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ABSTRACT: In this communication two techniques for fast assessment of maximum envelopes are presented. The forward
problem is solved by a semianalytic solution, presented by one of the author and coworkers in 2006. The modal equations of
motion are solved in closed form in the time domain. This analytic treatment permits some analytical tools for fast evaluation of
maximum response. At the first explored option, the analytic train speed sensitivity is developed. The evolution of the maximum
values (displacements or accelerations) is carried out through analytic sensitivity. This causes less number sampled speeds,
giving an approximate hermitic cubic spline for the envelope. This technique avoids computing the solution at intermediate
speeds, which reduces the global computing times. The second technique explores the analytic transformation of the signal in its
quadrature form, by way of the Hilbert transform. Analytic Principal Cauchy Values are integrated, avoiding singular integrals
or regularization techniques. The amplitude signal is proposed as a transformed signal to sample. The main advantage comes
from the frequency contents of the transformed signal, which can be downsampled with large time steps. This causes time
reductions. Numerical examples are shown in which, particularly in the context of low damping rates, the reduction in
computing times dramatically decreases.
KEY WORDS: Bridge dynamics; Design envelopes; Hilbert transform, Traininduced vibrations.
1 INTRODUCTION
According to Eurocode [1], the design of a bridge for high
speed trains requires timedomain numerical assessment for
both Service and Ultimate Limit States. The evaluation is
carried out with timedomain solutions. Among the different
models, the linear solution based on modal superposition
covers the wider range of applicable cases. The train is
modelled as a tandem system with constant loads traversing
the structure at constant speed. The aim of the envelope
analysis is the detection of the maximum response (given in
terms of accelerations or displacements) at each passage
speed, for each train and postprocessing point selected. To
cover mass and frequency variations, in ballasted tracks, three
different mass (nominal, decreased and increased 30%) are
considered, which generates three times number of direct
evaluations.
The numerical evaluation of the envelopes is time
consuming, due to several factors: i) the number of modes
considered, up to 30 Hz, increased with the complexity and
number of degrees of freedom (from BernoulliEuler or
Timoschenko beam models, to plate shells or solid elements
at general 3D structures); ii) time steps are sensitive to modal
damping ratio, requiring less time step size when the damping
ratio decreases, as occurs in composite or steel bridges; iii)
speed steps decreases when damping ratio decreases, to
accurate assess the local maximum at envelope curves, iv) the
number of trains considered, in which the High Speed Model
Load (HSML), which cover 10 trains, is completed with other
train compositions; v) the number of postprocessing points
increases with the complexities of the spatial discretization.
To illustrate about the number of direct evaluations, the
dynamic analysis of a continuous 3 span bridge, under 12
trains (10 HSML, AVE and TALGO), speed interval running
from 20km/h to 420 km/h (1,2·Vmax) with speed step
km/h, would require the determination of 960 time
series and a posteriori maximum values detection at each post
processing point. This number increases as three ballast mass
hypotheses are considered. The number of direct simulations
rises to 2880. If low damping rate is considered, time step
would require km/h; the number of forward time
series evaluations rises to 14940.
The analytical integration of the timedomain solution,
proposed by MartínezCastro et al. [2] avoids the dependence
of timeintegration step. An important property is that results
in the time domain are exact, as the evaluation comes from an
stepped closedform analytical solution. The global computing
time dramatically decreases as a consequence of the sampling
time step, which is considerably 1 order of magnitude higher
than the one required for stable numerical integration. The
solution is approximated only in the spatial variables (modal
analysis). In this sense, the solution is considered to be semi
analytic. Generalization to plates and other spatial domains in
which the train traverses a class1 spline is carried out [3,4].
Unlike classical stepbystep methods, the semianalytical
solution gives an exact solution for the time domain through
an analytical expression. Therefore, all the applicable
mathematical treatment to analytical signals can be used.
Taking advantage of this condition, this research employs this
methodology in the definition of two approaches for fast
evaluation of maximum envelopes.
Two techniques for fast evaluation of design envelopes in highspeed train railway
bridges: Train speed sensitivity and the Hilbert Transform.
Alejandro E. MartínezCastro1, Enrique GarcíaMacías2,
1Department of Structural Mechanics and Hydraulic Engineering, ETS Ingenieros de Caminos, Canales y Puertos.
Av. Fuentenueva sn 18002. University of Granada, Granada, Spain.
21Department of Structural Mechanics and Hydraulic Engineering, ETS Ingenieros de Caminos, Canales y Puertos.
