e , as pe Eq. 11, he ampli ude o ei he R1o R2is mul iplied by he ac o
FP k =Pk/P1in o de o accoun o he di e en axle load modulus o he HSLM-A ains, Pkbeing he320
axle load o he k- h ain and P1= 170kN, which is he minimum alue. This co ec ed p oduc is shown
18
wi h a ed ci cle ha has a black bo de in he case o esonances o he i s an isymme ic mode and a
g ey bo de in he case o esonances o he second mode. Admi ing ha he numbe o loads is su icien
and ha he esonance s a e has eached a cons an ampli ude due o he p esence o damping, his may be
used o compa e he ela i e ampli udes o he accele a ion a esonance induced by di e en ains on he325
i s wo modes o ib a ion. This is o cou se an es ima ion ha only akes in o accoun he geome y o
he composi ions (i.e. he leng hs o he passenge s’ coaches) and admi s simila modal damping a ios o
bo h modes, bu i allows a p elimina y p edic ion o which ain will induce he mos de imen al esonance
and which mode will be he one unde going i , aking in o conside a ion all cancella ion and maximum ee
ib a ion si ua ions. In Table 6 he alues o RnFP k and RnFP k/ω2
na e included as well o each ain and330
esonance speed. No ice ha he i s is p opo ional o he ampli ude o he accele a ions in ee ib a ion
and he la e o he ampli ude o he displacemen s as pe Eq. 11. Also, he o e all maximum alue o
each o hese a ios is highligh ed in bold. Acco ding o his, he maximum displacemen could occu when
ain A8 induces i s esonance o he undamen al mode (j= 1, n= 1), while he maximum accele a ion
may ake place a i s esonance o he second mode induced by ain A4. This ain should be one o 335
he mos agg essi e ains o he pa icula s uc u e and speed limi acco ding o he Se iceabili y Limi
S a e o a ic sa e y.
1
K
n
R
n= 1
n= 2
A5
A10
A9
A8
A7
A6
A5
A1-A6
A2
A3
A4-A10
A4
A3
A2
A1
A7
A8
A9
A4, =2n
R K ·P P( ) /
1
1j k1
R K ·P P( ) /
2
1j k1
Figu e 8: L¨ogde b idge. Rn s. K1 o n= 1,2 and he mos c i ical esonan non-dimensional speeds om HSLM-A ains
unde 300 km/h.
The esponse o he L¨ogde b idge o he ci cula ion o he en HSLM-A ains is now calcula ed and
p esen ed in he ange o a elling speeds [20,83.33] m/s wi h ∆V= 0.5 m/s (i.e. [72,300] km/h wi h
∆V= 1.8 km/h). The maximum esponse is ob ained conside ing he con ibu ion o he i s wo modes340
and he i s six modes a sec ions x/L = [0.25,0.5,0.75,1.25,1.5,1.75]. The o e all maximum accele a ion
akes place a mid-span o he second span x/L = 1.5 in bo h cases. In Fig. 9 he maximum ans e se
displacemen and accele a ion a e plo ed in absolu e alues a his mos c i ical sec ion e sus he non-
19
dimensional speed V/ 1d o he en HSLM-A ains. Plo s (a) and (b) a e calcula ed aking in o accoun
he i s wo modal con ibu ions (N= 2), while plo s (c) and (d) ep esen he esponse calcula ed wi h345
six modes (N= 6). The maximum esponse induced by he HSLM-A4 is ep esen ed wi h a ed ace.
The o e all maximum accele a ion eaches 6.01 m/s2 o N= 2, exceeding he limi o ballas ed acks
acco ding o s anda ds [23]. The e o e, his b idge may need o be imp o ed in o de o allow inc eased
ain speeds. These esul s a e consis en wi h hose p esen ed by Ande sson [33]. As p edic ed, o he
admi ed design eloci y he maximum esponse in e ms o accele a ions is due o a i s esonance o he350
second mode (V/ 1d= 1.56), and i is induced by he HSLM-A4 ain ( ed ace). As pe he displacemen ,
ain A8 (g een ace), oge he wi h A7 and A9, lead o he maximum displacemen a i s esonance o
he undamen al mode. No ice in Table 6 ha he h ee ains (A7, A8 and A9) p esen a e y simila
alue o R1FP k/ω2
1. In Fig. 9 i can also be obse ed ha ain A2 does no induce i s esonance o he
symme ic mode. Fo his ain L/dk= 2.26, e y close o he heo e ical alue 2.266 o cancella ion o he355
second mode i s esonance (see Table 3). Finally, i should be no ed ha he e ec o modes highe han
he second one is e y low, especially a esonance. In he displacemen esponse, he di e ence is almos
impe cep ible.
