Frequency characteristics of railway bridge

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Frequency characteristics of railway bridge response to moving trains

with consideration of train mass

Yong Lu a,c,⇑, Lei Mao a,c, Peter Woodward b,c

a Institute for Infrastructure and Environment, School of Engineering, The University of Edinburgh, The Kings Buildings, Edinburgh EH9 3JL, UK b Institute for Infrastructure and Environment, School of Built Environment, Heriot-Watt University, Edinburgh, UK c Joint Research Institute for Civil and Environmental Eng., Edinburgh Research Partnership in Engineering and Mathematics, UK

a r t i c l e i n f o

Article history:

Received 22 January 2012

Accepted 10 April 2012

Available online 20 May 2012

Keywords:

Railway bridge

Moving train

Moving mass

Frequency content

Critical speed

Resonance speed

a b s t r a c t

The dynamic response of railway bridges is known to be influenced by a combination of factors including

the bridge natural frequency, train speed, and bridge and carriage lengths. However, the intrinsic rela-

tionships among these parameters have seldom been elaborated in common dynamics terms so as to

enable more effective implementation in practice. This paper attempts to approach this classic problem

from a frequency perspective, by investigating into the frequency characteristics in thebridge response as

well as in the moving trainloads. In particular, the significance of the so-called ‘‘driving’’ and ‘‘dominant’’

frequencies arising from the moving load is examined. Based on numerical results and a securitization

using a generalised trainload pattern, it is demonstrated that the primary frequency contents in the train-

load, and consequently in the dynamic response of the bridge, is largely governed by the bridge-to-car-

riage length ratio. Namely, for short bridges (with a length ratio below the order of 1.5), well-distributed

frequency peaks occur at a number of dominant frequencies, whereas for longer bridges the main fre-

quency peak tends to concentrate towards the lowest dominant frequency. Such a characteristic affects

directly the resonance condition and resonance speeds for bridges of different length categories, and this

observation echoes well the predictions of the resonance severity using a so-called Z -factor. For the spe-

cial case of bridge response under a single carriage/vehicle, theinfluenceof the carriage mass is examinedin association with the concept of critical speed, and the abnormal acceleration spikes that could occur

when the vehicle moves at the critical speed is highlighted.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

The dynamic response of railway bridges is complicated due to

the involvement of moving loads and moving masses. Comparing

to road traffic, the trainload excitation is characterised by a unique

pattern of frequency spectrum, which directly affects the dynamic

response of the bridge. Moreover, the dynamic properties of rail-

way bridges, especially the natural frequencies of small- to med-

ium-size bridges, can be altered significantly due to moving

carriage masses.

Numerous publications exist in the literature regarding bridge

dynamic response and the train–bridge interactions. It is well

recognised that the dynamic response of a railway bridge is influ-

enced by a combination of factors, chiefly the bridge natural fre-

quency, train speed, and bridge and carriage lengths. However,

the intrinsic relationships among these parameters have generally

been implicitly expressed through dynamic formulations, whereas

specific guides for their application in practice are lacked. For

example, it is not straightforward to implement a general recom-

mendation that resonance could take place under certain norma-

lised speeds, without specific information with regard to the

resonance severity and an understanding of the trend of variations.

A seemingly effective way of approaching this subject is to re-

sort to the frequency analysis of the trainload in conjunction with

the frequency characteristics of the responding system. However,

studies stemming from a frequency perspective are still limited.

Those that fall into this category may be loosely divided into two

groups, one concerns the variation of the natural frequencies of

the responding bridge during the passage of a laden train/vehicle

(e.g. [1–5]), and the other deals with the frequency contents in

the trainload excitation and their general effect in the bridge re-

sponse (e.g., [5–8]). In particular, it has been demonstrated that,

in addition to the resonant frequencies, the primary frequencies

in the bridge response may be attributable to (a) the so-called

‘‘driving frequencies’’ associated with the duration of a vehicle

crossing the bridge [8], and (b) the so-called ‘‘dominant frequen-

cies’’ arising from the repeated loads (hence are related to the time

0141-0296/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2012.04.007

⇑ Corresponding author at: Institute for Infrastructure and Environment, School

of Engineering, The University of Edinburgh, The Kings Buildings, Edinburgh EH9

3JL, UK. Tel.: +44 131 6519052; fax: +44 131 6506781.

E-mail address: [email protected] (Y. Lu).

Engineering Structures 42 (2012) 9–22

Contents lists available at SciVerse ScienceDirect

Engineering Structures

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http://dx.doi.org/10.1016/j.engstruct.2012.04.007

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interval between two consecutive carriage loads) (e.g., [7,9]). De-

spite the identification of these frequency factors, the understand-

ing of their influence on the bridge response remains to be rather

general.

The present paper aims to provide a comprehensive evaluation

of the frequency characteristics of a railway bridge response under

trainload, paying special attention to examining the significance

and the variation trend of key frequency components in the re-

sponse arising from the trainload, namely the driving frequencies

and dominant frequencies mentioned above. To incorporate the

influence of the moving mass, the analysis is carried out using a fi-

nite element model, in which a moving vehicle is simulated with a

moving mass block which is coupled with the bridge via surface

contact. For simplicity while withholding the primary frequency

characteristics, the vehicle dynamics and track irregularities are

not considered.

It is particularly worth noting that the relative length of the

bridge with respect to the length of the carriage is found to be a

governing factor determining the characteristic patterns and hence

the frequency contents in the trainload excitation, and therefore

this length ratio is employed in the classification of the frequency

response characteristics. Following the establishment of the fre-

quency characteristics, the bridge resonance effect is evaluated

through a series of parametric calculations. The observations on

the resonance phenomenon are then correlated with a newly pro-

posed resonance severity factor, called the Z -factor [10], to provide

a complete framework for the understanding as well as quantifica-

tion of the bridge resonance under moving trains.

2. Background theories

2.1. Natural frequency of bridge–moving train system

When a train moves on a bridge, the frequencies of the bridge

will be affected due to the effects of train mass coupled with the

bridge through the suspension systems. When the train mass is

large with respect to the mass of the bridge, such effect can be-

come significant.

The natural frequencies of the bridge during the passage of a

train (or a single vehicle as a specialised case) may be established

on the basis of the dynamic equation for the bridge coupled with

the moving object, as follows:

mb@ 2wb

@ t 2 þ EI

@ 4wb@ x4

þ c b@ wb@ t

¼ P ð x; t Þ ð1Þ

where EI , mb, c b are the flexural stiffness, mass per unit length and

damping coefficient of the bridge, wb is the bridge vertical displace-

ment, and P ( x, t ) is the interacting force between the vehicle and the

bridge.