Av. Fuentenueva sn 18002. University of Granada, Granada, Spain.
email: amcastro@ugr.es, enriquegarma@correo.ugr.es
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June  2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 23119020; ISBN: 9789727521654
1309
At the first one, it is explored the direct analytic sensitivity
of the modal coordinate respect to train speed. For a fixed
speed, it is explored the way in which the local maximum
(accelerations or displacement) evolves in a small
neighbourhood. It is shown that the partial derivative of the
maximum response respect to the train speed permits the exact
evaluation of the envelope derivative respect to the train speed
At each sampled speed, both the maximum and its speed
sensitivity variation are obtained. This permits a local cubic
approximation of the maximum envelopes, to interpolate
maximum values by a simple cubic spline, avoiding time
series at intermediate speeds. Time reduction in this technique
is caused by decreasing the number of sampled speeds.
The second technique causes time reduction in computing
times coming from less number of steps required to sample
the maximum at each speed. The application of the analytic
Hilbert transform [5] is a common technique in signal
processing as a way to detect the analytic envelope of a signal.
The analytic envelope is a partial demodulated curve, which
contains only low frequency information. Sampling the
analytic envelope requires greater time step, typically in the
range 10 Hz (0.01 seconds) or less (depending on damping
ratio), for an original 30 Hz signal, which requires a time step
close to 0.003.
2. THE SEMIANALYTIC SOLUTION
In this section it is described the analytic timedomain
solution proposed in [2]. A local reference is
introduced, in which the origin is located at the first point of
the lane, stands for the location along the lane, and is the
vertical coordinate to describe displacements at the points of
the lane. A point load traverses the bridge moving at constant
speed as . The lane is divided in
elements, where local cubic interpolation (Hermite type) is
considered as the function basis to interpolate the solution. Let
us consider element ,
, with ! the
spatial length of element. The vertical displacement " is
expanded in the base of natural modes as
"
!
#
$
!
%
&
!
%
'
()*
%
+
,
(1)
with $% the nth timedependent modal amplitude, &%
the modal shape evaluated at point . Note that both functions
are locally evaluated at element e .The number of modes runs
from 1 to the maximum number considered. In general,
Eurocode [1] fixes this number as the corresponding mode
with 30 Hz. If a special discretization is considered at the
global spatial domain, function &%(x) represents the local
interpolation of the approximed field in a polynomial basis.
Particularly, when the local description is built from the four
Hermite cubic splines./ 01, Eq(1) can be written
as
"
!
#
$
!
%
#
2
!
%.
3
.
+
,

.
'
()*
%
+
,
( 2)
Coefficient matrix 2%.
! relates for element the spatial shape
functions with the modal coordinates. Such matrix can be
built once the generalized eigenvalues problem that defines
modes and natural frequencies at the 3D domain is solved.
Functions $%
! are closedform solved at each interval.
The solution is split into two terms: $%
! = $%
!45 $%
!6.
Considering the local time 7
, and omitting the
superscript , solutions for the homogeneous and particular
functions can be written as,
$
%
4
7
8
9
:
;
:
<
=
>
%
?@A
B
C%
7
5
D
%
AEF
B
C%
7
G
(3)
$
%
6
7
H
%
I
5
H
%
,
7
5
H
%
J
7
J
5
H
%
K
7
K
(4)
In Eq (3), BC% LM%J stands for the damped natural
angular frequency of nth mode; in Eq (4), the four
coefficients can be obtained in terms of 10 coefficients, non
dependent on the train speed , which can be evaluated and
store previous to the timedomain computation. Thus,
H
%
I
K
H
%
I,
5
J
H
%
IJ
5
H
%
IK
5
H
%
I3
H
%
,
J
H
%
,,
5
H
%
,J
5
H
%
,K
H
%
J
H
%
J,
5
H
%
JJ
H
%
K
H
%
K,
(5)
Expressions for these 10 coefficientes can be seen in [2]. In
Eq(3), constants >%D% are defined at each element with the
initial and interelement continuity conditions.
3 TRAIN SPEED SENSITIVITY FORMULATION
Direct derivation
3.1
The analytical solution represented by Eq (345) shows an
explicit dependence on the train speed . At the local solution
at element in Eq (1), the dependence on the speed can be
introduced, considering 7
"
!
#
$
!
%
&
!
%
'
()*
%
+
,
(6)
The sensitivity of the response respect to the train speed can
be obtained by the derivative evaluation,
N"
!
N
#
N$
!
%
N
&
!
