Dynamic analysis of a bridge structure exposed to high

Appendix A
Paper I
Paper I
79
Numerical modelling of dynamic response of high-speed
railway bridges considering vehicle–structure and
structure–soil–structure interaction
Paulius Bucinskas1 , Liuba Agapii1 , Jonas Sneideris1 , and
Lars Vabbersgaard Andersen2
1
Master student
professor
2 Associate
Department of Civil Engineering, Aalborg University, Denmark
Abstract
The aim of this paper is dynamic analysis of a multi-support bridge structure exposed
to high-speed railway traffic. The proposed computational model has a unified approach for simultaneously accounting for the bridge structure response, soil response
and forces induced by the vehicle. The bridge structure is modelled in three dimensions based on the finite element method using two-noded three-dimensional beam
elements. The track structure is composed of three layers: rail, sleepers and deck
which are connected through spring–dashpot systems. The vehicle travelling along a
bridge is idealized as a multi-degree-of-freedom system, modelled with two layers of
spring–dashpot suspension systems. Coupling the vehicle system and railway track is
realized through interaction forces between the wheels and the rail, where the irregularities of the track are implemented as a random stationary stochastic process. The
soil body is considered as a layered half-space employing the transfer-matrix method
to obtain the Green’s functions in frequency domain. An iteration procedure is proposed for simultaneously solving the time-domain solution of the multi-degree-offreedom vehicle affected by the bridge displacements and track unevenness together
with the frequency-domain solution of the bridge structure coupled with the subsoil.
The effects caused by different soil properties and stratification, as well as different
vehicle speeds are determined and compared.
Keywords: High-speed railways, railway bridge, multi-degree-of-freedom vehicle,
wheel–rail interaction, track unevenness, soil-structure interaction.
1
Introduction
Railway bridges are exposed to significant dynamic effects induced by passing trains.
The interest in dynamic behaviour of railway bridges has increased in recent years
1
due to development of high-speed railways in various countries. An example of this is
the long-term green plan for transportation in Denmark, part of which is the so-called
“Hour Model”. The aim is to decrease the travel time between the four major cities
of Denmark to one hour, which includes upgrading and constructing railway lines capable of handling train traffic speeds up to 250km/h. The dynamic effects caused by
increased traffic speed are higher and difficult to predict. Thus, more advanced numerical calculation methods have to be developed in order to properly evaluate the complexity of the structural response. The aim of this paper is to analyse a multi-support
high-speed railway bridge structure passed by a vehicle as illustrated in Figure 1. It
proposes a numerical approach to solving the coupled interaction of bridge and vehicle as well as bridge and subsoil simultaneously. A computational model including
a three-dimensional (3-D) bridge structure, soil coupled with bridge foundations, and
non-linear wheel–rail interaction emerges.
Most of the earlier analytical studies of structural dynamic behaviour are based on
Euler-Bernoulli beam theory. Meirovitch [1] and Fryba [2] proposed closed-form solutions of the dynamic problem with a simply supported beam subjected to a moving
constant force. Simple beam models based on finite-element formulation were used
by Thompson [3], Ripke and Knothe [4] and Gry [5] to investigate vertical and lateral
vibration at high frequencies. Later, Wu and Thompson implied double Timoshenko
beam [6] and continuously supported multiple-beam models [7] for dynamic analysis
of the railway track. Heckl [8] included in his studies non-linear properties of the pads
and ballast. A physical system with a single wheel and rail coupling was proposed by
Remington [9] and Grassie et al. [10]. At recent days, sophisticated models are employed to analyse vehicle-structure interaction. Lei and Noda [11], Uzzal et al. [12],
Zhai and Cai [13] used a multi-degree-of-freedom (MDOF) vehicle running on the
track modelled as three layers of rail, sleeper and ballast. The vertical dynamic interaction between wheel and rail was described by Hertzian non-linear elastic contact
theory.
Figure 1: Multi-support high-speed railway bridge subjected to a passing vehicle
2
The studies in the ground vibration field are usually done by using computational
models based on the finite element method (FEM), or alternatively, Green’s function
sometimes combined with the boundary-element method (BEM). A simple case of
structure–soil interaction was investigated by Metrikine and Popp [14]. They studied ground vibration by using a model based on the FEM in which the vehicle was
implemented as a two-degrees-of-freedom system coupled with a beam of infinite
length representing the track and a homogeneous half-space model representing the
subsoil. The same scenario was recreated by Lombaert [15] using a BEM model.
Sheng [16, 17] studied the ground vibration generated by a harmonic load that moves
along a railway track on a layered half-space. Andersen and Clausen [18] investigated a dynamic soil–structure interaction, more particularly the problem of a rigid
foundation on a layered subsoil.
Section 2 presents the FEM formulation used to model the bridge structure and the
MDOF vehicle. Dynamic modelling of the soil is described at Section 3 and Section 4
shows the solution procedure which involves two iteration schemes. Validation of the
computational model and the obtained results are given in Section 5 and Section 6,
respectively. Finally, conclusions are given in Section 7.
2
Finite element method in the computational model
In this paper FEM is used to model the bridge structure together with the vehicle
traversing it. The vehicle interacts with the bridge through interaction forces Pi , while
each bridge column is coupled with the soil through a connecting node. The governing equations for the track structure and pylons are discretized, using the standard
Galerkin approach with cubic interpolation functions for transverse displacements and
linear interpolation for longitudinal displacement and torsion. Damping is introduced
as Rayleigh damping. In the current section, dynamic modelling of the vehicle is
presented in Subsection 2.1. Further, dynamic modelling of the track is described in
Subsection 2.2, the formulation for the bridge supports and foundations is given in
Subsection 2.3, the coupling between the structure and the soil is explained in Subsection 2.4. Subsections 2.5 and 2.6 present wheel–rail interaction forces and vertical
track profile, respectively.
