Structural Analysis of Historical Constructions - Modena, Lourenço & Roca (eds) © 2005 Taylor & Francis Group, London, ISBN 04 15363799

Historical railway bridges: tests and numerical analysis

M. Ferraioli, P. Malangone, M. Rauci & A. Zambrano Second University of Naples, Ilaly

ABSTRACT: A procedure based on experimental and theoretical analyses to evaluate the response ofrailway bridges under dynamic loading is presented. In particular an ancient railway bridge under service on the AvelIino­Rocchetta-Foggia line in the Southem part ofItaly was considered. A simplified procedure to study the vehicle­bridge interaction with a simple continuous beam model is proposed. The calibration ofthe model was carried out at first with the study ofthe technical reports, the visual inspection, the material tests and the dynamic tests. Then an identification procedure to characterize the presence of structural non-linearities is applied. FinalIy, the calibrated model of the bridge was used to perform modal analysis, to evaluate the magnification facto r of the bridge and the acceleration response of the vehicle.

INTRODUCT ION

The procedures for the analysis of existing bridges are usualIy based on field experiments which are the most reliable mean to determinate their dynamic char­acteristics. In fact the dynamic properties are only marginalIy influenced by variations in loading while the static properties may sutTer strong variations with the load pattem. As a consequence, during the last years many studies focused on the possibility ofusing the vibration characteristics to evaluate the structural health. In the simplest form , the identification tech­niques generalIy use the natural frequencies and the mode shapes obtained with field tests (ambient vibra­tion, forced vibration, free vibration, tratTic vibration, earthquake response measurements). These dynamic properties alIow the control ofthe construction quality, the validation and the improvement ofstructural mod­eIs, the assessment of damage (Salawu & Williams 1995, Kou & De Wolf 1997, Capecchi & Vestroni 1999, Castiglioni et aI. 2002). The approaches based on the modal analysis techniques in frequency domain are stilI dominant in the model updating philosophy (Ewins 1986). GeneralIy, the finite element model of the structure is validated with the comparison between the eigenproperties deriving from the model and the eigenproperties calculated from the moda I tests. Then the interaction among the train, the support­ing track and the bridge has to be properly modeled. In fact this interaction generalIy produces the ampli­fication of displacements, strains and stresses in the structural members (Hurty & Rubinstein 1967). This interaction problem between vehicles and bridge has attracted much attention due to the large increase in the

proportion of heavy vehicles and high-speed vehic les in railway tratTic.

FinalIy, the dynamic nature ofthe train loading has to be considered because it produces some particular etTects. First of alI, the sudden changes in the bridge loading due to train speed create heavy inertial etTects on the structure. Then changes in the axial load on the bridge occur as a consequence both of the rough­ness of the railroad and of the irregularity of the train wheels. FinalIy, the repeated sequences of the loads due to the nearly constant spacing of the train ax les can produce both resonance phenomena and huge and dangerous vibrations. After all the dynamic behavior ofthe bridge is influenced by some main factors: a) the natural frequency ofthe structure; b) the wheel bases of the train; c) the train speed; d) the structural damp­ing; e) the type ofbridge span; f) the spacing oftrain axles; f) the track structure; g) the wheel imperfections; h) the roughness of the rail road .

In this paper some simple models for dynamic inter­action between vehicle and bridge were characterized to study the dynamic response ofthe bridge under the moving train loading, and to evaluate the acceleration inside the vehicle. The interaction between the mov­ing train and the bridge was modeled neglecting the rolIing and yawing of the railway vehicles, constrain­ing the rail on the bridge deck and considering the damping characteristics ofthe vehicle suspension. An equivalent beam model for the bridge was calibrated on the basis of the experimental results . This simple model accounts for the vehicle-bridge interaction, but cannot precisely represent two or three-dimensional behaviour, particularly in the case ofmoving train with paths that are not along the centerline of the bridge.

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For this reason also a three-dimensional finite element model of the bridge that considers moving forces but neglects the dynamic interaction was considered.