%
'
()*
%
+
,
(7)
Relative time 7
; thus, the partial derivative can be
evaluated as,
N
$
%
!
N
N
$
%
!
7
N
5
!
J
N
$
%
!
7
N7
(8)
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
1310
Considering the decomposition given $%
!$%
!45
$%
!6, the sensitivity evaluation is split into two derivatives.
The first one can be written as (omitting superscript e),
N$
%
4
7
N
89:;:<
=
O
%
?@A
B
C%
7
5
P
%
AEF
B
C%
7
G
(9)
Parameters O%, P% are determined from >%D%, including
the initial conditions and its derivatives at each element,
depending on the speed v. Note that Eq.(9) has the same form
than Eq. (3). Thus, the way to store and compute the
derivative of the homogeneous term $%
!4 is analogous to
the way in which the forward solution is computed and stored.
The second term is derived as,
N$
%
6
7
N
Q
%
I
5
Q
%
,
7
5
Q
%
J
7
J
5
Q
%
K
7
K
(10)
Q
%
I
R
J
H
%
I,
5
S
H
%
IJ
5
H
%
IK
Q
%
,
R
H
%
,,
5
S
H
%
,J
5
H
%
,K
T
Q
%
J
R
H
%
J,
5
S
H
%
JJ
T
Q
%
K
R
H
%
K,
T
(11)
Note that Eq.(10) is similar to Eq.(4). It can be seen that Eq.
(11) has been written analogous to Eq. (5). The only
restriction is U V. Thus, the way to compute and store the
partial derivative of the particular solution term $%
!6 is
analogous to the one used in the direct solution. It only
requires storing the original 10 terms, Eq (5).
Evolution of the maximum response.
3.2
At each trainspeed, the maximum and minimum values for
the relevant parameters (displacements or accelerations) are
searched by sampling the time series.
Let be W a function depending on the speed
=XIX,G and time =I,G. Typically, function W
represents oscillating displacements and acceleration time
series. For a fixed speed , a set of local extreme
valueWY occurs at times Y. Only one of them is
considered as the maximum value during the time interval. At
any local extreme WY with associated time Y, the partial
derivative respect to speed parameter can be evaluated at
time Y, with an analytical formula including only information
at time Y. By virtue of the envelope theorem [6], it can be
proof that such partial derivative describes the evolution of the
local maximum value.
N
W
Y
7
N
ZE[
\
]
I
W
Y
5
^
Y
5
^
Y
W
Y
Y
^
(12)
Note that, despite the times in which the local extreme is
reached differs by ^Y when speed changes from to 5^,
the partial derivative is evaluated with only data at time Y and
speed .
Equation (12) is valid at any local maximum; in the context
of time solutions for highspeed trains, the global maximum at
time interval is always a relative maximum, for which partial
derivative respect to time is zero. The envelope curve is built
by computing the maximum value for each train speed.
Particularization of Eq (12) at the time Yof the global
maximum at speed allows the evaluation of the slope of the
envelope curve. Note that, despite the evaluation of the
forward time serie to search the maximum value requires
sampling at the time interval, the evaluation of Eq. (12) only
requires the evaluation of the partial derivative at only time Y.
Note that the time domain solution is written in closedform.
The partial derivative respect to train speed is also a closed
form expression, see Eqs. (9) and (10). Thus, the semi
analytic formulation allows a fast evaluation of the maximum
response slope, just by evaluating at fixed time Y_ At this
time, the position of the train is defined and the analytic
derivation is computed.
At each speed , the proposed formulation permits
computing the pair WY`aY
` . Thus, information about the local
evolution of the maximum value is obtained. The usual way to
compute the envelope curve requires a fine enough sampling
speed interval, to catch the shape of the envelope curve; at
each speed, a time series are checked (one serie for each event
at each postprocessing point) to search the maximum value.
The envelope is evaluated by linear interpolation between
sampled points. The new approach replaces the linear
interpolation by cubic spline, built from the slope information.
This allows a greater speed interval of evaluation, avoiding
time series to be sampled at intermediate speeds.
4 SIGNAL ENVELOPE BY HILBERT TRANSFORM
Introduction.
4.1
This second technique explores the time reduction which
comes from a transformed signal, less oscillating than the
original one. This causes less time required for sampling the
maximum value at each fixed speed.
Given a time dependent signal , the quadrature signal
(or analytic signal)b is a twodimensional signal whose
value at some instant in time is specified by two parts, a real
part and an imaginary part [7].
b
5
/
c
(13)
Function c can be obtained from by different
techniques. The Hilbert transform is the usual way to obtain it.
