Dynamic amplification factor and response spectrum for the evaluation of vibrations of beams under...

Journal of Sound and <ibration (2001) 248(2), 267}288doi:10.1006/jsvi.2001.3787, available online at http://www.idealibrary.com on

DYNAMIC AMPLIFICATION FACTOR AND RESPONSESPECTRUM FOR THE EVALUATION OF VIBRATIONS

OF BEAMS UNDER SUCCESSIVE MOVING LOADS

E. SAVIN

Structural Dynamics and Coupled Systems Department, ONERA, BP72, 92322 ChaL tillon Cedex,France. E-mail: [email protected]

(Received 20 November 2000, and in ,nal form 5 March 2001)

Analytic expressions of the dynamic ampli"cation factor and the characteristic responsespectrum are derived for weakly damped beams with various boundary conditions subjectedto point loads moving at constant speeds. These coe$cients are given as functions of theratio of the span length to the loads wavelength, and the loads wavelength respectively. Theyallow a rapid calculation of the vibration amplitudes induced by a succession of movingloads on a beam. These results are particularly useful in the context of railway bridgespreliminary design and assessment of the expected maximum vibration levels underhigh-speed trains.

( 2001 Academic Press

1. INTRODUCTION

The literature dealing with moving loads on bridges is considerable and has been enrichedin the last few decades by the development of high-speed rail networks in continentalEurope and Asia. A comprehensive review of these works is out of the scope of this paper,and the various references below are just given as non-exhaustive indications for furtherreading. Nevertheless, it appears that most of the recent studies focus on numericalsimulations, possibly including the e!ects of train mass inertia, coupling with the train carssuspension systems, tracks sti!ness, damping and roughness, especially for ballasted tracks,or rail}wheel contact (see for instance, the recent works [1}6]). Comparatively few studiesconcentrate on analytical developments, although the analytical solutions of a beamtraversed by a single point load is well known (see references [7}11] and references therein)provided that simple assumptions are used for the description of the load, mainly neglectingits inertia e!ect. Fewer works deal with the possible dynamic ampli"cations induced bya regular succession of loads, as is the case for railway bridges when the frequency ofsuccession matches their natural frequencies [12}15]. Note that such analyticaldevelopments can also be performed up to a thorough level for the case of a moving load onan elastic half-space, with applications to the evaluation of the characteristics of guidedwaves induced by trains in the trackbed, ground vibrations in the environment or possibleinstabilities [16, 17]. Another major subject of interest has been the consideration of somerandomness in the train speed, loads spacing and/or amplitudes [11, 18, 19], trackroughness [20}22] or track sti!ness characteristics [23]. These latter cases are probably themost relevant ones when applied to modern railways although the most di$cult to handlefrom a theoretical point of view; on the contrary, randomness related to the loadsamplitudes, speed or arrival times, is more likely to occur for highway bridges.

0022-460X/01/470267#22 $35.00/0 ( 2001 Academic Press

268 E. SAVIN

Most studies focus on the assessment of the dynamic de#ection of a beam induced by themoving loads for application to structural strength design. Practically, this point is seldoma real concern for prestressed concrete railway bridges since, "rstly, they are primarilydesigned for heavy static loads, including the prescribed dead loads and train loads, andsecondly, viscous damping is often rather important (a minimum of about 5% of criticaldamping). Steel or composite steel/concrete bridges have much lower damping rates (about0)5%) but they are also usually sti! enough to withstand dynamic ampli"cations from trainloads. The main concerns for railway engineers are in the evaluation of riding comfort whentrains traverse bridges, running safety, and tracks stability and deformation. Riding comfortis basically related to the vertical and transverse accelerations experienced inside the traincars; running safety is ensured at least if a continuous contact can be kept between thewheels and the rails; rails deformation is a critical point for modern railways wherecontinuous welded rails are used [24], whereas it is necessary to avoid any possibledecohesion of the ballast whenever it is used. Therefore, vertical accelerations, rotations atabutments and supports, torsion or warping are the most relevant quantities to focus on.The usual criteria to be ful"lled for them may be found, for instance, in reference [25]. Thelast issue worth mentioning from an engineering point of view is the analysis of fatigue inoperational conditions, namely the repetition in time of such regular loadings at very highspeeds (more than 250 km/h) for each direction on the bridge. Composite bridges may beparticularly sensitive to this phenomenon which needs to be carefully evaluated for thestructure itself as well as its various connections [24}26]. Use of ballasted tracks andconcrete decks has the advantage of signi"cantly enhancing their behaviour [25].

The purpose of the present paper is to propose simple analytical tools for the evaluationof the transient dynamic ampli"cations induced by successive loads preliminary to anydetailed numerical studies whenever needed. The developments are restricted to boundedelastic media, and particularly beams, and oriented toward practical engineeringapplications for real cases of weakly damped structures. In section 2, we introduce our basicnotations and some general, well-known notions on vibrations of Euler}Bernoulli beams.These elements are used in section 3 to calculate the time response of such a beam modelsubjected to a succession of massless point loads and assess the dynamic magni"cation theycan produce. These preliminary considerations allow us to derive in section 4 a simpli"edmethod for the direct calculation of the expected maximum dynamic de#ection of the beamunder successive loads by considering only few simple parameters related to the bridge andthe train. In section 5, we give some indications on the possible modi"cations of the theoryconsidering the e!ects of shear deformation and rotatory inertia according to Timoshenko'sbeam model. We "nally draw some conclusions and recommendations in section 6.