The interacting force with the ith wheel–axle set may be ex-

pressed as [2]:

P ið x; t Þ ¼ d½ x ðVt aiÞ P 0;i mc @ 2wb

@ t 2 þ c c _wi þ kc wi

! ð2Þ

where d is the Dirac delta function, P 0,i is the static weight borne by

the ith wheel–axle set, a i is the distance between the first and ith

wheel–axle sets, wi is the displacement within the suspension

spring, mc denotes the effective mass that may be attributed to a

wheel–axle set, c c , kc are spring damping and stiffness of the vehi-

cle’s suspension system, respectively.

The solution of the motion equation can be obtained by modal

superposition. Denoting the nth mode shape as /n( x) and the gen-eralised modal coordinate as qbn(t ),

wbð x; t Þ ¼Xn

/nð xÞqbnðt Þ ð3Þ

For a simply supported bridge (beam), the mode shapes may be

expressed in a sinusoidal form, thus:

wbð x; t Þ ¼Xn

sinnp xLbqbnðt Þ ð4Þ

where Lb is the bridge length.Substituting Eqs. (2) and (4) into Eqs. (1), multiplying both sides

with sin (np x/Lb), and then integrating with respect to Lb yields:

€qbnðt Þ þ 2nbnxbn _qbnðt Þ þ x2

bnqbnðt Þ ¼ 2

mbLbP bnðt Þ ð5Þ

where xbn is the nth natural frequency, nbn is the corresponding

damping ratio, P bn(t ) is the generalised modal force and may be

expressed [1] as:

P bnðt Þ ¼XM i¼K

sinnpðv t aiÞ

LbP 0;ims

X1

k¼1

sinkpðv t aiÞ

Lb€qbkðt Þ þc c _wiþkc wi

" #

ð6Þ

A numerical integration method, such as the Wilson-h method,can be employed to obtain the bridge natural frequencies at each

time step. The results will allow the variation of the bridge fre-

quency to be plotted against time or the position of the moving

train.

2.2. Driving frequencies

The so-called ‘‘driving frequency’’ [8] is associated with the in-

verse of the time duration a vehicle crosses the bridge. Specialising

Eq. (2) into a single moving load,

P ð x; t Þ ¼ f c ðt Þdð x Vt Þ ð7Þ

where f c (t ) is the sum of the vehicle weight and the dynamic force

of the suspension system,

f c ðt Þ ¼ mc g þ kc ðwc wbÞ ð8Þ

where kc is the stiffness between moving mass and the bridge, wc

and wb are the dynamic deflections of the moving mass and the

bridge, respectively.

Substituting Eq. (7) into Eq. (5), and ignoring the damping term

yields:

€qbn þ x2

bnqbn ¼ f c ðt Þ

R L0 dð x Vt Þ/nð xÞd x

mbR L

0 /

2

nð xÞd xð9Þ

Combining with the motion equation of the moving mass, and

substituting Eq. (8), Eq. (9) may be re-written as [8]:

€qbn þ x2

bnqbn þ2x2c mc mbLb

sinnpVt Lb

X j

sin jpVt Lbqbj

2x2c mc mbLb

sinnpVt Lbqc ¼

2mc g

mbLbsinnpVt Lb

ð10Þ

If the mass of the passing vehicle is much less than that of the

bridge and hence may be ignored, the above equation reduces to:

€qbn þ x2

bnqbn ¼2mc g

mbLbsinnpVt Lb

ð11Þ

Assuming a zero initial condition, the solution to the above equa-

tion may be obtained as:

qbnðt Þ ¼

D

1 S 2n sin

npVt Lb S n sinðxbnt Þ

ð12Þ

10 Y. Lu et al. / Engineering Structures 42 (2012) 9–22



where D ¼2mc gL3

b

n4p4EI is the static deflection due to the passing vehicle

with respect to the nth mode, and S n ¼ npV Lbxbn

is a relative speed

parameter.

Finally, the displacement response of the bridge can be ex-

pressed [8] as:

wbð x; t Þ ¼ XnD

1 S 2

n

sinnp xLb

sinnpV Lbt S n sinxbnt ð13Þ

The above results indicate that frequencies at

f dr ¼ npV Lb

1

2p

¼ n V

2Lb

ð14Þ

will be present in the bridge dynamic responses, in addition to the

natural frequencies of the bridge structure. These frequencies are

the ‘‘driving frequencies’’.

It is also noteworthy that with the introduction of f dr , the rela-

tive speed parameter S n becomes: S n = 2p f dr /xbn = xdr /xbn. It is

therefore clear that when xdr approaches xbn (or f dr approaches

f bn), S n approaches 1, and this effectively constitutes the criterion

for the critical speed. From this point of view, the definition of

the ‘‘driving frequencies’’ is meaningful as it relates the critical

speed condition to a conventional resonance criterion.

2.3. Dominant frequencies

The repeated wheel–axle loads from a moving train at a specific

point of observation may be represented in the following simpli-

fied form, assuming one generic load per carriage (e.g., [11]):

P ðt Þ ¼XN c j

f c ðt Þdðt jt c Þ ð15Þ

where N c is the number of carriages, f c (t ) denotes the generic

wheel–axle load, t c represents the time interval between the re-

peated loads, t c = Lc /V , with Lc being the length of a carriage.

Apparently the load described by Eq. (15) is a series of N c peri-odic load pulses with a time interval of t c . Such loading will man-

ifest on the frequency spectrum as having peak frequencies at

n (1/t c ) = nV /Lc . For an actual train with multiple wheel-sets in

each carriage, studies (e.g. [9]) have shown that the frequencies

arisen from the overall trainload is still dominated by nV /Lc . There-

fore, it makes sense to define these frequencies as the apparent

‘‘dominant frequencies’’ in the trainload [9], which can be ex-

pressed as:

f do;n ¼ nV =Lc ð16Þ

It should be noted that the velocity ratios underlying the ‘‘driving

frequencies’’ and ‘‘dominant frequencies’’, i.e., V /2Lb and V /Lc , have

been examined in previous researches in connection with the study

of the bridge resonance under moving trainloads (e.g. [1,11–14]). Itis recognised that when these ratios or their multiples come closer

to the bridge natural frequencies, resonant response to a certain de-

gree will occur [7,11]. In spite of the general notion, however, few

researches have been carried out to investigate the relative signifi-

cance of these frequencies (including their different orders) under

different train–bridge parameter combinations and the subsequent

effect on the resonance amplitude.