2.1
Dynamic modelling of the vehicle
In this paper, the effects of a single passing locomotive are analysed. The locomotive is
simplified into a two-dimensional ten-degrees-of-freedom system, which involves two
layers of spring–dashpot suspension systems in which vertical translation and pitch
rotation of the bogie and vehicle body are included. The interaction forces between
the wheel and the rail are accounted for and denoted as Pi . The whole system is shown
in Figure 2.
3
Vehicle body
Mc Jc
K su,2
wt,2
t,2
K su,1
Wheel 4
z
x
C su,2
K su,2
Bogie 2
C su,1 K su,1
C su,1
Mw,4
Mw,3
Mt,2 Jt,2
K su,1
Wheel 2
C su,2
KH
P4
P3
2L1
c
wt,1
Mt,1 Jt,1
Bogie 1
C su,1 K su,1
C su,1
Mw,2
Mw,1
Wheel 3
KH
wc
Moving direction
t,1
ww,i
Wheel 1
r
KH
KH
P2
P1
2L 2
2L1
Track
unevenness
Figure 2: Multi-degree-of-freedom vehicle
The locomotive is considered as a rigid body with mass Mc and pitch moment of
inertia Jc . It is connected with two bogies through spring–dashpot systems. Mass and
pitch moments of inertia of the bogies are Mt,j and Jt,j , respectively, where j = 1, 2.
Each bogie is connected to two wheels using similar spring–dashpot systems and the
wheels are described by a wheel mass Mw,i , i = 1, 2, 3, 4. The translations of the
locomotive and bogies are defined by vertical displacements wc and wt,j , respectively,
while pitch rotation is described by angles ϕc and ϕt,j . The vertical displacements of
four wheels are denoted as ww,i . The vehicle is described by ten equations of motion.
Vertical displacement of the car body is affected by the secondary suspension system
which has stiffness Ksu,2 and damping Csu,2 :
dwc (t)
dt
dwt,1 (t)
dwt,2 (t)
d2 wc (t)
− Csu,2
− Csu,2
− Mc
= 0. (1)
dt
dt
dt2
2Ksu,2 wc (t) − Ksu,2 wt,1 (t) − Ksu,2 wt,2 (t) + 2Csu,2
The rotation of the car body is similarly affected by the secondary suspension system,
where L2 is half the length between the bogie centre lines:
dϕc (t)
dt
dwt,1 (t)
dwt,2 (t)
d2 ϕc (t)
− L2 Csu,2
+ L2 Csu,2
− Jc
= 0. (2)
dt
dt
dt2
2L22 Ksu,2 ϕc (t) − L2 Ksu,2 wt,1 (t) + L2 Ksu,2 wt,2 (t) + 2L22 Csu,2
The vertical displacements for bogie one and bogie two are influenced by the primary
suspension system with parameters Ksu,1 and Csu,1 and the secondary suspension system:
4
(2Ksu,1 + Ksu,2 )wt,1 (t) − Ksu,1 ww,1 (t) − Ksu,1 ww,2 (t) + (2Csu,1 + Csu,2 )
− Csu,1
dww,1 (t)
dww,2 (t)
d2 wt,1 (t)
− Csu,1
− Mt,1
= 0, (3a)
dt
dt
dt2
(2Ksu,1 + Ksu,2 )wt,2 (t) − Ksu,1 ww,3 (t) − Ksu,1 ww,4 (t) + (2Csu,1 + Csu,2 )
− Csu,1
dwt,1 (t)
dt
dwt,2 (t)
dt
dww,3 (t)
dww,4 (t)
d2 wt,1 (t)
− Csu,1
− Mt,2
= 0. (3b)
dt
dt
dt2
The rotations for bogie one and bogie two are affected by primary and secondary
suspensions, where L1 is half the length between two wheel axles of one bogie:
dϕt,1 (t)
dt
2
dww,1 (t)
dww,2 (t)
d ϕt,1 (t)
− L1 Csu,1
+ L1 Csu,1
− Jt,1
= 0, (4a)
dt
dt
dt2
2L21 Ksu,1 ϕt,1 (t) − L1 Ksu,1 ww,1 (t) + L1 Ksu,1 ww,2 (t) + 2L21 Csu,1
dϕt,2 (t)
dt
2
dww,3 (t)
dww,4 (t)
d ϕt,2 (t)
− L1 Csu,1
+ L1 Csu,1
− Jt,2
= 0. (4b)
dt
dt
dt2
2L21 Ksu,1 ϕt,2 (t) − L1 Ksu,1 ww,3 (t) + L1 Ksu,1 ww,4 (t) + 2L21 Csu,1
Vertical displacements of wheels one, two, three and four are influenced by the primary suspension. Each wheel is subjected to one quarter of the total force Ftotal =
Mtotal · g, where Mtotal is the total mass of a vehicle and g is a gravitational acceleration
constant:
dww,1 (t)
dt
2
dwt,1 (t)
d ww,1 (t)
1
− Csu,1
− L1 Csu,1 ϕt,1 (t) − Mw,1
= Ftotal + P1 (t), (5a)
2
dt
dt
4
Ksu,1 ww,1 (t) − Ksu,1 wt,1 (t) − L1 Ksu,1 ϕt,1 (t) + Csu,1
dww,2 (t)
dt
2
dwt,1 (t)
d ww,2 (t)
1
− Csu,1
+ L1 Csu,1 ϕt,1 (t) − Mw,2
= Ftotal + P2 (t), (5b)
2
dt
dt
4
Ksu,1 ww,2 (t) − Ksu,1 wt,1 (t) + L1 Ksu,1 ϕt,1 (t) + Csu,1
dww,3 (t)
dt
dwt,2 (t)
d2 ww,3 (t)
1
− Csu,1
− L1 Csu,1 ϕt,2 (t) − Mw,3
= Ftotal + P3 (t), (5c)
2
dt
dt
4
Ksu,1 ww,3 (t) − Ksu,1 wt,2 (t) − L1 Ksu,1 ϕt,2 (t) + Csu,1
5
dvw,4 (t)
dt
2
dwt,2 (t)
d ww,4 (t)
1
− Csu,1
+ L1 Csu,1 ϕt,2 (t) − Mw,4
= Ftotal + P4 (t). (5d)
2
dt
dt
4
Ksu,1 ww,4 (t) − Ksu,1 wt,2 (t) + L1 Ksu,1 ϕt,2 (t) + Csu,1
Applying gravity on the vehicle to the wheels results in a model where the vehicle
motion is considered in a reference state in which gravity is already applied. However,
the gravity has to be included to account for the travelling mass effect regarding the
bridge and soil response. The weight of the vehicle is uniform distributed between the
four wheels, sharing the same part of the load.