2. NOTATION AND GENERAL CONSIDERATIONS

The purpose of this section is to introduce the notation used throughout this paper, aswell as to recall brie#y the basic notions on beam vibrations and modal analysis which shallbe applied. We "rst consider the case of the Euler}Bernoulli beam model described by thefollowing partial di!erential equation:

oL2z

Lt2(s, t)!2f JoD

LLt

L2z

Ls2(s, t)#

L2

Ls2 ADL2z

Ls2(s, t )B"x (s, t), (1)

where z is the beam de#ection due to planar bending, which depends on the position s alongthe beam and on time t, o is the mass of the beam per unit length, D is its bending rigidity,

VIBRATION OF BEAMS UNDER MOVING LOADS 269

assumed constant along the beam, f is the critical damping rate, and x are the externallyapplied loads, which depend on the position and time. In the above equation the positions belongs to the set X"]0, ¸[ where ¸ is the span length of the beam, and t is in the set[0,#R[. The initial conditions at time t

0"0 are given by the functions f (s) and g(s):

z(s, 0)"f (s);Lz

Lt(s, 0)"g(s) ∀s3XM . (2)

Finally, the boundary conditions are written ∀t*0

z(0, t)"Z0; z (¸, t)"Z

1,

Lz

Ls(0, t)"h

0;

Lz

Ls(¸, t)"h

1,

!DL2z

Ls2(0, t)"M

0; !D

L2z

Ls2(¸, t)"M

1,

DL3z

Ls3(0, t)"¹

0; D

L3z

Ls3(¸, t)"¹

1

(3)

for, respectively, the de#ection, rotation (slope), moment and shear force of the beam at bothsupports.

The solution of this boundary value problem is projected on the eigenmodes basis,hereafter denoted by Mu

j(s)N

j*1, as

z(s, t)"`=+j/1

qj(t)u

j(s), (4)

where Mqj(t)Nj*1 are the generalized co-ordinates of the beam de#ection. In the following,

the eigenmodes are normalized with respect to the mass, that is, they satisfy theorthogonality condition

1

¸ PX

uj(s)u

k(s) ds"d

jk, (5)

where d is the Kronecker symbol. The associated eigenvalues are denoted by Ma2jNj*1 and

correspond to the eigenfrequencies MujNj*1 such that

uj"

a2j

¸2c

with c2"D/o. The generalized co-ordinates are solutions of the ordinary di!erentialequations:

qKj(t)#2fu

jqRj(t)#u2

jqj(t)"

1M PX

x(s, t)uj(s) ds,

qj(0)"

1

¸ PX

f (s)uj(s) ds ; qR

j(0)"

1

¸ PXg(s)u

j(s) ds

t'0(6)

for all integers j*1, M"o¸ being the total mass of the beam. The generalized loads aredenoted by x

j(t)":X x(s, t)u

j(s) ds. These equations can be generalized further by assuming

a di!erent critical damping rate fjfor each eigenfrequency, which is done in the rest of the

paper. The damped eigenfrequencies are denoted by MuDj

Nj*1 with uDj"u

jJ1!f2

j.

Figure 1. Layout of N successive point loads passing by a beam structure at constant speed v.

270 E. SAVIN

Four standard cases of boundary conditions will be considered in this study: the "rst oneis a simply supported (hinged}hinged) beam, the second one is a clamped}clamped beam,the third one is a clamped}hinged beam, and the last one a cantilever (clamped}free) beam.The analytical expressions for the corresponding normalized eigenmodes can be found inany textbook on structural dynamics (see reference [27] for instance). Although limited tobeams with a single degree of freedom for bending (the de#ection), the analysis developed inthis paper can be extended to vector-valued eigenmodes for multi-degree-of-freedombounded media. Thus, provided analytical expressions are known for the eigenmodes, thedevelopments presented below can be carried out for such structures as well (see reference[28] for the case of plates).

3. RESPONSE TO MOVING LOADS

3.1. DEFINITION OF THE MOVING LOADS

The externally applied loads consist in a succession of N concentrated point loads actingtransversally on the beam and moving at a constant speed v (see Figure 1). Their weights aredenoted by P

nand their relative distances with respect to the "rst load by d

n, for

n"1, 2,2 , N with the convention that d1"0. Therefore, x (s, t) can be expressed by

x (s, t)"N+n/1

Pn1[d

n/v, (d

n#¸)/v] (t)d (s!vt#d

n), (7)

where 1I(t)"1 if t3I, 1

I(t)"0 if t N I, and d is the Dirac measure. Introducing the circular

frequencies u6j"a

j(v/¸) corresponding to a single load travelling across the beam, the

generalized loads are given by

xj(t)"

N+n/1

Pn1[0, ¸/v]At!

dnv Bu

jAuN jAt!dnv BB . (8)

These expressions can now be used to solve equations (6) with the principle ofsuperposition, assuming the beam is at rest at the initial time t

0"0 when the "rst load

P1

reaches it, that is f (s)"g(s)"0 for s3XM .