2 A MODEL FOR DYNAMIC INTERACTlON BETWEEN VEHICLE AND BRIDGE

The dynamic response ofthe bridge is evaluated con­sidering a continuum beam model under the load pattem p(t, x) which is a function of time t and space x (figure 1). The beam is simple supported and has length L, density mass J.L(x) and constant flexura l iner­tia EI. The dynamic of the beam is described by the following differential equation:

84y 82y EI-4- + f-l-2 = p(t,x)

8 x 8t (I)

The solution ofthe homogenous equation is given by:

y (x, t) = i: An (t) ~n (x) (2) n: !

where AII(I) are the modal coordinates, while <PII(X) are the mode shapes obtained solving the following differential equation:

AJt} + OJ;AJt} = O

d4 M x} _ f-lOJ; Ao (x) = O dx4 EI 'f'n

(3)

Starting from eq. 3 the nth mode shape and modal frequency are given by:

(4)

(5)

p(',I)

pdx

.. , (Qú

\;, dt)

À

Figure I . Continuum beam mode l.

The moving vehicle is modeled with the sprung mass and the unsprung mass concentrated at the axle positions. The suspension device is represented by a spring and a viscous damper (figure 2). The dynamic equilibrium equations of the bridge in the presence of viscous damping can be written in terms of modal coordinates as follows:

;C (t) + 2çnOJ)n (t) + OJ; An (t ) =

F;~n (vt) F2~n (vtl )

= 81 f-l"L/2 +82 f-l.L/2 n =1, ..... ,00 (6)

where 12 = 1 + L, /v, while & 1 and &2 take into account the presence of the two vehicle systems on the bridge (with & 1 = &2 = 1 when the system is on the bridge, &1 = &2 = O otherwise). Substituting the modal shape functions <Pn(x) defined by eq. 4, eq. 6 becomes:

An (t) + 2çnOJnÀn (t) + OJ; An (t) =

=8 ~Sin(n 1r(vt )JF. +8 ~Sin(n 1r(vt2)JF (7) If-lL L 1 2f-lL L 2

The following notation is adopted: ~n is the nth moda I viscous damping ratio ofthe beam; aJn is the nth modal frequency ofthe beam, v is the constant velocity ofthe vehicle.

The forces Fl and F2 that the two moving axles transmit to the beam are given by:

F; = M" [(1 + Mvu Jg -Mvu I An (t)sin (n 1r( vt) J + Mvs M vs n=1 L

+~(YI - IAn (t)Sin(n 1r(vt)JJ+ Mvs "=1 L

+_c (YI-IÀn(t) Sin(n1r

(vt) JJJ (8) Mvs n=1 L

rx y

L • .. vI .. ..

LI .. .. Figure 2. The vehicle-bridge model.

1020

where .h and YI are the displacement and the velocity of the the first moving unsprung mass; Y2 and Y2: are the displacement and the velocity ofthe second moving unsprung mass; kv and c are the spring stiffness and damping coefficient of the vehicle suspension; Mvs and Mvu are the unsprung mass and the suspended mass of the moving vehicle, respectively.

The eguations that describe the dynamics of the two moving systems on the bridge are:

M,.,Y, +k, [y, -c,y,]+c[y,-c,y, J=o (lO)

M.,Y, +k,[y, -c,y,]+c[y, -c,y,J=o

The vertical displacements ofthe unsprung mass Mvs and sprung mass Mvu with reference to their respec­tive vertical eguilibrium positions are YI (I) and Y2(1), respectively. Substituting the expressions of the dis­placement response YI (I) and Y2 (/) given by eg. 2 in eg. 10 gives:

"tYt +~[y, -t;t,~(t)sm( n 7) ]+{y,-t;t,~(t)sin( n 7)]=0 "tY'+~[Y'-f7t,~(t)sm( '7 )}{y,-bit.~(t)1'7 )]=0

(11)

The system of eguations composed by egs. 7 and II can be written in state-space formulation as follows:

{i} = [N]{ z} +[ O]{g} (12)

where the matrices [N] and [O] and the vector {g} can be defined starting from egs. 7 and 11.

Eg. 12 is a system of 2(n + m) eguations, where n is the number of considered modes (in the exam­ined case n = 5) and m is the number of moving systems (m = 2). The matrices O and N include ali the terms of the system of eguation, and have dimension 2(n + m) x 2(n + m).

The vector z is composed both of the principal coordinates of the beam An(/) and of the vertical displacementYm ofthe suspended masses, as follows:

{i} = (A, Ã, A, ···A, y,"'Ym A, fi, ·· ·A, Y" "Ym}T (13)

{z} = (A, A., A, ···A, y,"'Ym A, Ã, ··.A, Y""Ym}T

In other words the method uses a modal superposi­tion technigue for the bridge components. The coupled system vectors {z} and {z} contain both physical and modal components. The physical components are the degrees of freedom of a vehicle modeled as lin­ear discrete mass-spring-damper systems. The modal components are the degrees of freedom of a linear beam model of the bridge.