2.2
Dynamic modelling of the track
The analysed bridge cross-section is shown in Figure 3. For the computational model
it is simplified so that only one railway line is modelled with rails, sleepers and ballast,
while the other line is accounted for by point masses Mot offset from the deck centre
line by the distance Wd . The two rails on the modelled line are taken as one beam
with double the bending stiffness Er Ir,y of one rail. The simplified system used for the
computational model is shown in Figure 4.
Rail
Sleeper
Ballast
Wd
Wd
Deck
z
y
Figure 3: Bridge cross-section
The rails are modelled as a discretely supported Euler-Bernoulli beam. The beam is
modelled in two dimensions (2D), more particularly in the x,z-plane, as the forces Pi
from the vehicle only act in the z direction. A number Nw of forces Pi act at distances
xw,i . Thus, the rail is described by: mass density ρr , cross-sectional area Ar , Young’s
modulus Er , moment of inertia around the y-axis Ir,y , and vertical displacement wr .
The rails are connected to Nsl sleepers at distances xsl,j through a spring–dashpot systems, with stiffness and damping parameters Kpd,j and Cpd,j , where j = 1, 2, ..., Nsl ,
to model the rail pads.
6
x,z-plane
x w,1
x w,2
x w,i
Pi
K pd,1
wsl,1
P2
Cpd,1
K pd,2
wsl,2
Cpd,2
Mst,1
z
Kb,1
C b,1
xsl,1
0.5M b,1
xsl,2
Rail
wr
P1
K pd,j
wsl,j
C pd,j
Sleeper
Mst,2
Kb,2
Mst,j
C b,2
wd
ud
Kb,j
Deck
C b,j
0.5M b,2
x
0.5M b,j
d
xsl,j
d
d
d
d,y
d,z
d,0
d
y,z-plane
wr
Rail
K pd,j
C pd,j
Sleeper
wsl,j
Mst,j
z
vd
y
wd
Kb,j
C b,j
d
Deck
0.5M b,j
Mot,j
Wd
Wd
Figure 4: Simplified railway track structure
The effects from concentrated forces and sleepers are accounted for by the Dirac
delta function δ(x),thus the equation of motion for the rail becomes:
N
sl
X
∂ 2 wr (x, t)
∂ 4 wr (x, t)
ρr Ar
+ Er Ir,y
+
δ(x − xsl,j ) Kpd,j (wr (x, t) − wsl,j (t))
∂t2
∂x4
j=1
!
Nsl
Nw
X
X
∂wr (x, t) dwsl,j (t)
+
δ(x − xsl,j ) Cpd,j
−
=
δ(x − xw,i (t)) Pi (t), (6)
∂t
dt
j=1
i=1
Sleepers have a vertical displacement wsl,j and are connected to the bridge deck
through another spring–dashpot system which models the stiffness Kb,j and damping
Cb,j of the ballast. It is assumed that half the ballast mass Mb,j moves together with
7
the sleeper mass Msl,j , thus the total mass moving with the sleeper becomes Mst,j =
0.5Mb,j + Msl,j . The equation of motion for the jth sleeper:
(Kpd,j +Kb,j )wsl,j (t) − Kpd,j wr (xsl,j , t) − Kb,j wd (xsl,j , t) + (Cpd,j +Cb,j )
− Cpd,j
dwsl,j (t)
dt
∂wd (xsl,j , t)
d2 wsl,j (t)
∂wr (xsl,j , t)
− Cb,j
+ Mst,j
= 0. (7)
∂t
∂t
dt2
In this paper, a bridge deck that has two railway lines is analysed. This means that
the railway track is at a certain distance Wd from the deck centre line. To model this
accurately, 3-D beam elements are used for the deck, which is described by the following parameters: mass density ρd , cross-sectional area Ad , Young’s modulus Ed , shear
modulus Gd , moment of inertia around the y-axis Id,y , moment of inertia around the
z-axis Id,z , polar moment of inertia Id,0 and torsional constant Td . The displacements
in the x, y and z directions are denoted as ud , vd and wd , respectively, and the rotation
around the x-axis is denoted θd . It is assumed that bending and axial deformations are
uncoupled, while the torsion is coupled to the bending in the x,z-plane through the
track structure. The deck bending in the x,z-plane is modelled as an Euler-Bernoulli
beam influenced by the Nsl of sleepers. Half of the ballast mass Mb,j moves with the
deck as well the mass from the other track Mot,j :
N
ρd Ad
sl
∂ 4 wd (x, t) X
∂ 2 wd (x, t)
∂ 2 wd (x, t)
+
E
I
+
δ(x
−
x
)(0.5M
+
M
)
d d,y
sl,j
b,j
ot,j
∂t2
∂x4
∂t2
j=1
+
Nsl
X
δ(x − xsl,j ) Kb,j wd (x, t) − wsl,j (t) − Wd θd (x, t)
j=1
+
Nsl
X
δ(x − xsl,j ) Cb,j
j=1
dwsl,j (t)
∂θd (x, t)
∂wd (x, t)
−
+ Wd
∂t
dt
∂t
!