3.2. DYNAMIC DEFLECTION FOR A SIMPLY SUPPORTED BEAM

The beam de#ection induced by each load can be decomposed into two steps: the "rst onecorresponds to the forced vibrations due to the load P

nconsidered, for d

n/v)t)(d

n#¸)/v,

and the second one corresponds to the free vibrations after the load Pnhas left the beam, for

t*(dn#¸)/v. Analytical expressions can be obtained directly from equation (6) by

integration; however, they are reproduced here only for the case of a simply supported beamfor illustration purposes. Expressions for the other boundary conditions considered in thisstudy may be found, for instance, in references [13, 29]. Introducing the factor i

j"u6

j/u

j,

sometimes called a speed parameter in the literature (see for instance reference [15]), thegeneralized co-ordinates for the de#ection of a simply supported beam due to the loadPn

are denoted by q(n)j

(t) for j*1 and are given byFirst step d

n/v)t)(d

n#¸)/v :

q(n)j1

(t)"J2P

nMu2

j

1

(1!i2j)2#(2f

jij)2

]G(1!i2

j) sinu6

jAt!dnv B!2f

jijcosu6

jAt!dnv B!i

je!f

ju

j(t!(d

n/v ))

]C(1!i2

j!2f2

j)

J1!f2j

sinuDjAt!

dnv B!2f

jcos u

DjAt!dnv BD H . (9)

Note that these terms do not correspond to purely forced vibrations as they also include freevibrations induced by the impact of the loads. However, they will be called &&forcedvibrations'' in the rest of the paper since they arise for the time intervals when the loadstraverse the beam. For weakly damped structures, equation (9) can be simpli"ed byneglecting the terms of orders greater than 1 in f

j. This leads to

q(n)j1

(t)KJ2P

nMu2

j

1

1!i2jCsini

ju

jAt!dnv B!i

jsinu

j At!dnv B e~f

ju

j(t!(d

n/v))D (10)

for ijO1, and

q(n)j1

(t)K!J2P

n2Mu2

jCujAt!

dnv B cos u

j At!dnv B!sin u

jAt!dnv B e~f

ju

j(t!(d

n/v))D (11)

for ij"1. The overall generalized co-ordinates for the beam forced de#ection induced by

the whole train of loads are

qj1

(t)"N+n/1

1[0,¸/v] At!dnv B q(n)

j1(t). (12)

Second step t*(dn#¸)/v :

q(n)j2

(t)"J2P

nMu2

j

ij

((1!i2j)2#(2f

jij)2 )

e!fju

j(t!(d

n#¸)/v)

]CQjcosu

DjAt!dn#¸

v B#AfjQ

jJ1!f2

j

#

QQj

uDjB sinu

DjAt!dn#¸

v BD , (13)

272 E. SAVIN

where the &&initial conditions'' for free vibrations after the load Pnhas left the beam are

Qj"e~f

jjn/i

jC(1!i2

j!2f2

j)

J1!f2j

sinAjnij {B#2f

jcosA

jnij {BD!2(!1)j f

j

QQj

uj

"(!1)j (1!i2j)!e!f

jjn/i

j

](1!i2

j!2f2

j) Acos A

jnij {B!

fj

J1!f2j

sin Ajnij {BB

#2fjAJ1!f2

jsin A

jnij {B#f

jcosA

jnij {BB

with ij {"i

j/J1!f2

jand a

j"jn for a simply supported beam. For weakly damped

structures, these expressions are simpli"ed to give

q(n)j2

(t)KJ2P

nMu2

j

ij

(1!i2j)e~f

ju

j(t!(d

n#¸)/v)

]CQjcosu

jAt!dn#¸

v B#QQ

ju

j

sinujAt!

dn#¸

v BD (14)

for ijO1 with

Qj"!sinA

jnijB e~f

jjn/i

j ,

QQj

uj

"(!1)j!cosAjnijB e~f

jjn/i

j .

The case ij"1 is examined in section 4.1. Finally, the overall generalized co-ordinates for

the free vibrations after the loads have left the beam one by one, are

qj2

(t)"N+n/1

H At!dn#¸

v B q(n)j2

(t), (15)

where H(t) is the Heaviside step function, H(t)"1 if t*0 and H(t)"0 if t(0. To concludethis subsection, it can be noted that an alternative representation of the solution to thesingle load problem for an undamped simply supported beam has been given by Steele [10]in terms of an in"nite series of in"nite integrals. It is claimed that its convergence isimproved for a high-speed parameter i

1as compared to the classical Fourier series

representation developed here. More recently, Pesterev and Bergman [30] have derived analternative series representation, exhibiting explicitly the quasi-static contribution to theresponse and converging more rapidly. The latter property can be helpful for the evaluationof spatial derivatives of the de#ection, in order to calculate the distributed e!orts in thebeam.


3.3. DYNAMIC AMPLIFICATION FACTORS

3.3.1. Forced vibrations of a simply supported beam

The forced vibrations of the beam, characterized by the generalized coordinates q(n)j1

(t) foreach load P

nand time d

n/v)t)(d

n#¸)/v, are seen from equation (9) or (10) to be

a combination of vibrations at the circular frequencies u6jfor a single load traversing the

beam, and uDj

for the damped eigenfrequencies of the beam. They have been extensivelystudied in the literature [7, 12, 25, 31, 32] and constitute the basis of the de"nition of thedynamic ampli"cation factors in some standard bridge design codes, such as the Eurocodeor the UIC (International Union of Railways) norms. These ampli"cation factors areusually given as functions of i

1when the analysis is restricted to the fundamental

eigenmode of the beam, that is when expansion (4) is truncated at j"1.We deduce from expression (9) that an apparent resonance occurs when i2

j"1!2f2

j,

that is when u6j"u

jJ1!2f2

jwhich is the well-known damped frequency of resonance of

the single oscillator (see for instance reference [27]). The maximum de#ection for thecorresponding loads speed is reached when t"(d

n#¸)/jv and is

max0)t!d

n/v)¸/v

q(n)j1

(t)"n2]