The system in eg. 13 can be solved with the recursive matrix formula:

( Z (I + dI)} = [ Ad 1 { Z (I)} + [ B d 1 {g} (14)

where

3 A CALIBRATION PROCEDURE BASED ON SYSTEM IDENTIFICATION

The continuum beam model of the iron bridge was calibrated with a specific procedure based on testing and data processing of recorded signals. At first , the main geometrical characteristics of the beams were derived from the study ofthe design reports and ofthe original drawings. Then the visual inspections conduct to estimate the status of the structure and to locate the changes from the original designo

Successively, from tensile tests on specimens extracted from the structure the Young Modulus, the yielding stress, the ultimate strength and the maxi­mum elongation can be calculated. Furthermore the metallurgic tests allow the characterization ofthe com­position ofthe material and its behaviour (isotropic or orthotropic). In the case of metallic structure the per­centage of certain metal can evidence the sensitivity to corrosion, and the possible type cracks due to the unhomogeneity of material.

Dynamic testing is the most important part of the experimental phase. Forced vibration induced by a mechanical eccentric mass vibrator and ambient vibration induced by traffic, train or other source of excitation can be recorded . The data can be acguired using piezoelectric accelerometers located on the bridge deck. Each acguisition system consists of an accelerometer and an integral charge amplifier. The measured signals have to be processed by an anti­aliasing filter, digitilized with AID converter and recorded into a specific laptop PC for storage and for successive analyses. The dynamic testing and data processing allow the estimation of the modal prop­erties of the structure. In particular, the first natural freguency and damping ratio can be obtained. The first moda I freguencies are detected using the spec­trum analysis (Ewins 1986, Liung 1997, Huang 2001 , Juang 1988). At first, a de-noising process ofthe signal

1021

is performed using the Wavelet decomposition . The power spectral density (PSD) is evaluated with the Welch periodogram method and then used to assess the frequency domain response funct ion. [n the case offorced vibrations induced bya mechanical eccentric mass vibrator the frequency domain transfer function can be easi ly obtained from the power spectral den­sity (PSD) of the recorded acceleration. [n the case of vibrations induced by the train transit the frequency response function is obtained starting from the free vibrations. In both cases the peak values of these functions allow to identifY severa I moda I frequencies.

A time doma in technique can be used fo r the esti ­mation of the modal damping ratio, since frequency domain technique is not able to give re liable results. In particular, the value ofthe damping ratio ~ is char­acterized from the free vibration response ofthe bridge when loaded at its main natural frequency by the mechanical vibrator. In such a way the hi gher fre­quency contributions weakly affect the response, and the maximum values ofthe time-history response can be matched with a function of the type y(t) = e-~wl. The damping ratio ~ can be finally calculated mini­mizing the error in the least square sense. On the other side the dynamic tests can give useful information on the damage state ofthe structure, and they can evidence irregu larities and non-lineari ties of the structure, and reliable diagnostic wamings.

The identification and the c1assification of struc­tural non-linearities can be carried out with a technique which uses the Hilbert transform in the time domain (De Stefano 2002). In particular, the signal obtained summing together the recorded acceleration under the tra in transi t and its Hilbert transform can be repre­sented on the complex plane. If circular orbits are obtai ned, the behaviour of the structure can be con­sidered linear under the loading case examined. Other wise some linearity is detected.

4 APPLICATION OF THE IDENTIFICATION PROCEDURE TO LAPIO BRIDGE

The vehicle-bridge interaction model and the calibra­tion procedure were applied to the real case of the Lapio Bridge. Built in 1896 and still in service at 22 km on the Avell ino-Rocchetta-Foggia line of the Ttalian Railways, the bridge has three spans (the main central one with a length of96 m) and a weight of257093 daN (Figure 3).

4. 1 Design report and visual inspection

The design report dated 1896 gives severa I information about both the detail ing of the connections and the des ign approach. The examination of members and connections of the bridge evidenced some damages deriving from corrosion.

4.2 Material Testing

In order to assess the Young modulus some tensile tests on specimens extracted from the structure were car­ried out. The value obtained (E = 20590 I N/mm2) is comparable with the Young modulus of steel, although the metallurgic tests gave evidence of the insuffi ­cient homogeneity of the iron. The ultimate strength is 405 N/mm2 , the yie ld stress is 307 N/mm2 , the elongation Agt is 28.75%, the Poisson ratio is 0.26. The Carbon percentage is 0.11 % -:- 0.12%, and the Chrom ium is absent.