= 0.
(8)
The rotations θd around the x-axis are described by Saint-Venant torsion, again influenced by Nsl sleepers and extra masses Mb,j and Mot,j :
Nsl
2
∂ 2 θd (x, t)
∂ 2 θd (x, t) X
2 ∂ θd (x, t)
Gd Td
−
ρ
I
+
δ(x
−
x
)(0.5M
+M
)W
d
d,0
sl,j
b,j
ot,j
d
∂x2
∂t2
∂t2
j=1
+
Nsl
X
δ(x − xsl,j ) Kb,j Wd wd (x, t) − Wd wsl,j (t) +
Wd2
θd (x, t)
j=1
+
Nsl
X
j=1
δ(x − xsl,j ) Cb,j
∂wd (x, t)
∂θd (x, t)
dwsl,j (t)
Wd
− Wd
+ Wd2
∂t
dt
∂t
!
= 0.
(9)
8
For the axial deformation ud of the deck, double mass Mot,j is considered to account
for two tracks:
Nsl
X
∂ 2 ud (x, t)
∂ 2 ud (x, t)
∂ 2 ud (x, t)
Ed Ad
−
ρ
A
+
δ(x
−
x
)2M
= 0. (10)
d d
sl,j
ot,j
2
∂x2
∂t2
∂t
j=1
The equation of motion for deck bending in the x,y-plane is based on Euler-Bernoulli
beam theory, with extra mass as in the previous equation:
Nsl
X
∂ 2 vd (x, t)
∂ 2 vd (x, t)
∂ 4 vd (x, t)
+
ρ
A
+
δ(x
−
x
)2M
= 0.
Ed Id,z
d
d
sl,j
ot,j
∂x4
∂t2
∂t2
j=1
2.3
(11)
Bridge supports and foundations
Pylons are modelled as 3-D two-noded beam elements, illustrated in Figure 5. The
pylons are described with the following parameters: mass density ρp , cross-sectional
area Ap , Young’s modulus Ep , shear modulus Gp , moment of inertia around the x-axis
Ip,x , moment of inertia around the y-axis Ip,y , polar moment of inertia Ip,0 and torsional
constant Tp . The displacements in the x, y and z directions are denoted as up , vp and
wp , respectively, and the rotation around the z-axis is denoted ψp . The bending in the
y,z and x,z-planes, axial deformation in the z direction and torsion around the z axis
are uncoupled, assuming small displacements. The equation of motion for bending in
the y,z-plane is:
∂ 4 vp (z, t)
∂ 2 vp (z, t)
+
E
I
= 0,
(12)
ρp A p
p
p,x
∂t2
∂z 4
and bending in the x,z-plane is governed by the equation:
∂ 2 up (z, t)
∂ 4 up (z, t)
ρp Ap
+ Ep Ip,y
= 0.
∂t2
∂z 4
(13)
Axial deformations in the z direction follows from:
Ep Ap
∂ 2 wp (z, t)
∂ 2 wp (z, t)
−
ρ
A
= 0,
p p
∂z 2
∂t2
(14)
and torsion around the z-axis is defined by the equation:
Gp Tp
∂ 2 ψp (z, t)
∂ 2 ψp (z, t)
−
ρ
I
= 0.
p p,0
∂z 2
∂t2
(15)
The foundations are assumed to be rigid, with dimensions Lf and Wf . Foundation
mass Mf , moments of inertia for roll Jf,x , pitch Jf,y and yaw Jf,z , are added to the node
connecting the structure and the underlying soil.
9
z
wp,1
wp,2
y
p,1
x
vp,1
Bridge support
p,2
up,1
vp,2
p
p
p
Mf , Jf,x , Jf,y , Jf,z
p
p,x
p,y
p,0
up,2
p
Soil surface
Wf
Lf
Rigid foundation
Connecting node
Figure 5: Pylons and foundations lying on the soil surface
2.4
Coupling between structure and soil
In order to couple the structure and soil, firstly structural stiffness [KB ], structural
damping [CB ] and structural mass [MB ] are converted to a complex impedance matrix
[ZB ] which has a different value for every circular frequency ω:
[ZB ] = −ω 2 [MB ] + iω[CB ] + [KB ].
(16)
√
where i is the imaginary unit, i = −1. The complex impedance matrix for the
footings [ZF ], is added to the structure at connecting nodes:
[ZB,11 ]
[ZB,12 ]
[Z] =
,
(17)
[ZB,21 ] [ZB,22 ] + [ZF ]
where [ZB,11 ] is the impedance for the degrees of freedom of the structure not connected to the soil, while [ZB,22 ] is for the degrees of freedom shared between the
soil and the structure. Further, symmetry of the FE system matrices implies that
[ZB,12 ] = [ZB,21 ]T .
2.5
Wheel–rail interaction force
The interaction between a wheel and the rail is modelled through the interaction force
Pi (t). If the wheel is on the track surface, the force is modelled as a non-linear
Hertzian force, and if the wheel lifts off the track, the force disappears:
(
3
KH | wrel,i | 2 for wrel,i ≤ 0
(18)
Pi =
0
for wrel,i > 0
where wrel,i = ww,i − wr,i − r(xw,i ). Here ww,i is the ith wheel traverse displacement,
wr,i is the traverse track displacement, r(xw,i ) is the vertical track irregularities at the
10
position of wheel number i, and KH is the Hertzian spring constant. The Hertzian
spring stiffness increases non-linearly with increasing deformations, thus it requires
an iterative solution which is described in Subsection 4.2.
2.6
Vertical track profile
High-frequency vibrations are excited by irregularities of the track. These irregularities are modelled as a stationary stochastic process and they may therefore be described by spectral power density functions. In this paper, only the vertical profile of
the rail is considered and denoted r(x).