The transformed signal can be described by complex variable
concepts: the instantaneous amplitude > and the
instantaneous phase d
b
>
.ef
(14)
The instantaneous amplitude > is also interpreted as the
analytic envelope of the signal. In this sense, and in the
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
1311
context of envelope computation, sampling the original signal
to obtain the maximum response is equivalent to sampling the
analytic envelope and finding its maximum value. The
analytic envelope contains information about the amplitude of
the signal, neglecting the oscillations due to phase change.
The analytic solution provided by the semianalytic permits
the analytic treatment required to obtain the complexvalued
signal, and from it, the amplitude signal. In what follows, the
Hilbert transform and the analytic signal is presented, for the
semianalytic solution.
Hilbert Transform of the semianalytic solution.
4.2
The Hilbert’s transformation c of a function is
defined by the convolution between the Hilbert’s transformer
,
ghf and the functionW.
i
j
k
c
l
h
m
n
n
o
8
o
pn
(15)
Due to singularity at n, the integral has to be considered
as a Cauchy’s principal value. The HT of the realvalued
function extending from q to 5q is another real
valued function defined by Eq. (15). Physically, the HT is an
equivalent to a special kind of a linear filter where all the
amplitudes of the spectral components are left unchanged, but
their phases shifted bylTS.
The Hilbert Transform of the semianalytic solution is
carried out by analytic integration represented in Eq. (15).
Note that the Cauchy Principal Value is solved in closed form
by direct integral evaluation, without cumbersome
transformations required to avoid singularities in numerical
integration.
To obtain the analytic signal, the direct method consists on
the integration represented by Eq. (15). Alternatively, when
the signal can be split as the product of a low varying signal
times a fast varying signal, the Hilbert transform can be
computed as the product of the slow varying signal times the
Hilbert transform of the fast varying signal, by virtue of the
Bedrosian Identity [5].
The Hilbert transform of the homogeneous signal Eq. (3)
can be integrated as:
$
c
%
4
7
S
h
/
h
h
B
O
r
h
B
r
h
M
r
h
>
r
5
/
h
D
r
hP/s/ h 7hjBOr/h BrhMr
k
t
>
r
/
h
D
r
h
Sh/h7hBOr
h
P/
s
/
h
7
B
O
r
5
/
h
B
r
h
M
r
t
(16)
$
c
%
6
7
h
u
ZE[
]
q
$
c
%
4
7
ZE[
]
q
$
c
%
4
7
(17)
With P/v the exponential integral function, defined as:
P/
v
m
8
f
o
8
w
p
(18)
The evaluation represented by Eq. (16) is computationally
inefficient. Alternatively, the evaluation can be carried out in
real variable by considering the form of Eq. (3) as the product
of a slow varying signal times a fast varying function. Thus,
function P/v is replaced by functions x/v and y/v, with
x
/
v
m
AEF
z
I
p
(18)
y
/
v
m
?@A
z
I
p
5
Z@{
v
5

(19)
With  the Euler gamma constant.
The Hilbert transform of the particular solution term can be
evaluated in closedform. From Eq. (4), the Hilbert Transform
can be written as:
$
c
%
6
7
h
l
ZE[
6
]
o
$
}
r
7
ZE[
6
]
8
o
$
}
r
7
(20)
With:
$
c
%
6
7
~
7
h•~Hr
5jRHr
S5 R7
5Hr
RSS575 7Sk€ Hr
V
57jHr
S5Hr
R7k 0•‚ 7‚
(21)
One of the most important properties of Hilbert transform is
that only the local characteristics of the signal a fixed time are
required to define the analytic envelope. In the context of the
timedomain solution for high speed trains, this means that the
integral represented by Eq. (15) can be integrated in a reduced
interval around the time of evaluation. To obtain the envelope
signal, the contribution of the loads close to a certain time
interval, related with the distance between loads, can be
reduce the integration to a local integral in a finite interval
(window filter).
$
%
ƒ
„
l
h
m
n
n
f
…
†
T
8
f
8
†
T
pn
(22)
This integral provides a faster evaluation of the Hilbert
transform.
Once the Hilbert transform is stored for a single load, the
solution for a set of moving loads at constant train speed can
be fast evaluated. The analytic envelope is computed as the
module of complex signal. Numerical tests confirm that the
sampling frequency reduces fom 300 Hz to 50 Hz or less. This
causes global time reductions to obtain the design envelopes.