3. Numerical model and verification

In this section, a finite element model is set up to represent a

generic bridge–moving train system. The model is subsequently

employed to investigate the effects of the trainload excitation fre-

quencies on the bridge response under different combinations of the influencing parameters.

Fig. 1 schematically illustrates a generic finite element configu-

ration for the bridge (represented by a beam) and moving load/

mass system. The bridge is modelled by solid elements to facilitate

the attachment of the moving objects through a contact algorithm,

which is available in common general-purpose FE codes, such as

ANSYS11 used in the present study. It is possible to model the de-

tailed wheel–axle sets and the associated suspension system in

each carriage by discrete mass-spring sets, and attach the springs

at the bottom ends to the bridge surface via contact, as depicted

in Fig. 1a.

For the main purpose of evaluating the primary frequency char-

acteristics, a simplified scheme with each carriage being repre-

sented by a single mass and a point load is deemed to suffice, as

shown in Fig. 1b. Such a moving mass model will enable a more

convenient observation of the key frequency characteristics associ-

ated with the movement of the carriage load and the carriage mass

in the bridge response. Similar simplification has been considered

in previous studies, e.g. [15]. It should be anticipated, however,

that the detailed response time histories and some secondary

frequency contents could be affected, as will be examined in the

verification analysis that follows.

A simply-supported railway bridge of 8-m length, which was

actually measured in a field study, is considered as a prototype

for the basic model set-up and validation. The mass per unit length

of the bridge is estimated to be 2 103 kg/m, and the flexural

rigidity (EI) is assumed such that the resulting fundamental natural

frequency of the bridge model matches that of the measured

bridge, which is approximately 14 Hz. A 1% damping ratio is con-

sidered in the model.

To avoid unnecessary complication, each mass block is

modelled essentially as a point mass with a nominal size of

0.1 0.1 m in the FE model. A particular amount of mass can be

obtained by specifying an appropriate density for the mass block.

The force exerted on the bridge by a carriage is achieved by impos-

ing a vertical load on the moving mass. The vertical load is assigned

independently, such that the effects of the moving load and mov-

ing mass may be investigated separately when needed. For exam-ple, in order to examine the adequacy of ignoring the moving mass

while considering the moving load, a zero mass may be assigned

while keeping the moving vehicle load.

Fig. 1b shows schematically the generic train–bridge model. It is

noted that modelling of multiple carriages (moving masses) is rea-

lised by starting the mass units one after another at pre-calculated

time intervals. The static deflection caused by the bridge self-

weight is not considered in the analysis.

(a)

V

Lc

Moving mass coupled with bridge viacontact, possibly involving springs

(b) Lc

mct=Lc /V

Pc

V

Fig. 1. Schematic of FE model for bridge dynamic response under moving train: (a)

Considering wheel-set configuration; (b) simplified with equally-spaced movingmass/load.

Y. Lu et al. / Engineering Structures 42 (2012) 9–22 11



As verification of the above simplified modelling scheme, the

model is firstly employed to analyse the response of the 8-m bridge

subjected to moving train as recorded from the field measurement

exercise, Fig. 2a, and the modelling results are compared with the

recorded responses. The measured responses included the dis-

placement and acceleration time histories at the mid-span of the

bridge. A typical recorded scenario involved 8 carriages, with the

distance between adjacent carriages being approximately 20 m,

and the speed of the train was approximately 45 m/s. The mass

of the carriages is approximated such that the predicted response

amplitudes match the measured peak responses. Fig. 2b compares

the simulated and measured mid-span displacement time histories

and their corresponding frequency spectra.

The characteristics of the numerical results resemble well the

measured bridge response. As can be expected from simplifying

each carriage into one single mass in the FE model, the numerical

time histories show only one peak during the passage of each car-

riage, while the measured response exhibits double peaks. On the

frequency spectra, the first few dominant frequencies in the

numerical results correlate well with their experimental counter-

parts; however the numerical results exhibit a few more frequency

peaks into the higher order range. This may be attributable to some

significant free vibration following each segment of forced re-

sponse from numerical results, whereas the measured displace-

ment dies out quickly after each excitation, due probably to the

low sensitivity of the displacement transducers to the high fre-

quency components in the response.

For further verification of the simplification of each carriage

into a single mass/load in representing the primary frequency

characteristics in the bridge response, another analysis is per-

formed in which each carriage is simulated via two wheel sets

(two mass blocks in the model). The resulting response time histo-

ries and the frequency spectra are compared with those using the

generalised single mass per carriage scheme in Fig. 3. As can be

seen, although the refined model simulates the bridge response

more realistically in the time domain, the frequency components

are very similar to those from the generalised model. It can there-fore be concluded that the generalised model is suitable for an

investigation into the frequency characteristics concerning the pri-

mary train–bridge parameter combinations.

4. Frequency characteristics of bridge response under a single

moving vehicle

As discussed in Section 2.2, the bridge dynamic response to a

single moving vehicle will contain the ‘‘driving frequencies’’ dueto the excitation of a single load pulse. Such ‘‘driving frequencies’’

become obvious if the moving load pulse is looked upon qualita-

tively as a half cycle of a sine function of duration t b, leading to

equally spaced apparent frequencies at an interval of 1/2t b = V /

2Lb, as expressed in Eq. (14). Moreover, the primary frequency as

in a half cycle sinusoidal excitation is obviously concentrated at

the first order, i.e. f dr 1 = V /2Lb. It can be envisaged that as this fre-

quency approaches the bridge fundamental frequency, the dy-

namic response of the bridge will increase. This leads to the

concept of the critical speed.

Critical speed is a term used to describe the condition under

which the bridge approaches the maximum (global) response

when subjected to a single moving load. Without considering the

moving mass, it can be shown that (e.g. [16,17]):

V cr ¼ 2 f b1Lb ð17Þ

where V cr is the critical speed, f b1 is the bridge fundamental fre-

quency, Lb is the bridge length.

With the identification of the driving frequencies associated

with a single carriage, especially the basic driving frequency

f dr 1 = V /2Lb, the above critical speed may be looked upon as a reso-

nance condition such that the primary excitation frequency

matches the natural frequency of the bridge, f dr 1 = f b1, and hence

Eq. (17).