4.3 Dynamic tests and system identification

In figure 4 the transfer function modulus TF obtai ned under the mechanical vibrator at different loading fre­quencies is shown. The results give evidence that the maximum values of the TF modulus are obtained at the frequency of 2.64 Hz for the bridge. In f igure 5 the frequency response function of the free vibra­tion response (at each accelerometer location) after the tra in transit is shown. Also in this case the fist

Figure 3. The study case: Lapio bridge (1 896).

07

2 .64 Hz 0 .6

05

04 "-f-

03

0.2

0 .1

! + . O2 4 6 8 10 12

frequency [Hz]

Figure 4. Transfer Function (bridge excited by mechanical eccentri c mass vibrator).

1022

flexural frequency is 2.64 Hz. Furthermore other fre­quencies corresponding to high mode shapes have been detected.

In table I the natural frequencies obtained from the dynamic response under the mechanical vibrator (f5 )

and from the free vibration afterthe train transit (fT ) are shown. Furthermore, the frequencies obtained from an optimized three-dimensional FEM model ofthe bridge (Ferraioli et aI. 2003) are reported. In figure 6 the first three mode shapes of the FEM model are shown.

The value of the damping ratio ç is characterized from the free vibration response of the bridge when loaded at its main natural frequency by the mechanical eccentric mass vibrator.

In figure 7 the free vibration response is shown. In the same figure the fitting curve y (I) = e-~Wf match­ing the maximum values is plotted . The damping ratio

2

1.8

1.6

1.4

1.2 o Cf) O-

0.8

0.6

0.4

0.2

o

I 2.64 Hz aC.3 aC.4

- aC. 5 - - aC.6

- - aC. 7 - aC. 8 - - aC.9

o 2 4 6 8 10

frequency [Hz]

Figure 5. PSD function (bridge excited by train transit).

Table I. Frequencies and mode shapes

fs fT

Mode no. [Hz) [Hz)

I 1.26 2 2.64 2.64 3 2.92 4 3.03 5 6 4.88 7 5.66 8 9

10 11 12 13 14 15 16 17

7.13 8.10 9.23

fFEM [Hz]

1.28 2.64 3.10 3.42 4.53 4.99 5.24 5.46 5.96 6.77 6.86 6.92 6.92 6.93 7.36 8.23 8.76

Mode shape

Transversal bending Longitudinal bending Transversal bending Torsion Bending yz Transversal bending Local Local Local Local Local Local Local Local Transversal bending Longitudinal bending Longitudinal bending

TRIDIMENSIONAL FEM MODEL OF THE BRIDGE

MODE 1: f=1.28 Hz - TRANSVERSAL BENDING

MO DE 3: f=3 .10 Hz - TRANSVERSAL BENDING

Figure 6. First three mode shapes of the FEM mode!.

0 .06

0.04

í 0 .02

c .Q O l'í Q)

Qi

" -0.02 " '"

-0.04

-0.06 10 15 20 25 30 35 40

time [sI

Figure 7. Damping identification from free vibration response.

1023

O.OS

0.06

0.04

0.02

Z'

>=" O I

--{l.02

--{l .04

--{l.06

--{l.OS '--~-~-~-~-~~-~----' --{l .os --{l.06 --{l .04 --{l.02 o 0.02 0.04 0.06 O.OS

Re(Y(t))

Figure 8. Lineari ty test with the Hilbert transformo

ofthe first mode shape is calculated as the mean value on several recorded signals. The value obtained for the damping factor is l; = 0.020.

The last step of the identification procedure is the analysis of the linearity with the Hilbert transformo In figure 8 the orbits in the complex plane of the signal obtained using the Hilbert technique are shown. Circu­lar orbits prove the linear behaviour ofthe bridge under the train loading. As a consequence, a linear model of the bridge can be considered. The discussion on lin­earity detection is reported in the paper of Ferraioli et aI. 2003.

5 EVALUATION OF THE DYNAMIC MAGNIFICATION FACTOR

The calibration of the continuum beam model was obtained on the basis ofthe first longitudinal bending frequency, and considering the Young modulus of the material deriving from the tensile tests on the extracted specimen. For the Lapio Bridge the first experimental bending frequency is

(OI = 2.64 cyc/sec = 16.58 radlsec

From the theoretical expression of the first natural fre­quency given by eq. 5 the following equivalent inertia can be defined:

For the spring stiffness kv and the damping coefficient c of the vehicIe suspension the folIowing values were considered:

Locomotive ALN663

l ffiffi ffiffi

J 12.451 18.39 12.451

~~ ~~ 5875 5875 5875 5875

Figure 9. Geometric configuration oflocomotive ALN663.