J2Pn

Mu2j

for the undamped case. In fact, this value is only a pseudo-resonance: true resonance occursat least for a whole period, but in this case the vibrations at the circular frequency u6

jare

de"ned just over an half period since 0)t!dn/v)¸/v. The overall maximum value of

q(n)j1

(t) is rather obtained for ijK0)61 and is about

max0)t!d

n/v)¸/v

q(n)j1

(t)K1)73]J2P

nMu2

j

(16)

for a simply supported beam (see references [12, 31]). These values for each generalizedco-ordinate have to be compared with the static de#ection of such a beam due to a singleload P at a distance l from either support:

zstat

(s)"2P

M

`=+j/1

1

u2j

sinAjnl

¸B sinAjns

¸B , s3XM .

Furthermore, the e!ects of the N loads on forced vibrations do not cumulate since theaction of each load is limited to time intervals of length ¸/v which do not overlap except forfew successive loads. Indeed, for typical span lengths and trains, only 2 or 3 loads would beon the bridge at the same time; from a conservative point of view, the maximumampli"cations outlined above may be multiplied by the maximum possible number ofloads on the bridge, which is limited to only few of them. This means that, as regardsforced vibrations, the maximum dynamic de#ection can be evaluated by multiplying themaximum static de#ection observed under the train layout considered, by a dynamicampli"cation factor U(i

1) possibly varying with speed (as in references [12, 33]) but never

greater than about 1)7 for a simply supported beam. Qualitatively, similar conclusionscan be drawn for other types of boundary conditions although this last maximumampli"cation factor is generally slightly smaller (see for instance, the results obtained inreferences [13, 29]). U(i

1) is plotted in Figure 2 for the four types of boundary conditions

considered here, and according to the approximate formula proposed in reference [33] forcomparison.

Figure 2. Dynamic ampli"cation factor U (i1) as proposed in reference [33] (===) and for a simply supported

beam (***), clamped}clamped beam (} } } } }), clamped}hinged beam (} ) } ) } ) } ) and cantilever beam(====).

274 E. SAVIN

It is also worth noting at this stage that, for typical engineering structures, such as railwaybridges, the span lengths are of the order of several tens of meters, and the fundamentalfrequency is about several Hertz. Thus, the resonance frequency for a single load travellingacross the bridge corresponds to a train speed of about several hundred meters per second(or of the order of 1000 km/h), which is unrealistic for the actual vehicles.

3.3.2. Free vibrations

The free vibrations of the beam, characterized by the generalized co-ordinates q (n)j2

(t) foreach load P

nand time t*(d

n#¸)/v, are vibrations at the damped eigenfrequencies of the

beam only. From equation (13) or (14) for weak damping, it is seen that each q(n)j2

(t) can berewritten as

q(n)j2

(t)"1

Mu2j

ijR

j(i

j, f

j)

[(1!i2j)2#(2f

jij)2]

]Pnsinu

Dj At!dn#¸

v#/

j(i

j, f

j)B e~f

ju

j(t!(d

n#¸)/v), (17)

where

R2j"2 CQ2

j#

1

(1!f2j) AfjQj

#

QQj

ujB2

D


depends on ijand f

j, and

/j"arctan A

J1!f2jQ

jfjQ

j#QQ

j/u

jB

is an additional phase term which is independent of the current load characteristic Pn

ordn

as well. Therefore, extending the above analysis to all kinds of boundary conditionsconsidered in this study, the generalized co-ordinates of the beam free vibrations have theform

qj2

(t)"1

Mu2j

ijR

j(i

j, f

j)

[(1!i2j)2#(2f

jij)2]

]N+n/1

PnHAt!

dn#¸

v B sinuDj At!

dn#¸

v#/

j(i

j, f

j)B e~

fju

j(t!(d

n#¸)/v)

,

(18)

where Rjdepends on the type of boundary conditions. The beam free motion is thus given

by the product of an ampli"cation factor which depends on ij(and damping) only, and of

damped free vibrations at the eigenfrequencies of the beam whose amplitudes and phasesare fully characterized by the speed and layout of the moving loads. These terms are studiedin detail in the next section, introducing the notions of dynamic ampli"cation coe$cient,train characteristic response spectrum and a new parameter called the loads wavelength.Note that here, contrary to the case of forced vibrations, the e!ects of all loadscumulate*however, with di!erent phases*since they arise on overlapping time intervals ofin"nite lengths, and may thus induce non-negligible de#ections as compared to the ones dueto the forced vibrations for weakly damped structures. These residual free vibrations [12]for successive loads have often been disregarded in the literature.