The power spectral density S(k) used to generate the track profile is a function of
the wavenumber k. S(k) can be expressed as:
φ(k) = 2S(k),
(19)
where φ(k) is the one-sided spectral density, defined for the range k > 0. The onesided spectral density applied in the present study is given in the form:
φ(k) = Alow
kc2
,
(kr2 + k 2 )(kc2 + k 2 )
(20)
where Alow , kc , kr are constant factors describing the track unevenness. Sample functions of stochastic excitation profiles r(x) may be generated using inverse Fourier
transformation method:
Nk −1
1 X
An eikn x .
(21)
r(x) = √
2π n=0
This method considers Nk number of discrete wavenumbers kn . Coefficients An are
complex numbers that are assigned random phase angles ϕn , which are uniformly
distributed in the range [0, 2π]. They are found as follows:
A0 = 0,
(22)
p
p
An = 2 S(kn ) ∆k · sin(ϕn ) + i 2 S(kn ) ∆k · cos(ϕn ) n = 1, 2, 3, ..., Nk ,
variable ∆k is the wave number k step size.
3
Dynamic model for the soil
This paper utilizes a semi-analytical approach to model the dynamic soil behaviour. In
this chapter a short introduction is given, based on the work by Andersen and Clausen
[18]. Subsection 3.1 describes the Green’s function and gives the fundamental assumptions used, Subsection 3.2 introduces the equations of motion for the stratum,
while Subsection 3.3 outlines an approach to obtain the impedance of the footings.
11
3.1
Response of a layered half-space
The fundamental assumption is that the soil is modelled as a horizontally layered
half-space with each soil layer consisting of homogeneous linear visco-elastic material. The displacements of the soil surface in time domain and in Cartesian space
{uss }(x, y, t) can be related to the load on the surface {pss }(x0 , y 0 , t0 ) through the
Green’s function tensor [gss ](x − x0 , y − y 0 , t − t0 ):
Z t Z ∞ Z ∞
{uss }(x, y, t) =
[gss ](x − x0 , y − y 0 , t − t0 ) {pss }(x0 , y 0 , t0 )dx0 dy 0 , dt0 ,
−∞
−∞
−∞
(23)
where the vector of the soil surface displacements {uss } contains the displacements in
all three directions, {uss } = {uss vss wss }T . However, a closed-form solution to the
Equation (23) is not possible; but it can be established in the frequency and wavenumber domain, assuming the response of the stratum is linear:
{Ūss }(kx , ky , ω) = [Ḡss ](kx , ky , ω){P̄ss }(kx , ky , ω),
(24)
where {Ūss }, [Ḡss ] and {P̄ss } are triple Fourier transforms of {uss }, [gss ] and {pss },
respectively. Thus, the transform of the vector of the soil surface displacements,
{Ūss } = {Ūss V̄ss W̄ss }T .
3.2
Equations of motion for stratum
The stratum consists of J horizontally bounded layers, each defined by Young’s mod(j)
(j)
(j)
(j)
ulus Es , Poisson’s ratio νs , mass density ρs and loss factor ηs , the layers have
(j)
depths Hs , j = 1, 2, ..., J. The layered half space is illustrated in Figure 6.
Surface
x
Layer j
Hs
(j)
y
z,z (j)
Figure 6: Layered half-space model
Equations of motion for the layers are given in terms of Cauchy equations, without
(j)
(j)
(j)
body forces, where us , vs and ws are displacements for layer j in x, y and z
directions, respectively:
12
(j)
(j)
(j)
(j)
∂σxx
∂σyx
∂σzx
∂ 2 us
+
+
= ρ(j)
,
s
∂x
∂y
∂z
∂t2
(j)
(j)
(j)
(25a)
(j)
∂σyy
∂σzy
∂ 2 vs
∂σxy
+
+
= ρ(j)
,
s
∂x
∂y
∂z
∂t2
(j)
(j)
(j)
(25b)
(j)
∂σyz
∂σzz
∂ 2 ws
∂σxz
+
+
= ρ(j)
.
s
∂x
∂y
∂z
∂t2
(25c)
(j)
Assuming hysteretic material dissipation defined by the loss factor ηs , the dynamic stiffness of the homogeneous material in layer j may be described in terms of
complex Lamé constants:
(j)
λ(j)
s =
(j)
(j)
νs Es (1 + i sign(ω)ηs )
(j)
(j)
µ(j)
s =
(j)
(1 + νs )(1 − 2νs )
,
(26a)
(j)
Es (1 + i sign(ω)ηs )
,
(26b)
(j)
2(1 + νs )
where sign() is the sign function. Applying Fourier transforms over the x and y coordinates and time t, the Navier equations in frequency and wavenumber domain are
obtained:
2
(j)
(j)
(j) d
2
2
(j)
(j)
¯
(λ(j)
+
µ
)ik
∆
+
µ
−
k
−
k
= −ω 2 ρ(j)
(27a)
x s
s
s
s
x
y Ūs
s Ūs ,
dz 2
2
(j)
(j)
(j)
(j) d
2
2
(j)
¯
(λs + µs )iky ∆s + µs
− kx − ky V̄s(j) = −ω 2 ρ(j)
(27b)
s V̄s ,
2
dz
2
¯ (j)
(j)
(j) d∆s
(j) d
2
2
(j)
(λs + µs )
+ µs
− kx − ky W̄s(j) = −ω 2 ρ(j)
(27c)
s W̄s .
dz
dz 2
¯ (j)
where the triple Fourier transform of the dilatation ∆
s is defined by the equation:
(j)
dW̄s
(j)
(j)
(j)
¯ (j)
¯ (j)
,
∆
+
s = ∆s (kx , ky , z , ω) = ikx Ūs + iky V̄s
dz
(28)
and the triple Fourier coefficients of the displacements are functions of the horizontal
wavenumbers, the vertical postion, and the cyclic frequency:
Ūs(j) =Ūs(j) (kx , ky , z (j) , ω),
(29a)
V̄s(j) =V̄s(j) (kx , ky , z (j) , ω),
(29b)
W̄s(j) =W̄s(j) (kx , ky , z (j) , ω).