+ m(x), EI(x)

L .. vt .. ..

Figure 10. Model A) fo r veh icle-bridge interaction.

After its calibration the vehicIe-beam model was used to evaluate the acceleration and displacement time­history responses both of bridge and of vehicIe. Fur­thermore, the dynamic amplification facto r under the train crossing as a function ofthe velocity was charac­terized. This is an important task in order to investigate the bridge resistance and stability to dynamic loads quickly increasing as an effect of the traffic rise.

The dynamic ampl if ication was characterized with a magnification facto r <I> defined as the ratio between the peak vertical displacement 8dyn at the mid-span of the bridge under the train dynamic loading and the ver­tical displacement Ostatic at the mid-span ofthe bridge under the train static loading.

The locomotive ALN663 which is in service on the Avell ino-Rocchetta-Foggia line was considered in the analysis.

This locomotive has a length of23.54 m and 4 axles each with a weight of 5875 daNo The spacing ofaxles is 2.45 m between the fi rst and the second, 20.84 m between the first and the last (figure 9). The dynamic interaction between bridge and vehicles was simu­lated with a continuum beam model for the bridge and with different models for the vehicle. In particular, the following models were considered: A) moving force (figure 10); B) moving mass (figure 11); C) sprung mass and unsprung mass connected by a spring (fig­ure 12); D) sprung mass and unsprung mass connected

1024

Mvfr--: ~~========~======!l~(X~)~, E~I=(X~)===~~

rX

by a spring and a dashpot (figure 13); E) two moving systems each modeled with sprung mass and unsprung mass connected by a spring and a dashpot (figure 14). The systems A), B), C) and D) can be derived from model E). As a consequence the equations of motion of all these systems can be obtained from eq.ll . In particular, the solution for the moving force model A) can be obtained by the close formula solution.

y L

vt .. Figure I I. Model B) for vehicle-bridge interaction.

f.1(X) , EI(x)

vt .. Figure 12. Model C) for vehicle-bridge interaction.

!l(X) , EI(x)

... vt .. Figure 13 . Model D) for vehicle-bridge interaction.

L

v i

LI

Figure 14. Model E) for vehicle-bridge interaction.

..

..

The dynamic magnification facto r was evaluated for the different interaction models and for velocity of locomotive ranging in [O; 300] kmJh with step 5 km/h. The results obtained were compared with the solu­tion obtained with the formula reported in Eurocode 1 and in the Italian Railways recommendation. In these codes, the following expressions for the magnification facto r - respectively in the case of good maintenance condition and in the case ofbad maintenance condition ofthe tine - are proposed:

rpreal = 1 + rp' + 0.5rp"

rpreal = 1 + rp' + rp"

where

, K rp = I-K +K 4

(16)

( 17)

(18)

rp" = ~[56e-( ~)' + 50( noL<I> - I Je-(~)' 1 (19) 100 80

In eqs. 18, 19 the coefficient k is defined by:

K= __ v_ 2·L·no

(20)

v is the velocity of the train in m/s; no is the first bending modal frequency of the bridge, L<I> is the characteristic length which depends from the type of bridge (Itatian Railways recommendation 1995), its constrains and its behaviour in the rail road plane.

The coefficient ex in eq . 19 is a function of the velocity as follows :

v$22m/s=>o.=~ v>22m/s=>o.=1 (21) 22

In figures 15 and 16 the dynamic magnification factor as a function ofthe velocity ofthe train is shown. The results obtained with the models A), B), C), D) and E) are compared with the results deriving from the appl ication of the code eqs. 16, 17. The magnification facto r curve obtained with the moving force model A) seems to have the same trend of the code curve. However this model neglects the vehicle-bridge inter­action, and underestimates the magnification factor. The curves obtained from the models C) and D) are almost coincident. This results derives from the low

1025

1.2

il 1.15 '" u.. c o ., '" u

1.1 "" 'i': C)

'" ::::; u 'E 1.05 '" c >-o

10

- A B C D code

100 150 200 Velocily [Km/h]

250 300

Figure 15 . Dynamic magnification factor as a function of velocity: comparison between simplified models A), B), C), D) and code formula.