A particular usefulness of viewing the critical speed from the

perspective of resonance is that, when the moving mass of the car-

riage is taken into account, the critical speed will automatically de-

crease as the natural frequency of the bridge decreases with the

increase of the moving mass. On this basis, the modified critical

speed may be estimated by introducing a modified ‘‘effective’’

(a)

(b)

0 1 2 3 4 5-1

0

1

2

3

4

Time (s)

D e f l e c t i o n ( m m )

Experimental

Numerical

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Frequency (Hz)

F F T a m p l i t u d e

Experimental

Numerical

Disp transducer Accelerometer

Fig. 2. Comparison between numerical and experimental results: (a) Measured 8-m railway bridge; (b) time histories and frequency spectra.




natural frequencywhenthe moving carriage mass is involved. As an

approximation, the modified natural frequency when the moving

mass reaches the mid-span of bridge, f b1m, may be employed, thus:

V cr ;m ¼ 2 f b1mLb ð18Þ

More detailed discussion on the determination of the critical speedwhen a significant moving mass is involved is presented elsewhere

[10], and it has been demonstrated that using f b1m to represent the

effective mass-loaded frequency can yield a reasonable estimation

of the mass-loaded critical speed to an accuracy within about

10%, even when a very large moving mass is involved.

To give an example, Fig. 4 illustrates the amplitude deflection

(mid-span) versus speed relationship as computed using the 8-m

bridge model described in Section 3, for (1) case-1: without consid-

ering the vehicle mass, and (2) case-2: considering a vehicle mass

of 250% of the bridge mass, respectively. To allow for a direct com-

parison, the moving load is kept the same in both cases, being

400 kN. The critical speed according to Eq. (17) is calculated as

224 m/s for case-1 using the bare bridge frequency f b1 = 14 Hz,

and this agrees well with the numerical results without consider-ing the moving mass. However, it deviates considerably from the

actual critical speed for case-2 when a heavy vehicle mass is in-

volved. On the other hand, when the mass-loaded frequency

f b1m 6 Hz is considered, the critical speed according to Eq. (18)

is found to be about 100 m/s, which matches well the correspond-

ing numerical result as indicated in Fig. 4.

Although from Fig. 4 the amplitude deflection of the bridge does

not appear to increase significantly upon the ‘‘resonance’’ at the

critical speed, the bridge acceleration, and to a lesser extent the

velocity, tends to exhibit drastic increase under such a condition.

Figs. 5 and 6 show the bridge mid-span deflection, velocity and

acceleration time histories when the vehicle moves at about the

critical speed, considering a small and very large vehicle mass

(case-1 and case-2 mentioned previously), respectively. For a com-

parison, the responses when the vehicle moves at selected reduced

speeds are also included.

For case-1 (Fig. 5), it can be observed that when the vehicle

moves at the critical speed, the mid-span deflection reaches the

maximum at the time when the vehicle leaves the bridge (0.035 s

in this case). At this point the bridge mid-span velocity approaches

zero; however, the acceleration (slope of the velocity) appearsto in-

crease drastically and reaches about 20 g around the end of the

forced response phase. When the vehicle moves at a reduced speed,

e.g. half of the critical speed herein as shown in Fig. 5b, no acceler-

ation spike at the end of the forced response occurs.

For case-2 (Fig. 6) where the influence of a large vehicle mass is

taken into account, the general trend is similar to the situation de-

scribed above for case-1. In fact, under the critical speed the slope

of velocity around the time when the vehicle leaves the bridge be-

comes even steeper than in case-1, prompting a maximum acceler-

ation to reach as high as 50 g. Such acceleration spikes tend to

disappear when the speed is reduced, and when the speed is

reduced to 1/4th of the critical speed, the maximum acceleration

reduces to below the order of 1.0 g.

It should be noted that in practice the critical speed V cr = 2 f b1Lb

under a single moving vehicle is not normally attainable in road

traffic. However, it is possible to reach such speeds in the case of

railway bridges especially with high speed trains. As will be dem-

onstrated in the sections that follow, when multiple carriages are

involved the frequency characteristics and subsequently the reso-

nant phenomena will be controlled by the repeated carriage loads,

i.e. the ‘‘dominant frequency’’ instead of the ‘‘driving frequency’’

effect.

5. Frequency characteristics of bridge response under multiplecarriages

For the dynamic response of railway bridges, the most impor-

tant dynamic characteristic is anticipated to associate with the re-

peated load caused by the passage of multiple carriages. As

mentioned in Introduction, the repeated carriage loads give rise

to the so-called ‘‘dominant frequencies’’ in the trainload, and such

frequencies play an important role in determining the bridge dy-

namic response as discussed generally in some previous studies

(e.g., [1,9,10,13,18]).

In the present numerical investigation, ten moving blocks are

considered to simulate a trainload. Each block is assigned a stan-

dard carriage mass of 40 tons, while a 400 kN load is applied on

each moving block. The distance between two adjacent blocks is20 m, representing the length of a typical carriage.

0 1 2 3 4 5-1

0

1

2

3

4

Time (s)

D

e f l e c t i o n ( m m )

One wheelaxle model

Two wheelaxles model

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Frequency (Hz)

F

F T a m p l i t u d e

One wheelaxle model

Two wheelaxles model

Fig. 3. Comparison of results using two different FE model settings.

0 100 200 3000

10

20

30

40

50

Speed (m/s)

M a x d e f l e c t i o n ( m m )

← I ← II

Negligible mass

250% bridge mass

Fig. 4. Variation of amplitude deflection of the bridge with the vehicle speed.Vertical line I: predicted by V cr = 2 f b1mLb; line II: predicted by V cr = 2 f b1Lb.




The 8-m prototype bridge is firstly used to investigate the

bridge responses and the corresponding frequencies in a short-

span bridge scenario. Other bridge cases with a medium-span

length and a long-span length will be modelled subsequently, so

as to observe the effect of different bridge-to-carriage length ratios.

5.1. Frequency characteristics of a short bridge scenario (8-m length)

In such a short bridge scenario where the bridge length is smal-

ler than the carriage length, only one moving block will act on the

bridge during each forced response period.