4. ASSESSMENT OF MAXIMUM VIBRATION LEVELS

Equation (18) suggests that the leading e!ect of the loads on the vibrations of the beam istheir frequency of succession, that is the ratio of their speed to a &&typical'' distance betweenthem. In fact, it will be shown that this e!ect is more likely to induce resonance of the beamthan the forced circular frequency u6

jattached to a single load, for the reasons mentioned

above in sections 3.3.1 and 3.3.2. For a meaningful analysis of this e!ect on the generalizedco-ordinates for free vibrations of the beam, it is suggested that a new parameter beintroduced, the so-called wavelength of the loads, de"ned by j"2n v/u. Its values at theeigenfrequencies of the beam are denoted by j

j"2n v/u

j. Another interesting point is to

introduce the well-known simpli"ed formula for the fundamental frequency of a beam inbending which is [12]

f1K0)18S

g

zM

, (19)

where g"9)81 m/s2 is the acceleration due to gravity and zM

is the beam maximum staticde#ection due to its own weight. Equation (19) is a posteriori valid for any kind ofboundary conditions (the coe$cient for a cantilever beam is slightly greater than 0)18 butthis value is conservative as it is used hereafter). Since i

j"(a

j/2n) (j

j/¸), q

j2(t) can now be

276 E. SAVIN

rewritten as

qj2

(t)"z1]r

jA2n¸

jj

, fjB]R

jAt!¸

v; j

j, f

jB , (20)

rj(u, f) is a dynamic ampli"cation coe$cient de"ned by

rj(u, f)"

n2]

a41

a4j

]u2

[(u2!a2j)2#(2fa

ju)2]

RjA

aj

u, fB , (21)

Rj(t; j, f ) is a train characteristic response function associated with the free vibrations of the

beam de"ned by

Rj(t ; j, f)"

1

(0)18n)2

]1

jN+n/1

PnHAuj

t!2ndnjB sin J1!f2Auj

t!2ndnj#/M

jBe~f(ujt!2nd

n/j) (22)

with /1j"u

j/j. These last two functions are studied in detail in the following subsections.

Finally, z1

is the beam maximum static de#ection due to uniform distributed load of 1, thatis z

1"z

M/og .

4.1. DYNAMIC AMPLIFICATION COEFFICIENT

The dynamic ampli"cation coe$cient r1

de"ned above is plotted in Figure 3 for di!erentvalues of damping and the four types of boundary conditions considered in this study. r

2is

also plotted in Figure 4 (note the di!erence of scales as compared to Figure 3). It is seen that,for weakly damped structures, these coe$cients are almost independent of the criticaldamping rate f

j. Thus, the dependence on f

jwill be ignored for them in the rest of the paper.

They are also rapidly decreasing when j increases; thus, a truncation at the order j"1 isfully justi"ed.

Following these conclusions, we introduce a slightly di!erent dynamic ampli"cationcoe$cient denoted by a(¸/j), independent of f and used hereafter to assess the maximumde#ection due to free vibrations. It is de"ned by

aA¸

jB"r1 A2n

¸

j, 0B]max

s3X1u1(s). (23)

Figure 5 displays a for the four cases of boundary conditions considered in this paper. Itsanalytical expression for a simply supported beam is for instance,

a (u)"K2 cos nu

1!4u2 Kfor uO1

2and a (1

2)"n/2 (which is not the maximum). The formulas for other boundary

conditions have been derived in reference [13]. They show that the shorter the span lengthis, the larger the dynamic ampli"cation coe$cients are. Furthermore, the ampli"cationcancels out*or almost cancels out*for some particular ratios of the span length to thewavelength, whereas it has local maxima for some other ratios. In the case of a simplysupported beam, these ratios are of the form (2m!1)/2 and Km (but never exactly),

Figure 3. Dynamic ampli"cation coe$cient r1(2n¸/j, f) for a simply supported beam, clamped}clamped beam,

clamped}hinged beam and cantilever beam, and various damping rates f (=== ), f"0)5%;====, f"1%;+ ' + ' + ' +, f"2%; ****, f"3%; } } } } }, f"4%; } )} ) } ) , f"5%).


respectively, where m'1 is an integer. For instance, for a train with a regular distanced between its successive bogies, dynamic e!ects on bridges are dramatically reduced byconsidering span lengths of order ¸K1)5 d. These observations are well known by railwayengineers and were made in other previous analytical studies [12, 13, 15].

4.2. TRAIN RESPONSE SPECTRUM

The response spectrum of an externally applied load x(t) is the maximum response withtime of a single oscillator subjected to this load [27], namely,

p(u, f)"1

MuD

maxt*0 K P

t

0

sinuD(t!q) e~fu (t!q)x(q) dq K ,

where u is the circular eigenfrequency of the oscillator, f is its critical damping rate and

uD"u J1!f2 . The response spectrum of the load for f"0 is usually close to the

amplitude of its Laplace transform; however, it remains a one-to-one transform which doesnot permit the recovery of the load from the knowledge of its spectrum. The so-calledprimary response spectrum is the maximum response of the single oscillator for theduration of the loading, whereas the secondary spectrum is the maximum response after itsend (damped free vibrations). Clearly, from the above analysis the secondary spectrumdominates for successive moving loads and needs to be evaluated.

Figure 4. Dynamic ampli"cation coe$cient r2(2n¸/j, f ) for a simply supported beam, clamped}clamped beam,

clamped}hinged beam and cantilever beam, and various damping rates f (=== , f"0)5%;====, f"1%;+ ' + ' + ' +, f"2%; ****, f"3%; } } } } }, f"4%; } )} ) } ) } ) , f"5%).

Figure 5. Dynamic ampli"cation coe$cient a (¸/j) for a simply supported beam (===), clamped}clampedbeam (==== ), clamped}hinged beam (*** ) and cantilever beam (} } } } }).