1
V d/1
V d/
N=6
w L(1.5 ) [m]
max N=2
w L(1.5 ) [m]
max
N=2
2
a L(1.5 ) [m/s ]
max
N=6
2
a L(1.5 ) [m/s ]
max
(a)
(c)
(b)
3.5 m/s2
3.5 m/s2
(d)
Figu e 9: L¨ogde b idge. (a)-(b) Maximum displacemen and accele a ion a x= 1.5L(N= 2). (c)-(d) Maximum displacemen
and accele a ion a x= 1.5L(N= 6). HSLM-A ains, Vmax = 300 km/h.
20
4.2. Case 2. F¨o sl¨o b idge in Sk˚ane, Sweden
As a second example, he case o a p e-s essed conc e e ailway b idge om he Wes Coas line loca ed360
be ween he ci ies o Go henbu g and Copenhagen is p esen ed. A modi ied e sion o he eal s uc u e is
analysed, wi h wo iden ical spans o 23.5 m and a uni o m c oss-sec ion wi h he p ope ies lis ed in Table
8. The b idge is composed o wo s uc u ally independen single- ack decks as shown in Fig. 10. A modal
damping a io o 1% is assigned o each mode as pe [30].
3.5 m 2.25 m0.2
Figu e 10: F¨o sl¨o b idge. Ele a ion iew and c oss-sec ion.
The i s wo na u al equencies o he b idge calcula ed analy ically a e 5.01 Hz and 7.83 Hz, espec i ely.365
Again, he heo e ical esonan equencies a e compu ed o he i s wo modes and he i s esonan o de s.
In his case s udy he na u al equencies a e highe han in he p e ious one. Fo his eason, he c i ical
eloci ies leading o i s esonance o he i s wo modes exceed he maximum design speed o 300 km/h
assumed o all he HSLM-A ains. Again, in Table 7 he highes a ainable esonan eloci ies o he i s
and second modes ha e been included, along wi h he a ios RnFP k and RnFP k/ω2
n, p opo ional o he370
accele a ion and displacemen ampli udes in ee ib a ion, espec i ely. The highes alues o hese wo
a ios a e highligh ed in bold. In his case when ain A10 induces a second esonance o he undamen al
mode, o ha pa icula speed he ee ib a ion ampli udes bo h o he displacemen s and he accele a ions
a e maximum. The e o e his ain could be one o he mos agg essi e.
In Fig. 11 he highes a ainable non-dimensional esonan speeds o each ain a e ep esen ed wi h375
e ical solid aces o modes n= 1 (black) and n= 2 (g ey). Again, he non-dimensional ampli ude o he
ee ib a ions, Rn o each ain in each mode, is ma ked wi h a ci cle a e applying he co ec ing ac o
FP k. In his case, all he ains in he HSLM model a e capable o inducing a second-o de esonance o
he i s an isymme ic mode, bu only he i s ou ha e a su icien ly low cha ac e is ic dis ance o induce
second-o de esonance o he i s symme ic mode below 300 km/h. I can be e i ied g aphically ha ain380
HSLM-A10 is he one leading o a highes alue o RnFP k, in pa icula o he i s mode (n= 1), as i s
associa ed second esonance speed coincides wi h he hi d local maximum o he ee ib a ions o n= 1.
In wha ollows, he esponse o he b idge is ob ained nume ically unde he en HSLM-A ains. Fig.
21

T dk(m) L/dk
n= 1 n= 2
j V
1j(km
h)K
1jR1FP k R1FP k/ω2
1j V
2j(km
h)K
1jR2FP k R2FP k/ω2
2
A1 18 1.31 2 162.3 0.191 0.26 2.58 ·10−42 253.7 0.299 0.36 1.49 ·10−4
A2 19 1.24 2 171.4 0.202 0.08 8.24 ·10−52 267.7 0.316 0.59 2.43 ·10−4
A3 20 1.18 2 180.4 0.213 0.38 3.85 ·10−42 281.8 0.332 0.31 1.27 ·10−4
A4 21 1.12 2 189.4 0.223 0.52 5.29 ·10−42 295.9 0.349 0.12 5.00 ·10−5
A5 22 1.07 2 198.4 0.234 0.37 3.78 ·10−43 206.7 0.244 0.14 5.71 ·10−5
A6 23 1.02 2 207.4 0.245 0.15 1.50 ·10−43 216.1 0.255 0.45 1.86 ·10−4
A7 24 0.98 2 216.4 0.255 0.16 1.59 ·10−43 225.5 0.266 0.50 2.08 ·10−4
A8 25 0.94 2 225.5 0.266 0.44 4.42 ·10−43 234.9 0.277 0.26 1.08 ·10−4
A9 26 0.90 2 234.5 0.277 0.69 6.98 ·10−43 244.3 0.288 0.11 4.34 ·10−5
A10 27 0.87 2 243.5 0.287 0.77 7.79 ·10−43 253.7 0.299 0.44 1.84 ·10−4
Table 7: F¨o sl¨o b idge highes a ainable esonan speeds and ee ib a ion ampli udes o n= 1,2 unde HSLM-A ains.