Figure 1 show the acceleration time series and the analytic
envelope, computed from both complete or shifted integral. It
is shown that window filter is a powerful method to compute
the analytic envelope.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
1312
Figure 1: Window filter effect in Hilbert transform
6 NUMERICAL TEST: THE SESIA VIADUCT.
In this section both techniques are tested in the context of a
composite steelconcrete bridge for high speed trains. This
bridge is analysed in the context of Operational Modal
Analysis and fatigue expected life in recent references (see
[8], [9] and [10]. This case is particularly well suited to show
why fast techniques are required. The spatial solution is
sensitive to threedimensional configuration. The damping
rate considered is low (M V_‡ constant damping rate). The
material and spatial configuration causes an elevated number
of time series requires for accurate evaluation of envelopes.
The Sesia viaduct is situated on the Italian high speed
railway line between Torino and Milano. It is a steelconcrete
composite bridge composed of seven spans, 46m long simply
supported each, resulting in a total length 322m. The steel box
girder is divided into two segments C1 and C2, which are
connected by full penetration butt welding. At each span, the
structural steel includes 15 cross diaphragms at interval
3.11m, which support the necessary lateral stiffness to limit
the distortion of the steel box girder. The twinbox steel
girder, with an overall width of 10.2m and a depth of 3.35m,
is covered by a 13.6m wide and 0.4m thick concrete slab,
which is connected to the top flanges of the steel box girder
by shear studs. Ballasted double tracks with UIC60 rails are
supported at every 0.6m distance by prestressed concrete
sleepers.
Figure 2: Figure 4: Sesia Viaduct FEM model
The results are compared in terms of maximum response
envelope. In the context of the first technique, the approximed
acceleration envelope is tested for different speed increments
and time steps for evaluation.
Figure 3 show the results obtained with speed intervals 10
and 20 km/h., in comparison with the envelope obtained by
SemiAnalytic methodology and a velocity step of 1 km/h. It
is observed that, at the resonance speed, the less relative error
is reported. It is due to the modal contribution which is
defined by the square of the natural frequency in the
derivative of accelerations, thus, at resonant velocities the
contribution is focused on a very few modes of vibration and
the errors are quite small.
Figure 4 shows the sensitivity of the approximed envelope
to time step used to evaluate the slope. Note that Equation
(12) requires that partial derivative respect to time must be
zero at the maximum value. When sampling at fixed time
stems, this requirement is violated, causing error in the
Figure 3: Approximed vs exact envelope evaluati
on with
speed sensitivity formulation. Sesia viaduct.
Figure 4: Evaluation time step effect on the approximed
envelope. Sesia viaduct.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
1313
derivative evaluation. Note that, at speeds different to the
resonance, more relative errors are reported. It is also due to
the damping assumption (constant rate 0.5%). Different
damping assumption, as Raileigh model, provides better
results. Nonetheless, the observed error at resonance speed is
less than 1%, which is the relevant value to be obtained from
envelope curves.
In Figure 5, the envelope is assessed by the analytic
envelope proposed by the Hilbert transform. Different time
steps are tested. It is observed that for coarse sampling times,
local errors are obtained. That is why the curve for 20km/h
registers peaks under the real curve. Nevertheless, the fitting
of the proposed curves with the real one improves at resonant
points.
Figure 5: Envelope tested with Hilbert transform. Sesia
viaduct.
7 CONCLUDING REMARKS
Two new techniques for fast assessment of maximum
envelopes in dynamic response of bridges under the passage
of moving loads were presented. Both solutions explore the
basic analytic definition of the timedomain solution involved
in the semianalytic solution.
At the first one, a metamodel based on train speed
sensitivity was developed. The derivative of response with
velocity was analytically defined in order to define a cubic
spline as the approximed envelope. At the second one, the
time series of response are replaced by the analytic envelope,
computed from the Hilbert’s transformation. The singularity
caused by Cauchy Principal value are solved by direct analytic
integration. Sampling at the analytic envelope requires less
sampling period, which implies less computing times to find
the maximum value. Numerical tests were used to validate
this two proposed techniques obtaining good results in all of
them. However, it has been exposed the limitations in both of
them: the train speed sensitivity metamodel is sensitive to
time stepping because of the accurate determination of the
time in which the maximum value appears. At the contrary,
the Hilbert Transform metamodel permits higher time steps.
ACKNOWLEDGMENTS
This work is part of the project TEP5066, ‘Monitorización
structural predictive en puentes ferroviarios de alta velocidad’,
Junta de Andalucía (Spain). The financial support is gratefully
acknowledged.
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1314