278 E. SAVIN


Here we de"ne the secondary train response spectrum as the maximum of Rj(t; j, f)

with time and the number of loads, to account for the di!erence in phase betweenthem

pj(j, f )" max

1)Ns)N

maxt*0

1

(0)18n)2

1

j KNs

+n/1

PnHAuj

t!2ndn

jB]sin J1!f2 Auj

t!2ndn

j#/M

jB e~f(ujt!2n d

n/j) K . (24)

This series is still quite cumbersome and (very) costly to evaluate numerically. However, theresponse spectrum as it is de"ned above is well approximated by p (j, f) where

p (j, f )"1

(0)18n)2max

1)Ns)N

1

j KNs

+n/1

PnexpC2nA!f#i J1!f2B

dnjD K (25)

with i"J!1. The series in the above expression is nothing but the Laplace transform ofthe generalized loads x

j(t) evaluated at the poles p

j"!f

ju

j#iu

Djof sub-critically

damped single oscillators with circular eigenfrequencies ujand critical damping rates f

j.

This formula has the advantage of being independent from the geometrical and mechanicalcharacteristics of the beam, apart from its critical damping ratio, although it is a relevantestimation of its true response spectrum. That is why from now on we will choose it to de"nethe train characteristic response spectrum p(j, f ) used to assess the maximum de#ection ofthe beam under moving loads. Figure 6 displays this response spectrum for the example ofa French TGV-A train whose layout is given in reference [25]. A &&typical'' distance betweenthe loads is dK19m (to be compared with the constant length of the cars d"18)7 m whichis di!erent from the length of the power cars) and maximum ampli"cations are observed forj"d and d/2. This observation can be generalized to other actual train layouts. Thecorresponding loads speeds are v"d]f

1and (d/2)]f

1, where f

1is the fundamental

eigenfrequency of the beam, and are often called &&critical'' speeds. However, the beamresponse at these speeds remains always bounded, whereas true critical speeds would rathercorrespond to the limits of instability for the steady state response of in"nite or semi-in"nitesystems (see for instance references [10, 16, 17, 23]). In agreement with Adams [14], we notethat they can be much smaller than the speed corresponding to i

1"1 for which we have

v"2¸]f1.

To get further insight on the response spectrum, let us consider the case wheredn"(n!1)d and P

n"P (d and P are constant), that is a succession of regularly spaced

and identical loads, and f"0. Then when j"d/m, where m is a positive integer, thetrain spectrum is p"(1/(0)18n)2) (mNP/d). Thus, it is linear with respect to the numberof loads and maximum for a frequency of succession of these loads which matches thefundamental frequency of the bridge or its fractions; of course, this linear dependencefor f"0 is increasingly damped out when f increases. The critical speeds are in thiscase v

m"(d/m)]f

1, and theoretically identical results hold for all eigenfrequencies of

the beam. If j"2d then the response spectrum is simply proportional to a single loadp"(1/(0)18n)2) (P/2d). These observations are con"rmed by Figure 7 where the responsespectrum for the train used by Yang et al. [15] and having a more regular arrangementof loads has been displayed. Its characteristics are as follows: d

2n"nd#(n!1)e

and d2n`1

"n (d#e), with N"10 (corresponding to 5 cars), d"18m, e"6 m, andPn"22]103kg.

Figure 6. TGV-A characteristic response spectrum p (j, f) for various critical damping rates of the beam (===,f"0)5%; ====, f"1%; + ' + ' + ' +, f"2%; ****, f"3%; } } } } }, f"4%; } ) } ) } )} ) , f"5%).

Figure 7. Response spectrum p (j, f ) for the train considered by Yang et al. [15] and various critical dampingrates of the beam (===, f"0)5%; ===== , f"1%; + ' + ' + ' +, f"2%; ****, f"3%; } } } } },f"4%; } ) } ) } ) } , f"5%).

280 E. SAVIN


4.3. SIMPLIFIED CALCULATION OF THE MAXIMUM RESPONSE OF THE BEAM

4.3.1. Proposed formulas

From the previous considerations and putting together equation (4) up to the "rst orderand equation (20), the maximum dynamic de#ection of a beam under successive movingloads is

zmax

KU A¸

j1B]z

stat#aA

¸

j1B]p(j

1, f)]z

1, (26)

where zstat

is the maximum static de#ection of the beam under the arrangement of loadsconsidered, z

1is its maximum static de#ection under a uniform unit load, ¸ is the span

length, f is the critical damping rate, j1

is the wavelength of the loads de"ned as the ratio oftheir speed to the fundamental frequency of the beam; "nally, U)1)7 is the dynamicampli"cation factor for the forced vibrations de"ned in the usual design codes and such that

limjP0

UA¸

jB"1.

The maximum vertical acceleration of the beam can be evaluated from the above using thepseudo-acceleration spectrum associated with a given train characteristic responsespectrum p (j, f):

cmax

Ka A¸

j1B]u2

1p (j

1, f)]z

1. (27)

Note that from the formulas de"ning a (¸/j) and p(j, f), it can be seen that

limjP0

aA¸

jB p (j, f )"0

as expected: dynamic ampli"cation vanishes when the train speed is zero.Other quantities of interest for engineering applications, such as the ones mentioned in

the introduction, can be derived by the same procedure using essentially equation (4). Note,however, that in doing so, it must be kept in mind that the convergence of the seriesdeteriorates by successive spatial derivatives and then higher order terms shall be added.For instance, estimates of the maximum rotations at the hinged supports may be obtainedfrom the above equations scaled by the span length ¸ with a modi"ed accordingly(maximum of u@

1(s) for s"0 or ¸ instead of the maximum of u

1(s)). Besides, torsional

motions are described by similar di!erential equations and mode shapes (depending on theboundary conditions) while the loads are multiplied by their excentricity with respect to thetorsion centre of the beam.