12 shows he maximum displacemen and accele a ion esponses o he F¨o sl¨o b idge. The esponse is
ob ained o he en HSLM-A ains a sec ions x/L = [0.25,0.5,0.75,1.25,1.5,1.75]. The esponse o he385
b idge is e alua ed o each ain in he ange o speeds [20,83.33] m/s in speed inc emen s o ∆V= 0.5 m/s
(i.e. [72,300] km/h and ∆V= 1.8 km/h). The o e all maximum accele a ion akes place a he mid-span
sec ion o he second span, as in he p e ious example. In Fig. 12 he maximum ans e se displacemen
and accele a ion a e plo ed in absolu e alues a his mos c i ical sec ion e sus he non-dimensional
speed V/ 1d o he en ains. The esponse in Fig. 12(a) and 12(b) is calcula ed aking in o accoun he390
con ibu ion o he i s wo modes, while ha in Fig. 12(c) and 12(d) is compu ed aking in o conside a ion
he con ibu ion o he i s six modes o ib a ion.
The maximum accele a ion eaches 2.89 m/s2when aking in o accoun six modes (2.86 m/s2i only he
i s wo a e conside ed), below he ecommended limi o a ic sa e y on ballas ed acks. The maximum
accele a ion is associa ed o a second esonance o he undamen al mode (i.e. V/ 1dk= 0.5), as p edic ed395
in he p e ious analysis o he le el o ee ib a ions. The ain leading o he maximum accele a ion
esponse is A10, wi h L/dk= 0.87, close o he heo e ical a io o he maximum second esonance o he
undamen al mode (see Table 2) and also consis en wi h wha is shown in Table 7 and Fig. 11. This ain
is also he one esponsible o he maximum displacemen , which is also caused by he second esonance
o he undamen al mode. Only a ew ains a e able o induce second esonance o he second mode (i.e.400
V/ 1d= 0.78) and he esponse in e ms o accele a ions is much lowe han ha induced by he mos
agg essi e ain in he i s mode. In Fig. 12 i can also be obse ed ha ain A2 does no induce second
esonance o he undamen al mode. Fo his pa icula ain and b idge L/dk= 1.24, e y close o he
22
n= 1
n= 2
1
K
n
R
A4
A1
A2
A3
A4
A5
A3
A2
A10-A1
A10-A9
A10, =1n
A9-A8
A8-A7
A7-A6
A6-A5
R K ·P P( ) /
1
1j k1
R K ·P P( ) /
2
1j k1
Figu e 11: F¨o sl¨o b idge. Rn s. K1 o n= 1,2 and he mos c i ical esonan non-dimensional speeds om HSLM-A ains
unde 300 km/h.
heo e ical alue o 1.25 o cancella ion o his pa icula esonance (see Table 2). I is impo an o s a e,
again, ha he maximum esponse is mainly go e ned by he i s wo modes o ib a ion o he wo-span405
beam, and ha he con ibu ion o highe modes in he accele a ion esponse a esonance is negligible.
Finally, in o de o summa ize he s eps applied in his sec ion o p opose he pa icula ain leading o
he maximum displacemen and accele a ion o he b idges a esonance a low cha is included in Fig. 13.
5. Conclusions410
In his wo k he e ical esponse o wo-span uni o m con inuous beams unde mo ing equidis an loads
is in es iga ed. The main p ac ical applica ion is o assess he maximum accele a ion in ailway b idges
induced by ains a elling a esonan speeds, and i s ela ion wi h he le el o ee ib a ions le by each
axle load and go e ned by he exis ence o cancella ion and maximum ee ib a ion phenomena.