(a)

0 0.05 0.1-0.04

-0.02

0

0.02

0.04

Time (s)

D e f l e c t i o n ( m )

0 0.05 0.1-4

-2

0

2

4

Time (s)

V e l o c i t y ( m / s )

0 0.05 0.1-300

-150

0

150

300

Time (s)

A c c e l e r a t i o n ( m / s 2 )

(b)

0 0.06 0.12-0.04

-0.02

0

0.02

0.04

Time (s)


0 0.06 0.12-4

-2

0

2

4

Time (s)


0 0.06 0.12-300

-150

0

150

300

Time (s)


Fig. 5. Bridge mid-span responses at representative speeds, ignoring moving mass; dashed line marks end of forced response: (a) At critical speed (224 m/s); (b) at half

critical speed (112 m/s).

(a)

0 0.08 0.16-0.04

-0.02

0

0.02

0.04

Time (s)


0 0.08 0.16-4

-2

0

2

4

Time (s)


0 0.08 0.16-600

-300

0

300

600

Time (s)


(b)

0 0.1 0.2-0.04

-0.02

0

0.02

0.04

Time (s)


0 0.1 0.2-4

-2

0

2

4

Time (s)


0 0.1 0.2-300

-150

0

150

300

Time (s)

(c)

0 0.25 0.5-0.04

-0.02

0

0.02

0.04

Time (s)

D e f l e c t

i o n ( m )

0 0.25 0.5-0.3

-0.15

0

0.15

0.3

Time (s)


0 0.25 0.5-10

-5

0

5

10

Time (s)



Fig. 6. Bridge mid-span responses at representative speeds, considering a heavy (250%) moving mass; dashed line marks end of forced response: (a) At near critical speed

(100 m/s); (b) at half critical speed (50 m/s); (c) at quarter critical speed (25 m/s).




Three different train speeds are examined, namely 20 m/s,

50 m/s and 100 m/s. Given a carriage length of 20 m, these speeds

will correspond to dominant frequencies of around 1n, 2.5n and 5n

Hz (n = 1,2,. . .), respectively. Note that the fundamental natural

frequency of the bare bridge is 14 Hz, and the critical speed based

on a single moving carriage would be about 100 m/s taking into ac-

count the mass-loaded natural frequency of about 6 Hz. Figs. 7 and

8 show the bridge displacement and acceleration responses, along

with the corresponding FFT spectrum curves.

As can be seen, both the displacement and acceleration fre-

quency spectra exhibit clear spectral peaks at the anticipated dom-

inant frequencies. From the mid-span deflection response it can be

observed that pronounced resonance effect occurs under the speed

of 50 m/s, apparently due to the repeated carriage loads or the

‘‘dominant frequencies’’. On the other hand, at the speed of

100 m/s which marks the critical speed under a single moving load

as discussed earlier in Section 4, no successive increase of the re-

sponse occurs between carriage loads, and the maximum response

is essentially governed by that induced from a single carriage. This

comparison clearly demonstrates a very different resonance phe-

nomenon in close association with the ‘‘dominant frequencies’’,

and the significance of the ‘‘driving frequency’’ tends to diminish.

Further discussion of the resonance under repeated loads will be

given later in connection with the examination of the composition

of the dominant frequencies for different bridge–carriage length

ratios.

It should be noted that, although the significance of the crit-

ical speed is rendered less significant in the overall dynamic re-

sponse, when such a speed is reached the abnormal acceleration

spikes still occur in this short-bridge scenario, see Fig. 8c. This is

consistent with the observation from the single vehicle load

case.

5.2. Frequency characteristics of a medium bridge scenario (40-m

bridge)

A 40-m bridge is considered to represent a ‘‘medium length’’

bridge in the sense that more than one carriage may act on the

bridge at one time. To be consistent with the increase in the bridge

length as compared to the 8-m bridge, the mass per unit length is

increased by 1.6 times, yielding a carriage–bridge mass ratio of

30%, while the stiffness is adjusted such that the resulting bridge

fundamental frequency is approximately 10 Hz. The damping

ratios are kept as 1%. Figs. 9 and 10 present the bridge dynamic

responses and their FFT curves for the same trainload at three dif-

ferent speeds.

The increase of the bridge length does not affect the trainload

dominant frequencies as these frequencies are associated only

with the speed and the carriage length. Recall that these frequen-

cies are 1n, 2.5n and 5n Hz, respectively. The basic driving fre-

quency, which is presumably to be of lesser effect, would be

0.25, 0.625, and 1.25 Hz, for the three speeds respectively, while

the critical speed associated with a single carriage would be a

rather theoretical value of 800 m/s for this bridge and hence is

not examined.

Similar to the 8-m bridge case, a number of dominant frequency

peaks can be observed in the responses. The bridge deflection does

not exhibit much dynamic response when the speed is low (20 m/s

in this case) and the deflection history closely resembles the pat-

tern of the motion of the trainload. With the increase of speed to

100 m/s, the deflection apparently increases. From the FFT curves

it can be observed that the peaks around the natural frequency

of the bridge (about 10 Hz) tend to be more significantly amplified,

and this may be attributable to the fact that the natural frequency

of the bridge falls within the more powerful first few dominant

(a)

0 2 4 6 8 10-0.06

-0.03

0

0.03

0.06

Time (s)


0 5 10 15 200

0.01

0.02

Frequency (Hz)


(b)

0 1 2 3 4-0.06

-0.03

0

0.03

0.06

Time (s)

D e

f l e c t i o n ( m )

0 10 20 30 400

0.01

0.02

Frequency (Hz)


(c)

0 0.5 1 1.5 2-0.06

-0.03

0

0.03

0.06

Time (s)


0 10 20 30 400

0.01

0.02

Frequency (Hz)


Fig. 7. Displacement time histories (left) and FFT spectra (right) for the 8-m bridge under different train speeds: (a) Speed = 20 m/s (expected f do,n at 1 Hz interval); (b)speed = 50 m/s (expected f do,n at 2.5 Hz interval); (c) speed 100 m/s (expected f do,n at 5 Hz interval).




(a)

0 2 4 6 8 10-10

-5

0

5

10

Time (s)

A

c c e l e r a t i o n ( m / s 2 )

0 5 10 15 200

0.5

1

Frequency (Hz)


(b)

0 1 2 3 4-200

-100

0

100

200

Time (s)

0 10 20 30 400

20

40

60

80

Frequency (Hz)


(c)

0 0.5 1 1.5 2-500

-250

0

250

500

Time (s)

0 10 20 30 400

50

100

150

Frequency (Hz)

F F T a m p l i t u d

e


A c c e l e r a t i o n ( m

/ s 2 )

Fig. 8. Acceleration time histories (left) and FFT spectra (right) for the 8-m bridge under different train speeds: (a) Speed 20 m/s (expected f do,n at 1 Hz interval); (b) speed

50 m/s (expected f do,n at 2.5 Hz interval); (c) speed 100 m/s (expected f do,n at 5 Hz interval).