(29c)
Combining Equations (27) with continuity conditions for displacements and equilibrium conditions for traction at the interfaces between adjacent layers leads to a
transfer matrix for the strata. Following the methodology of Andersen and Clausen
[18], the Green’s function matrix [Ḡss ] for a layered half-space can then be found in
closed form.
13
3.3
Dynamic stiffness of rigid surface footings
In this paper, the surface footings are assumed to be rigid. Each footing has six degrees of freedom, three translational and three rotational, as shown in Figure 7. In the
frequency domain the degrees of freedom are related to the corresponding forces and
moments via the impedance matrix [ZF ], with dimensions 6NF × 6NF , where NF is the
total number of footings:
[ZF ]{DF } = {FF }.
(30)
The displacement vector {DF } contains Fourier transforms over time of footings displacements and rotations, similarly, the vector {FF } contains the corresponding Fourier
transforms of the forces and moments with respect to time:
T
{DF } = DTf,1 DTf,2 ... DTf,NF ,
T
{FF } = FTf,1 FTf,2 ... FTf,NF .
(31)
Where the vectors {Df,n } and {Ff,n } for one footing read:
T
{Df,n } = Uf,n Vf,n Wf,n Θf,n Φf,n Ψf,n ,
T
{Ff,n } = Pn,x Pn,y Pn,z Mn,x Mn,y Mn,z .
(32a)
(32b)
In order to compute the non-zero components of the impedance matrix [ZF ], the distribution of the contact stresses at the interfaces between the footings and the ground
due to a given rigid body displacements has to be determined. However, the Green’s
function provides the displacement field for a known stress distribution. To obtain the
footing impedance matrix the following approach is taken:
Step 1. The displacement corresponding to each rigid body mode is prescribed at
Nf points distributed uniformly at the interface between the footings and the
ground. For NF identical footings, the number of points per footing is Nf /NF .
Step 2. The Green’s function matrix is evaluated in the wavenumber domain.
Step 3. The wavenumber spectrum for a simple distributed load with unit magnitude
and rotational symmetry around a point on the ground surface is computed.
Step 4. The response at point n to a load centred at point m is calculated for all
combinations of n, m = 1, 2, ..., Nf . This provides a flexibility matrix for the
footings.
Wf,1
f,1
z
Vf,1 y
Wf,2
f,1
Wf,NF
Vf,2
Uf,1
f,1
f,NF
f,2
f,2
Vf,NF
f,NF
Uf,NF
Uf,2
f,2
Figure 7: Degrees of freedom for rigid surface footings
14
f,NF
x
Step 5. The unknown magnitudes of the loads applied around each of the points are
computed. Integration over the contact areas provides the impedance [ZF ]. The
obtained impedance of the footings can then be added to the structure at the
connecting nodes, as described in Subsection 2.4.
4
Computational model solution procedure
The solution involves two iterative procedures: a global scheme for convergence of
time-domain vehicle and frequency-domain structure–soil presented at Subsection 4.1
and the local scheme for the wheel–rail interaction presented at Subsection 4.2.
4.1
Frequency-time-domain iteration
The impedance matrix for the footings is obtained in frequency domain. Thus the
bridge structure coupled with the soil needs to be analysed in frequency domain.
However, to obtain correct wheel–rail interaction forces, a time-domain solution is
needed as described in Subsection 4.2. To simultaneously account for wheel–rail
and structure–soil interaction a global iteration procedure is proposed. Assuming that
bridge displacements [dm
B ] at all time steps at global iteration step m are known, the
calculation procedure is:
Step 1. Calculate vehicle displacements [dm+1
] and wheel–rail interaction forces [fvm+1 ]
V
for all time steps using known bridge displacements matrix [dm
B ] which reads:
0,m
1,m
Nt ,m
[dm
(33)
}(t)
B ] = {dB }(t) {dB }(t) ... {dB
where Nt is a number of time steps. This involves the local iteration procedure
described in Subsection 4.2.
Step 2. The wheel–rail interaction forces [fVm+1 ] from the vehicle are applied on the
rail, thus obtaining the force matrix [fBm+1 ] for the bridge–soil system. Then
the obtained forces are Fourier transformed with respect to time to get the force
matrix in frequency-domain [FBm+1 ].
Step 3. Solve structure–soil system in frequency-domain to obtain bridge displacement {DBj,m+1 } for one frequency ω j , using vector {FBj,m+1 } taken from matrix
[FBm+1 ]:
{DBj,m+1 } = [Z]−1 {FBj,m+1 }.
(34)
Matrix [Z] is a complex impedance matrix for the structure–soil system described in Subsection 2.4. These calculations are done for every frequency till
displacement matrix [Dm+1
] is computed. The displacement matrix [Dm+1
] for
B
B
global iteration step m + 1 is defined:
t ,m+1
[Dm+1
] = {D0,m+1
(35)
}(ω) {D1,m+1
}(ω) ... {DN
}(ω) .
B
B
B
B
Step 4. Inverse Fourier transform the displacement matrix [Dm+1
] from frequencyB
domain to time-domain [dm+1
].
B
15
Step 5. Check for convergence of the solution:
k[dm+1
]k/k[dm
B
B ]k < glob
(36)
where k k is the Euclidean norm of a matrix and glob is maximum global iteration error. If the Equation (36) is fulfilled, the computation is stopped; if not it
continues to the next iteration step, by repeating from Step 1.