4.3.2. Numerical example

We consider the case of a simply supported beam whose characteristics are chosen inaccordance with the example examined by Yang et al. [15], and focus on the time history ofthe de#ection at mid-span for di!erent speeds of the loads. The plots are obtained using theexact analytical formulas given in section 3.2 and the summation of "ve eigenmodes. Thespan length is ¸"20m, its density is o"34)088 kg/m and its bending rigidity isD"1)12]1011Nm2, such that its fundamental eigenfrequency is f

1"7)1Hz. The static

de#ection under a uniform unit load of 100 kN/m is z1K1)9 mm. The critical damping rate

is chosen as f"2)5%. The loading train is the one described in section 4.2 whose spectrum

Figure 8. Time history of the de#ection at mid-span of a simply supported beam at various critical speeds of thetrain. Train and beam characteristics are taken from reference [15].

282 E. SAVIN

is given in Figure 7. From the examination of this spectrum, it is seen that possiblemaximum ampli"cations can occur at j

1K23, 8 and 6 m corresponding to train speeds

v"589)7, 205)1 and v"153)8 km/h respectively. Time history curves of the mid-spande#ection for v"153)8 and 589)7 km/h are plotted in Figure 8, together with the curves forv"615)5 km/h, corresponding to i

1"0)6, and v"10 km/h, corresponding to the

so-called crawling (quasi-static) de#ection. On the other hand, Figure 5 gives the values¸/j

1"m!1

2where dynamic ampli"cations can theoretically be cancelled. They

correspond to the cancellation speeds v"341)9 km/h for m"2, v"205)1 km/h for m"3(which is a critical speed as well), v"146)5 km/h for m"4 and v"114 km/h for m"5.Figure 9 displays the time history de#ection at mid-span for these four speeds. It is seen thatdynamic ampli"cations are e!ectively cancelled out since the maximum de#ectionsobserved are comparable to the maximum crawling de#ection plotted in Figure 8. Forinstance, if m"2 then i

1"1

3and UK1)45 whereas the observed ampli"cation is

zmax

K1)3]zstat

. Regarding the critical speeds, for instance, v"589)7 km/h, we haveUK1)7, whereas aK0)9 and pK140 kN/m. Thus, z

maxK3)2 mm from equation (26),

compared to zmax

K2)7 mm from Figure 8. We also observe that at v"153)8 km/h, thedynamic ampli"cation coe$cient a is negligible and UK1)15, so that dynamicampli"cations are signi"cantly reduced.

4.3.3. Comparison with experiments

The proposed formula for the expected maximum acceleration is compared withmeasured accelerations on an existing railway bridge in France (data provided by the

Figure 9. Time history of the de#ection at mid-span of a simply supported beam at various cancellation speedsof the train. Train and beam characteristics are taken from reference [15].


French National Railways, SNCF). This simply supported structure is labelled OA71)18and its useful characteristics are as follows: its span length is ¸"17)4 m, its fundamentaleigenfrequency is f

1"4)5 Hz, its critical damping rate is estimated at f

1"4%, and its mass

per unit length is o"16)3]103 kg/m. The bridge was traversed by a TGV-A typehigh-speed train at the speed v"260 km/h and the maximum acceleration experienced atmid-span was c

exp"3)5 m/s2.

For these data, the loading wavelength j1

is j1"16 m leading to aK0)5 for the dynamic

ampli"cation coe$cient and pTGV

K95 kN/m for the train spectrum. Applying equation(27), the expected maximum acceleration is found to be c

maxK3)8 m/s2 which compares

reasonably well with measurements.We conclude from these examples that the proposed formulas for the assessment of

maximum vibration levels give accurate results, although slightly conservative with respectto the numerical simulations based on the exact analytical expressions, and the measuredacceleration for the particular experiment described previously. The discrepancies can easilybe explained by invoking the simplifying assumptions introduced as and when required forthe derivation of equation (26).

5. INFLUENCE OF SHEAR DEFORMATION AND ROTATORY INERTIA

In this section, we brie#y review the in#uence of transverse shear deformation androtatory inertia on the results obtained so far, using the more exact Timoshenko beam

284 E. SAVIN

model. The developments below are intended to give some indication on how the formulasof section 4.3 may be tentatively modi"ed or not in accordance with this alternative model.

Euler}Bernoulli beam models are valid only if the slenderness ratio r/¸ is small, where r isthe radius of gyration. These models are also often inaccurate for beams with a constantcross-section S and mass per unit length o. A Timoshenko beam theory taking into accountshear deformations and rotatory inertia is more adapted to such cases and when theslenderness ratio is not too small. Introducing the bending angle h (s, t) along the beam, thebasic coupled partial di!erential equations considered for h and z are

x (s, t)!oL2z

Lt2(s, t)"

LLs Ck@GSAh(s, t)!

Lz

Ls(s, t)BD!2fJoD

LLt

L2z

Ls2(s, t),

oI

S

L2hLt2

(s, t)"k@GSALz

Ls(s, t)!h (s, t)B#D

L2hLs2

(s, t),

(28)

where o is the mass per unit length of the beam, S is its cross-section area and I is its bendinginertia, D"EI is the bending rigidity and E is Young's modulus; G"E/2(1#l) is theshear modulus, l the Poisson coe$cient, and k@ is the usual shear reduction coe$cientdepending on l and the cross-section geometry. The same boundary conditions and initialconditions as in section 2 are applied. We also consider that the beam is at rest at t"0( f (s)"g(s)"0 for s3XM ).