(a)

0 2 4 6 8 10 11-0.04

-0.02

0

0.02

0.04

Time (s)


0 5 10 15 200

5x 10

-3

Frequency (Hz)


(b)

0 1 2 3 4 4.4-0.04

-0.02

0

0.02

0.04

Time (s)

D e

f l e c t i o n ( m )

0 5 10 15 200

5x 10

-3

Frequency (Hz)

F F T

a m p l i t u d e

(c)

0 0.5 1 1.5 2 2.2-0.04

-0.02

0

0.02

0.04

Time (s)


0 5 10 15 200

5x 10

-3

Frequency (Hz)


Fig. 9. Displacement time histories (left) and FFT spectra (right) for the 40-m bridge under different train speeds: (a) speed = 20 m/s (expected f do,n at 1 Hz interval); (b)speed = 50 m/s (expected f do,n at 2.5 Hz interval); (c) speed = 100 m/s (expected f do,n at 5 Hz interval).




frequencies. As a result, increased degree of resonance occurs. This

observation is supported by the significantly increased accelera-

tion amplitudes as the speed increases. It should be noted thatthe speed of 100 m/s is still far below the critical speed (800 m/s)

for this 40-m bridge, hence no acceleration spikes occur.

5.3. Frequency characteristics of a long bridge scenario (80-m bridge)

An 80-m bridge is employed to represent a ‘‘longer’’ bridge such

that multiple carriages can act on the bridge simultaneously. The

mass and stiffness of the bridge are increased from the previous

40-m bridge, resulting in a fundamental frequency of about 5 Hz,

and a carriage-to-bridge mass ratio of 8%. The damping ratio is kept

at 1%. Figs. 11 and 12 depict the bridge dynamic responses and cor-

responding FFT curves.

Comparing to the 40-m bridge, it can be observed that the res-

onance effect, herein at around 5 Hz, is more significantly excited.This is deemed to be attributable to the fact that, with increase of

the bridge-to-carriage length ratio, the excitation frequencies tend

to further concentrate to the lowest dominant frequencies. As the

natural frequency of the bridge also reduces, resonance becomes

inevitably more significant.

Summarising the frequency characteristics from the above anal-

yses under multi-carriage trainload excitation, it may be concluded

that (a) the dynamic responses are governed by the multiple-car-

riage dominant frequencies, while the driving frequencies which

arise from an individual moving load do not appear to have a sig-

nificant effect; and (b) with the increase of the bridge length, the

frequency peaks as can be observed from the FFT of the response

tends to be increasingly concentrated to the lowest dominant fre-

quencies, and as the natural frequency of the bridge also reduces,the frequency peak at the natural frequency tends to be more

significantly amplified. To better explain these observations, it is

necessary to look into the variation of the frequency spectrum in

the excitation as the bridge-to-carriage length increases, whichwould also provide insight into the trend of the resonance effect.

6. Effect of bridge-to-carriage length ratio on frequency

characteristics of trainload

For a typical trainload involving multiple carriages, it has been

demonstrated that the dominant frequencies are a primary charac-

ter both in the excitation (with repeated carriage loads) as well as

in the bridge response. It has also been shown that, as the bridge

length increases, the primary frequency contents in the response

tend to become increasingly concentrated in the lower few domi-

nant frequencies. In this section, we shall attempt to gain insight

into this trend by analysing the variation of the frequency contents

in different trainload scenarios.Considering the fundamental bending mode response of the

bridge, the load exerted by each carriage as it moves from one

end of the bridge to another may be generalised as a half sine

pulse, with a duration of T b as depicted in Fig. 13. When multiple

carriages are involved, the load will consist of a series of such

pulses with an interval between consecutive pulses being T c . For

a train speed V , T b = Lb/V , T c = Lc /V , where Lb, Lc are the bridge length

and the carriage length, respectively. The overall pattern of the

trainload, and hence the frequency distribution, depends only on

the relative values of T b and T c , which in turn depends on the length

ratio aL = Lb/Lc . It is noted that a similar approach in characterising

various trainload scenarios was used in [10] for a different purpose.

In what follows, we shall examine the variation of the frequency

spectrum for different characteristic aL values. For convenience of presentation, we assume a fixed velocity of 50 m/s and a carriage

(a)

0 5 10-10

0

10

Time (s)

A

c c e l e r a t i o n ( m / s 2 )

0 5 10 15 200

1

2

3

Frequency (Hz)


(b)

0 1 2 3 4-10

0

10

Time (s)

0 5 10 15 200

2

4

6

Frequency (Hz)


(c)

0 0.5 1 1.5 2-50

0

50

Time (s)

0 5 10 15 200

10

20

Frequency (Hz)



A c c e l e r a t i o n ( m / s

2 )

Fig. 10. Acceleration time histories (left) and FFT spectra (right) for the 40-m bridge under different train speeds: (a) Speed = 20 m/s (expected f do,n at 1 Hz interval); (b)

speed = 50 m/s (expected f do,n at 2.5 Hz interval); (c) speed = 100m/s (expected f do,n at 5 Hz interval).




(a)

0 3 6 9 12

-0.1

-0.05

0

0.05

0.1

Time (s)


0 5 10 15 20

0

0.005

0.01

Frequency (Hz)


(b)

0 1 2 3 4 5-0.1

-0.05

0

0.05

0.1

Time (s)


0 5 10 15 200

0.005

0.01

Frequency (Hz)


(c)

0 0.5 1 1.5 2 2.5-0.1

-0.05

0

0.05

0.1

Time (s)


0 5 10 15 200

0.005

0.01

Frequency (Hz)

F F T a m p l i t u

d e

Fig. 11. Displacement time histories (left) and FFT spectra (right) for the 80-m bridge under different train speeds: (a) speed = 20 m/s (expected f do,n at 1 Hz interval); (b)

speed = 50 m/s (expected f do,n at 2.5 Hz interval); (c) speed = 100 m/s (expected f do,n at 5 Hz interval).