4.2
Wheel–rail interaction iteration
In order to determine the wheel–rail interaction force Pi , an iteration procedure is
required as the force is non-linear, see Subsection 2.5. Thus, the Newmark second–
order time integration scheme is modified to include an iteration to find the force
acting in every time step. Coefficients β and γ are chosen as 1/4 and 1/2, respectively,
for the time integration to be unconditionally stable. The discretized displacements,
j
velocities and accelerations of the MDOF vehicle are stored in vectors {djV }, {ḋV }
j
and {d̈V }, respectively. Assuming that at time step j they are known, the calculation
procedure can be written as:
j+1
Step 1. Calculate predicted values for velocities {ḋV,∗ } and displacements {dj+1
V,∗ } at
time step j + 1:
j+1
j
j
{ḋV,∗ } = {ḋV } + {d̈V }∆t,
(37a)
1 j
j
j
2
(37b)
{dj+1
V,∗ } = {dV } + {ḋV }∆t + {d̈V }∆t .
2
Begin iteration inside time step j + 1, using as an initial guess for the displacements in iteration k = 0:
{dj+1,0
} = {dj+1
(38)
V
V,∗ }.
Step 2. Calculate the predicted force Pij+1,k+1 for iteration k + 1 and place it in the
global force vector {fVj+1,k+1 }:
(
3
j+1,k
j+1,k
≤0
KH | wrel,i
| 2 for wrel,i
j+1,k+1
(39)
Pi
=
j+1,k
0
for wrel,i > 0
j+1,k
j+1,k
j+1
j+1
where wrel,i
= ww,i
− wr,i
− rj+1 . The rail displacements wr,i
are taken
from the bridge displacement matrix [dm
],
as
described
in
Subsection
4.1.
B
Step 3. Determine acceleration, velocity and displacement vectors for time step j + 1
and iteration k + 1:
j+1,k+1
{d̈V
j
} = {d̈V }+[M̂V ]−1 {fVj+1,k+1 }
j
j+1
−[MV ]{d̈V }−[CV ]{ḋV,∗ }−[KV ]{dj+1
V,∗ } ,
j+1,k+1
j+1
j+1,k+1
j {ḋV
} = {ḋV,∗ }+γ {d̈V
}−{d̈V } ∆t,
j+1,k+1
j {dVj+1,k+1 } = {dj+1
}−{d̈V } ∆t2 ,
V,∗ }+β {d̈V
where [M̂V ] = [MV ] + γ[CV ]∆t + β[KV ]∆t2 .
16
(40a)
(40b)
(40c)
Step 4. Check for convergence:
k{dj+1,k
}k
V
k{dj+1,k+1
}k
V
< loc ,
(41)
where loc is the local iteration tolerance. If the Equation (41) is fulfilled then
continue to the next time step (from Step 1) if not continue to the next iteration
step by repeating from Step 2.
4.3
Full solution procedure
A flow chart for the solution of the entire computational model is shown in Figure 8.
Where TV is the time necessary for the vehicle to pass the bridge.
Input
Stif fness [KV], damping
[CV ], mass [MV], force
{f V }matrices for vehicle
Footing impedance
matrix, [ZF]
Stif fness [K B], damping
[CB ], mass [MB]
matrices for bridge
Simulate
track profile, r
Bridge-soil
impedance matrix, [Z]
Next global iteration step
.
..
Find {dV,*},{d V,*}
for vehicle
Wheel on track?
(ww,i -wr,i -r)<0
True
False
..
Find {dV},{dV},{dV}
for vehicle
Wheel-rail interaction force
Pi =0
Local convergence
Next local iteration step
Next time step
.
Wheel-rail interaction force
Pi =KH (|ww,i -wr,i-r|) 3/2
True
t
Tv
local
<
local, set
False
True
False, finish vehicle calculations
Apply Pi on the
structure, [fB]
Solve
DB} [Z]-1 FB}
Fourier transform
[fB] [FB]
False
Display results
True
Inv. Fourier transform
[D B] d B]
Global convergence
global
<
global, set
Figure 8: Numerical model solution procedure
17
5
Validation of computational model
In the present section, validation of the computational model is presented. Firstly, the
results of the FEM model are compared with a closed-form solution given by Fryba
[2] in Subsection 5.1. Subsequently, in Subsection 5.2 the iteration scheme used in
this paper is compared with the iteration scheme given by Lei and Noda [11].
5.1
Validation of the finite-element model
Analytical
20
−1
15
10
−2
5
0
0
x coordinate [m]
0
25
0.1
FEM
20
15
0.2
0.3
0.4
0.5
0.6
Time [s]
Position of maximum deflection at instant time
Vehicle position
Maximum absolute deflection
10
−3
0
−1
−2
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Position of maximum deflection at instant time
Vehicle position
Maximum absolute deflection
Displacement [mm]
25
Displacement [mm]
x coordinate [m]
To validate the accuracy of the FEM solution it is compared with an analytical solution
derived by Fryba [2]. A simplified model is constructed, in which a single constant
force travels across a simply supported beam. The simply supported beam has the following parameters: mass density ρ = 2400kg/m3 , cross-sectional area A = 7.52m2 ,
length L = 28.4m, Young’s modulus E = 34 · 109 Pa, moment of inertia around the
y-axis Iy = 4.07m4 , damping ratio ξ = 0.01 and the constant force F = −6.8 · 105 N.
Figure 9 shows the behaviour of a beam for a load travelling at a speed of 47m/s.
The top graph is the analytical solution and the bottom one is the FEM results. Both
solutions are based on Euler-Bernoulli beam theory, but the damping is introduced
−3
Figure 9: Analytical (top) and FEM (bottom) solution for simply supported beam
18
differently. In the analytical solution, the damping is dependent on the mass, while
in the FEM solution Rayleigh damping is used, i.e. damping is dependent on the
mass as well as the stiffness of the system. This particular speed is chosen as the
pre-calculations showed it as one of the critical speeds. Despite the different damping
approaches, the overall behaviour of the beam is very similar in the solutions, the
maximum difference between the displacements being around 1.4%. Analyses carried
out for other vehicle speeds show a maximum difference not exceeding 1.5%. Based
on this, it is concluded that the proposed iteration scheme is valid.