The conservative spectral problem for the beam de#ection is given by the equation

d4u

ds4(s)#u2A

1

c2p

#

1

c2sB

d2u

ds2(s)!

u2

c2 A1!u2c2

c2pc2sB u(s)"0, s3X (29)

and boundary condition (3), with the following parameters:

c2"D

o, r2"

I

S, c2

p"

ES

o"

c2

r2, c2

s"

k@GS

o.

The eigenmodes have expressions of the form

uj(s)"sinAa`j

s

¸B#AjsinhAa~j

s

¸B#BjcosAa`j

s

¸B#CjcoshAa~j

s

¸B (30)

up to a normalization coe$cient, and the eigenvalues are given by

Aa$

j¸ B

2"

u$2j2 C$A

1

c2p

#

1

c2sB#SA

1

c2p

!

1

c2sB2#

4

u$2j

c2D (31)

provided that u$

j(c

s/r"u

c.

The forced de#ection of a simply supported Timoshenko beam excited by a single movingload is studied for instance in reference [34]. In this case, a

j"jn and the two sets of circular

eigenfrequencies are

u$2j

"c$2

ju2

j"

u2j

2t2jC1#

c2s

c2pA1#

1

t2jB$SC1#

c2s

c2pA1#

1

t2jBD

2!

4c2s

c2pD (32)

with tj"a

jr/¸ (Rayleigh's coe$cient). Mu

jNj*1 are the circular eigenfrequencies for the

Euler}Bernoulli beam model, which is recovered when ucP#R, so that u$

jPu

j. The

normalized eigenmodes for the beam de#ection Muj(s)Nj*1 and bending angle Ms

j(s)Nj*1

Figure 10. Coe$cients 1/k$

jc$2j

for a circular (*** ) and a rectangular (=== ) Timoshenko beam withl"0)2.


are given by

uj(s)"S

2

k$

j

sinAajs

¸B,

sj(s)"S

2

k$

jA1!

c2p

c2s

c$2j

t2jB

aj

¸

cosAajs

¸B ,

(33)

where k$

j"1#t2

j(1!(c2

p/c2

s)c$2

jt2j)2; they satisfy the orthogonality condition

1

¸ PX

uj(s)u

k(s) ds#

r2

¸ PX

sj(s)s

k(s) ds"d

jk.

Then it can be shown that the generalized co-ordinates satisfy the ordinary di!erentialequations:

qKj(t)#2f$

ju$

jqRj(t)#u$2

jqj(t)"

xj(t)

Mk$

j

, t'0,

qj(0)"0, qR

j(0)"0

(34)

with f$

j"f

j/k$

jc$

j. Therefore, all the results obtained so far are formally unchanged and

the maximum response of the beam can be evaluated using equation (26) for the same

286 E. SAVIN

structure modellized by an Euler}Bernoulli beam, corrected by a factor 1/k~1

c~21

*1 (forthe lowest eigenfrequency):

zmax

K

1

k~1

c~21CUA

¸

j1B]z

stat#aA

¸

j1B]p(j

1, f)]z

1D . (35)

In Figure 10, we have plotted the coe$cients 1/k$

jc$2j

as functions of tj

for a Poissoncoe$cient l"0)2 and circular and rectangular cross-sections, for which [35]

k@"6(1#l)7#6l

, circular cross-section,

k@"10(1#l)12#11l

, rectangular cross-section.

Similar analyses can be performed for the in#uence of an axial load or elastic foundation,and comparable conclusions are reached.

6. CONCLUSIONS

In this paper, the exact analytical solution for an Euler}Bernoulli beam traversed bya succession of massless point loads has been given. The dynamic ampli"cationscorresponding to the forced response and free vibrations were both analyzed in detail. Inparticular, free vibrations were shown to be generally non-negligible as compared to theforced ones, if the number of loads is of the order of several tens as is the case in railwayapplications. Based on these results, simple formulas involving a minimum of soundlyparameters were derived for the calculation of the maximum expected dynamicampli"cations induced by the loads. They show that for some particular span lengthsrelated to the wavelength of the loads, these e!ects can be dramatically reduced and eventheoretically cancelled. Therefore, they may be especially useful in the context ofpreliminary design of railway structures modelled by beams. Finally, tentative formulaswere also given to account for transverse shear deformations and rotatory inertia accordingto the Timoshenko beam model.

The simpli"ed formulas for beams obtained in this work cannot be applied directly tosome particular structures such as skewed or cable-stayed bridges for instance. In theselatter cases, three-dimensional models are needed and can only be solved numerically.However, the results obtained so far could easily be extended, in our opinion, to plates sinceanalytical expressions of the eigenmodes are available for most cases of boundaryconditions and geometrical layout used in practical applications. We also note that in theevaluation of the dynamic ampli"cation e!ects, structural strength may be less a concern forrailway bridges than riding comfort and safety standards for the tracks in operationalconditions. The formulas given in this paper are adapted to the evaluation of the maximumdeformations related to the criteria applicable for accelerations in the train cars oralignment of the tracks.

ACKNOWLEDGMENTS

This study was carried out while the author was a$liated to the Systra Group, mostly in1994. The author thanks V. T. Doan and D. Mion of the Engineering Direction at SNCF forsome valuable discussions and useful suggestions.


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