(a)

0 5 10-20

-10

0

10

20

Time (s)


0 5 10 15 200

1

2

Frequency (Hz)


(b)

0 1 2 3 4 5-50

-25

0

25

50

Time (s)

0 5 10 15 200

2

4

Frequency (Hz)


(c)

0 0.5 1 1.5 2 2.5-100

-50

0

50

100

Time (s)

0 5 10 15 200

5

10

Frequency (Hz)




Fig. 12. Acceleration time histories (left) and FFT spectra (right) for the 80-m bridge under different train speeds: (a) speed = 20 m/s (expected f do,n at 1 Hz interval); (b)speed = 50 m/s (expected f do,n at 2.5 Hz interval); (c) speed = 100 m/s (expected f do,n at 5 Hz interval).




length of 20 m, with ten carriages. The theoretical ‘‘dominant fre-

quencies’’ are thus fixed at (V /Lc )n = (1/T c )n = 2.5nHz, where n is a

positive integer.

(i) aL = Lb/Lc ? 0: for example, let T b = 0.02 s, T c = 0.4 s

(Lb = ‘‘1 m’’, Lc = 20 m, V = 50 m/s)This scenario represents a

lower-bound aL situation with essentially a ‘‘point’’ bridge,

such that the load resembles a string of impulses, as shown

in Fig. 14i. The frequency spectrum is a typical repeated

impulse character with almost equal peaks for all orders of

the ‘‘dominant frequencies’’ at a constant interval of 1 /

T c = 2.5 Hz.

(ii) aL < 1.0: for example, let T b = 0.16s, T c = 0.4 s (Lb = 8 m,

Lc = 20 m, V = 50 m/s) This case is analogous to the analysed

8-m short bridge, with the load pattern as depicted in

Fig. 14ii. It can be clearly seen that, as the duration T b

increases with respect to the time interval T c , the spectralpeaks at higher order ‘‘dominant’’ frequencies reduce

sharply.

(iii) aL = 1.0: for example let T b = 0.4s, T c = 0 .4 s (Lb = 20 m,

Lc = 20 m, V = 50 m/s) This is a dividing scenario where the

bridge length is equal to the carriage length so that the con-

secutive load pulses arrive one immediately after the other.

The frequency spectrum still shows the character of equally

spaced peaks at an interval of 2.5 Hz, but it is increasingly

dominated by the first few frequencies.

(iv) aL > 1.0: for example let T b = 2 s, T c = 0.4 s (Lb = 100 m,

Lc = 20 m, V = 50 m/s) As depicted in Fig. 14iv, in this case

the generalised individual carriage load pulses overlap one

another; for this particular example there are 4 carriages

acting on the bridge simultaneously. The peak frequenciesat the expected intervals of 2.5 Hz are visible, with concen-

tration of spectral power at the lowest dominant frequency

of 2.5 Hz. Note that the spectrum in the near-zero frequency

range is the result of the overall trapezoidal load shape in

the positive domain, and it represents a strong presence of

the quasi static component.

In light of the above observations, much of the characteristics in

the bridge responses described in Section 5 becomes explicable

from the view point of frequency contents in the trainload excita-

tion. Namely,

(a) The 8-m bridge case falls into the loading scenario-(ii),

though the frequency axis can expand or contract as thespeed of the train varies. The frequency spectra of the

displacements for the 20 m/s and 50 m/s speeds show clear

consistency with the frequency spectra of the generalised

load patterns, with higher peaks appear at the lowest few

‘‘dominant frequencies’’. The frequency spectra of the accel-

eration responses also exhibit clear peaks at the dominant

frequencies, but over a wider frequency range, i.e., involves

higher order as well as lower order dominant frequencies.

(b) For longer-length bridges, as in the cases of 40 m and 80 m

bridge analysed in Section 5, the overlapping of individual

carriage load pulses results in the energy in the dynamic

excitation being concentrated towards the lowest dominant

frequency, causing the dynamic deflection to exhibit pri-

mary frequencies at the first 1–2 dominant frequencies.

7. Discussion on the bridge resonance and effects of moving

mass and bridge-to-carriage length ratio

The resonance phenomenon can be expected to associate clo-

sely with the ‘‘dominant frequencies’’. Existing studies tend to sug-

gest that when any of the ‘‘dominant frequencies’’ coincides with

the (fundamental) natural frequency of the bridge, resonance

would occur, thus,

f do;n ¼ nV =Lc ¼ f b1 ð19Þ

or

V re ¼1

n f b1Lc ð20Þ

Having identified the varying patterns of the frequency spec-

trum of the trainload as a function of the bridge-to-carriage length

ratio, as discussed in Section 6, it may be envisaged that shorter

bridges are more prone to resonance because it is possible for dif-

ferent orders of the dominant frequencies, and hence different

speeds, to excite resonance. On the other hand, for longer bridges

it would primarily be the first dominant frequency that may excite

significant resonance. This can be observed from Fig. 15, which

depicts the variation of the 8-m and 40-m bridge mid-span deflec-tions with moving speed. For a comparison, the curve for the 8-m

bridge under a single carriage is also included in Fig. 15a.

It can be observed that in the case of the 8-m bridge, two signif-

icant resonance peaks appear, and the ‘‘resonance’’ at the two dif-

ferent speeds is actually excited by the second and first order

dominant frequency, respectively. Namely, at the resonance speed

of 60 m/s, it is f dr 2 = 2 (V /Lc ) = 2 3 Hz f b1m (mass-loaded natu-

ral frequency equal to about 6 Hz), while at the speed of 120 m/s, it

is the f dr 1 = 1 (V /Lc ) = 1 6 Hz f b1m. On the other hand, only one

significant resonance, at around 190 m/s, is found in the 40-m

bridge scenario, and it is caused by the first dominant frequency

being coincident with the natural frequency of the mass-loaded

bridge. This confirms the previous argument that in the case of

longer bridges, the spectral energy from the excitation will be pri-marily concentrated towards the first dominant frequency to cause

significant resonance; whereas in shorter bridges, the energy is

distributed among a few dominant frequencies, and thus signifi-

cant resonance may be excited due to one of these dominant fre-

quencies, leading to multiple resonance speeds.

Fig. 16 depicts the bridge deflection time histories and their FFT

spectra corresponding to the resonance peaks in Fig. 15. The results

further illustrate that a series of dominant frequencies indeed exist

in the 8-m bridge response, while only the fundamental dominant

frequency is identifiable in the 40-m bridge response.

A more rigorous derivation of the resonance effect when a train-

load is moving at a potential resonance speed is presented in [10].