1.2 ,----_--_ --_--_- -_--,

-§

~ 1 15 I n" DE I 2 l'l - code

~ 1 1 g>

::::; //",-- \ //

/ ,",/~

/ /.,. .-"' ~

u

.~ 1.05

S ---10~~~~-~1~OO=--~1~570-~2~O~O-~2~5~O~-~300

Velocity [km/hl

Figure 16. Dynamic magnification factor as a function of velocity: comparison between simpl ified models D) and E) and co de formula.

damping ratio ~ assumed for the vehicle suspension . On the other side, using the mode] with two moving systems (figure 16) generally produces magni fication factors ]ower than the model with one moving system.

In figure 17 the comparison in terms of dynamic magnification factor among the model A), the finite element model with one only moving force (FEM-I) and the code formula is carried out. The equivalent beam model with one moving force gives a good approximation of the response of the three-dimensi­onal FEM model without interaction. Furthermore the model A) generally overestimates the magnification factor. The curve obtained from the co de formula seems to be derived from the interpolation of the results obtained from model A).

In figure 18 the dynamic magnification factors obtained from the fin ite element model without inter­action with one moving force (FEM-I), and with four moving forces (FEM-4) are compared. In the case of force distributed on four axles the increase of peak

1.18

1.16

il 1.14 '" u.. § 1.12 ~ ~ 1.1 'i': g> 1.08

::::; .~ 106

~ 1.04 >-o

1.02

10~~~~-~1~00~-~1~5~0-~2~0~0-~2~5~O--3~00 Velocily [km/h]

Figure 17. Comparison between mode!. A), FEM-l and code formula.

il ro u..

1.2

§ 1.15 ., '" u ~ 1.1 C)

'" ::::; .1< 1.05 E '" c >­o

-code ,- , -, FEM,1 •.•• / - .- FEM,4 .

.' , ,/

L_=.-=;;:::-.-... ' ·",::: /v-----.~~ "'...".. ':

0.950 ----50----100---150--200 250

Velocily [km/h]

Figure 18. Comparison between FEM-I and FEM-4.

300

0.1 r------------------,

c: o

~

0.05

o

~ -0.05

~ - 0.1

\'r=ci l=...QJ

- 0.15 '---~-~---'=='---~-~-~----' O 0.5 1.5 2

Time [sec] 2.5 3 3.5

Figure 19. Comparison between model C) and model D) in terms ofvehicle acceleration response,

vertical displacement at the mid-span as an effect of dynamic amplification is less evident.

In figures 19 and 20 the comparison in terms of vehicle acceleration response is shown. The analysis

1026

0.08r-;------,----~-------===

~ É. c:: o

0.06

~ -002 11>

1j - 0.04 u « - 0.06

- 0.08 " ';

-0.10~----:--~--:::----=-3----:-4 ------:'.

Time [sec]

Figure 20. Comparison between model D) and model E) in terms of vehicle acceleration response.

was carried out considering the peak velocity of the train on lhe Avellino-Rocchetta-Foggia, that is v = 100 km/h. In the presence oflow damping ratio for the vehicle suspension the models C) and D) gives very close acceleration responses (figure 19). Furthermore, figure 20 shows that the number of moving systems modeling the vehicle greatly affects the acceleration response.

6 CONCLUSIONS

A procedure based on experimental and theoretical analyses for the study of existing iron railway bridges was described in detail and then applied to an iron railway bridge of the nineteenth century.

Both ambient vibration and force vibration tests were carried out in order to identify the modal fre­quencies and damping ratios. The Hilbert transform was applied to validate the linear elastic model used in the theoretical analyses.

The moda I properties deriving from the identifi­cation procedure were used to calibrate a continuum model ofthe bridge to be considered for the evaluation ofthe dynamic magnification facto r and ofthe accel­eration response of the vehicle. The magnification factors obtained with models that neglect the vehicle­bridge interaction are underestimated. These results evidence the importance ofusing vehicle-bridge inter­action models, even ifthey are approximated and only the bending behavior is considered.

The damping of the vehicle suspension does not affect the dynamic response of the bridge. On the contrary it plays a leading role in the acceleration of the vehicle mass. As a consequence, the damping of the suspension is an important parameter whenever the acceleration inside the vehicle should be account, in particular for the study of the comfort of the passenger in high speed lines.

ACKNOWLEDGEMENT

The authors gratefully acknowledge Df. A. D' Aniello, engineer director of the Ferrovie dello Stato Italiane, for his support to perform the tests on the bridge.

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