Fi s , he p oblem o a uni o m wo-span con inuous beam is o mula ed, and a uni less exp ession o 415
he ee ib a ion ampli udes le by he ci cula ion o a single load a elling a cons an speed in a gene ic
mode is ob ained in e ms o a speed pa ame e . The exis ence o cancella ion and maximum ee ib a ion
condi ions is p o en, bo h o symme ic and an isymme ic modes, and he condi ions o bo h si ua ions
a e p esen ed and analysed. Second, he o ced ib a ions o he beam unde sequences o equidis an
ains a e analysed nume ically, and L/d a ios a e de i ed leading o cancelled o maximum ampli ude420
esonances. Due o he non-dimensional na u e o he o mula ion used, hese a ios a e applicable o any
s uc u e. The abo e men ioned de i a ions enable us o unde s and wha ype o esonance will lead o
he highes accele a ion esponse o he b idge and wha ype o ain may be he mos de imen al o a
ce ain design eloci y, limi ing he analysis o exclusi ely geome ical conside a ions. Finally, he p e ious
23
1
V d/1
V d/
N=6
w L(1.5 ) [m]
max N=2
w L(1.5 ) [m]
max
N=2
2
a L(1.5 ) [m/s ]
max
N=6
2
a L(1.5 ) [m/s ]
max
(b)(a) 3.5 m/s2
3.5 m/s2(d)(c)
0.005
0.004
0.003
0.002
0.001
0.000
0.005
0.004
0.003
0.002
0.001
0.000
Figu e 12: F¨o sl¨o b idge. (a)-(b) Maximum displacemen and accele a ion a x= 1.5L(N= 2). (c)-(d) Maximum displace-
men and accele a ion a x= 1.5L(N= 6). HSLM-A ains, Vmax = 300 km/h.
1
K
2
R
2
2
w
Pk
F
2
R
2
2
w
Pk
F
2
RPk
F
2
RPk
F
1
R
1
2
w
Pk
F
1
R
1
2
w
Pk
F
1
RPk
F
1
RPk
F
max{ , } max{ , }
max
max
max
V
1j_min
1j_min
V< V 1j
Kn( =1)
1j
Kn( =1)
1j
Kn( =2)
1j
Kn( =2)
V< V
k N=1,...,
k N=1,...,
max
k
T w( ) max
k
T a( )
12
, k k
d , P
®
®
Eq. (13)
Eq. (8)
Figu e 13: S eps applied o iden i y ains leading o he maximum displacemen and accele a ion a esonance.
24
heo e ical de i a ions a e applied o wo case s udies o wo b idges om he Swedish ailway ne wo k and425
hei applicabili y is shown.
The main conclusions de i ed om he esea ch conduc ed a e:
•When a load mo ing a cons an speed a els on a wo-span con inuous beam, he le el o ee
ib a ions le by he load in a ce ain mode may be maximum o negligible, depending on he a io
be ween he load eloci y and he beam equency. These condi ions o maximum ee ib a ion and430
cancella ion can be ob ained analy ically in a non-dimensional o ma .
•Linea eloci ies ha cancel ou he ee ib a ions in he i s an isymme ic mode also cancel ou he
esponse o he emaining an isymme ic modes. This does no occu among he symme ic modes o
among hese and he undamen al one.
•When esonance is induced on a wo-span con inuous beam o b idge by a ain o equidis an loads,435
i s ampli ica ion will depend on he le el o he ee ib a ions associa ed o he pa icula eloci y.
Mo eo e , depending on he a io be ween he leng h and he ain cha ac e is ic dis ance, he esponse
a esonance may be a he p ominen o almos impe cep ible.
•The analy ical p edic ions o he L/d a ios leading o maximum esonance o i s cancella ion, ob ained
in he absence o damping and admi ing sepa a e modal con ibu ions, show hemsel es o be excellen 440
es ima es o he eal alues when se e al modal con ibu ions and modal damping a e conside ed.
•The maximum accele a ion esponse in a wo-iden ical-span ailway b idge is mos ly go e ned by
he i s an isymme ic and i s symme ic modes. I a b idge unde goes esonance o he i s wo
modes, he one leading o he maximum accele a ion will depend on he maximum design speed. The
non-dimensional ee ib a ion ampli udes a he ac ual esonan speeds may be used o es ima e445
he pa icula ain, esonance o de and mode numbe leading o he o e all maximum accele a ion
esponse in he s uc u e.
The p e ious conclusions cons i u e a heo e ical basis limi ed o he e ec o he ain axles dis ibu ion.
Addi ional phenomena may modi y he maximum accele a ion esponse, such as he e ec s o ehicle- ack-
b idge o soil-s uc u e in e ac ion, which a e beyond he scope o his s udy.450
Appendix A
De ini ion o High Speed Load Model-A om Eu ocode [30]:
25