It demonstrates that under such a resonance condition, the sever-

ity of resonance as measured by the normalised increase of theconsecutive deflection amplitudes (squared), called the Z -factor,

(a) Lb << Lc

T bT c

T b = V / Lb; T c = V / Lc

Time

Load

Load

Time

(b) Lb > Lc

Fig. 13. Generalisation of the trainload patterns.




(i)

0 1 2 3 40

0.5

1

1.5

Time (s)

L o a d

0 10 20 300

0.02

0.04

0.06

Frequency (Hz)


(ii)

0 1 2 3 40

0.5

1

1.5

Time (s)

L o a d

0 10 20 300

0.2

0.4


(iii)

0 1 2 3 40

0.5

1

1.5

Time (s)

L o a d

0 10 20 300

0.2

0.4

Frequency (Hz)


(iv)

0 2 4 6 80

2

4

Time (s)

L o a d

0 10 20 300

0.02

0.04

Frequency (Hz)

F F T a m

p l i t u d e

Frequency (Hz)

Fig. 14. Different patterns of generalised trainload (left) and corresponding frequency spectra (right): (i) T b = 0.02s, T c = 0.4s (aL = 0.05); (ii) T b = 0.16s, T c = 0.4s (aL = 0.4);

(iii) T b = 0.4s, T c = 0.4s (aL = 1.0); (iv) T b = 2.0 s, T c = 0.4s (aL = 2.5).

(a) (b)

50 100 150 200 250 3000

20

40

60

80

100

Speed (m/s)


Single carriage

Multiple carriages

50 100 150 200

20

40

60

80

100

Speed (m/s)


Multiple carriages

Fig. 15. Amplitude response (mid-span deflection) versus carriage moving speed: (a) 8-m bridge; (b) 40-m bridge.




is essentially controlled only by the bridge-to-carriage length ratio,

aL = Lb/Lc . Z can be expressed as:

Z ¼ 2nLb=Lc

ð2nLb=Lc Þ2 1

" #2

1 þ cos 2pnLbLc

ð21Þ

where n is the order of the dominant frequency. A plot of the Z -fac-

tor is given in Fig. 17. It can be clearly identified that for the 8-m

bridge discussed herein, where aL = 0.4, significant resonance could

happen upon two n-values, with n = 2 giving rise to the highest res-

onance and n = 1 the next, thus according to Eq. (20) leading to two

significant resonance speeds of around 60 and 120 m/s, respec-

tively; For the 40-m bridge, aL = 2, sensible resonance would only

occur with the first order dominant frequency, with a speed equal

to 200 m/s (a more accurate value would be 190 m/s considering

an effective mass-loaded bridge frequency of 9.5 Hz as shown in

[10]). These results are very consistent with the observations made

earlier based on the numerical simulation results. In general, short-

er bridges with a bridge–carriage length ratio up to 1.5 appear to be

more susceptible to the resonance effect.

8. Conclusions

The dynamic response of a railway bridge to trainload excita-

tion is largely influenced by the frequency characteristics of the

imposed trainload, particularly the so-called ‘‘dominant frequen-

cies’’ in the general case with multiple carriages, and the indicative

‘‘driving frequencies’’ when only a single vehicle/carriage is

involved.

For the special case with a single carriage, the dynamic effect

may be conveniently described in association with the basic driv-ing frequency, which is effectively a nominal frequency of the load

pulse, i.e., V /2Lb. Subsequently, the classical definition of the critical

speed becomes a straightforward ‘‘resonant’’ condition such that

the above driving frequency matches the bridge natural frequency.

With the consideration of the moving carriage mass, it is observed

that the critical speed reduces as the moving mass increases, and

the modified critical speed can be well predicted using the

resonance condition mentioned above, with however the incorpo-

ration of an effective mass-loaded bridge frequency.

For typical trainload with multiple carriages, the dynamic effect

is overwhelmed by the repeated carriage loads; thus the ‘‘domi-

nant frequencies’’ at an interval of V /Lc , which may be viewed as

the normalised speed with respect to the carriage length, becomes

a governing factor. Examination of the frequency spectrum revealsthat the distribution of the spectral peaks among different orders

of the dominant frequencies is not uniform and it depends closely

upon the bridge-to-carriage length ratio, aL = Lb/Lc . As aL increases

the spectral peaks become increasingly more concentrated at the

lowest few dominant frequencies, and when aL is greater than

1.5, the frequency spectrum is effectively dominated by the basic

frequency at V /Lc .

The resonance phenomenon of the bridge response to a typical

trainload with multiple carriages is closely associated with the

above mentioned frequency characteristics. For short bridges such

as the 8-m bridge considered in the paper, significant resonance ef-

fect could occur at more than one resonance speeds, whereas for

longer bridges such as the 40-m bridge, significant resonance

may occur only at one speed when the first order dominant fre-quency matches the (mass-loaded) natural frequency of the bridge.

(a)

0 1 2 3 4-0.1

-0.05

0

0.05

0.1

Time (s)


0 10 20 300

0.005

0.01

0.015

0.02

Frequency (Hz)


(b)

0 0.5 1 1.5-0.1

-0.05

0

0.05

0.1

Time (s)


0 10 20 300

0.02

0.04

Frequency (Hz)


Fig. 16. Mid-spandeflection time histories (left) andtheir FFTspectra (right) for twobridge scenarios under resonance speed: (a)8-m bridgeunder resonance speed of 60 m/

s; (b) 40-m bridge under resonance speed of 190 m/s.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

←8-m bridge ←40-m bridge

n=1

n=2

n=3

n=4

Z

3

←8-m bridge ← 40-m bridge

n=1

n=2

n=3

n=4

Lb / Lc

Fig. 17. Resonance severity indicator, the Z -factor, versus as a function of bridge–carriage length ratio.




For an assessment of the severity of resonance at a potential reso-

nance speed, use can be made of the newly developed Z -factor.

Numerical results demonstrate consistent results with regard to

the resonance effect as compared with the prediction using the

Z -factor.

It is also noteworthy of the abnormal acceleration spikes that

tend to occur when the moving speed approaches the critical

speed. This phenomenon has particular implications for short

bridges, not only because of its relatively more significant effect

but also because of the increased possibility of reaching the critical

speed in such bridges, especially concerning high speed trains.

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http://dx.doi.org/10.1061/(ASCE)BE.1943-5592.0000336




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Frequency characteristics of railway bridge