5.2
Validation of iteration procedure
To validate the iteration solution described in Section 4, the results were compared
to the iteration scheme described by Lei and Noda [11]. Again a simplified model
was created, in which the vehicle described in Subsection 2.1 travels on a simply
supported beam with uneven vertical profile. The properties of the track unevenness
are as follows: kc = 0.8246rad/m, kr = 0.0206rad/m and Alow = 0.59233 · 10−6 rad·m.
The properties of the vehicle are given in Table 1.
Primary suspension stiffness
Ks,1
Secondary suspension stiffness
Ks,2
Primary suspension damping
Cs,1
Secondary suspension damping
Cs,2
Hertzian spring constant
KH
Mass of car body
Mc
Mass of bogie
Mt
Mass of wheel set
Mw
Pitch moment of inertia for car body
Jc
Pitch moment of inertia for bogie
Jt
Total force from static vehicle
Ftotal
Half length between two axles of one bogie L1
Half length between two bogies centre lines L2
3.28 · 106
1.31 · 106
9.00 · 104
3.00 · 104
8.70 · 1010
5.35 · 104
3260
2000
2.24 · 106
2.45 · 103
−6.80 · 105
1.5
7
N/m
N/m
N·s/m
N·s/m
N/m3/2
kg
kg
kg
kg·m2
kg·m2
N
m
m
Table 1: Properties of the vehicle
The iteration scheme used for validation solves both the upper and the lower structures in time domain. Firstly, the interaction force between the rail and the wheel
is applied to the lower structure, in this case the bridge. The accelerations, velocities and displacements are found using the Newmark second-order time integration
scheme. Using the newly obtained bridge displacements, the interaction forces are recalculated and applied to the upper structure, in this case the vehicle. By using again
Newmark integration, the accelerations, velocities and displacements are obtained.
Finally, convergence of the solution is checked within the current time step before
proceeding to the next time step. If there is no convergence, the steps are repeated.
19
Displacement [mm]
Displacement [mm]
0
−1
Time/frequency
Time/time
−2
0
0
0.5
1
Time [s]
1.5
2
−2
Time/frequency
Time/time
−4
0
0.5
1
Time [s]
1.5
2
Figure 10: Bridge displacements (top) and displacements for wheel one (bottom)
The top graph in Figure 10 shows calculated displacements at the middle of the
beam, while the bottom one shows the displacement of the wheel one of the MDOF
vehicle traversing the beam. The approach described in this paper is denoted as
time/frequency iteration, while the solution by Lei and Noda [11] is referred to as
time/time iteration. The results obtained by the two approaches are almost identical.
Thus the proposed iteration method can be used for the model with coupled soil–
structure system. As long as time is discretized in a proper manner and the considered
time interval is long enough to allow all vibrations to damp out, the transformations
between physical domain and Fourier domain will not introduce significant errors.
6
Results
In this section, the results obtained from computational models for various cases are
presented. Firstly, geometrical dimensions of the bridge structure, its specifications
and underlying soil properties are presented at Subsection 6.1. At Subsection 6.2
frequency response functions (FRFs) are compared between different model configurations, as well as the soil stratification effects for critical speeds. The effects of the
passing vehicle are presented at Subsection 6.3.
6.1
Parameters used in the computational model
The geometry of the bridge structure used in computational model is shown in Figure 11. The bridge has a total length of 200m, which is subdivided into eight spans,
20
each 25m long. Sleepers are spaced 0.6m apart. From each side of the bridge there is
25m of the track which is attached to a rigid surface. The bridge is supported by seven
6m tall pylons, each resting on a rigid surface footing.
0.6
Deck
0.6
Fixed support
25
z
25
25
25
25
Rigid surface
Response point 1
250
25
Rail
25
25
25
25
6
x
Soil surface
Response point 2
Excitation point
Figure 11: The bridge structure used in the computational model (measures in m)
The track structure including the rail, rail pads, sleepers and ballast is described
by parameters given in Table 2. The vertical track unevenness is implemented in the
model using the same parameters as given previously in Subsection 5.2. The bridge
deck properties are listed in Table 3. The specification for the pylons and footings are
given in Table 4.
Rail properties
Young’s modulus
Er
210 · 109
Mass density
ρr
8050
Moment of inertia around y-axis Ir,y 2.00 · 10−5
Cross-sectional area
Ar
0.0064
Rail pad properties
Stiffness
Kpd,i
1.2 · 108
Damping
Cpd,i 1.24 · 105
Sleeper properties
Mass
Msl,j
237
Spacing
Lsl
0.6
Ballast properties
Stiffness
Kb,j
2.4 · 108
Damping
Cb,j
5.88 · 104
Mass
Mb,j
683
Table 2: Properties of the track structure
21
Pa
kg/m3
m4
m2
N/m
N· s/m
kg
m
N/m
N· s/m
kg
Young’s modulus
Ed
Shear modulus
Gd
Mass density
ρd
Moment of inertia around y-axis
Id,y
Moment of inertia around z-axis
Id,z
Polar inertia moment
Id,0
Torsion constant around y-axis
Td
Cross-sectional area
Ad
Mass of second track
Mot,j
Railway track distance from deck centre line Wd
34 · 109
13 · 109
2400
4.07
52.2
56.3
1.79
7.52
920
2
Pa
Pa
kg/m3
m4
m4
m4
m4
m2
kg
m
Table 3: Properties of the bridge deck
Pylon properties
Young’s modulus
Ep
34 · 109
Shear modulus
Gp
13 · 109
Mass density
ρp
2400
Moment of inertia around x-axis Ip,x
4.57
Moment of inertia around y-axis Ip,y
0.42
Polar inertia moment
Ip,0
4.99
Torsion constant around z-axis
Tp
1.46
Cross-sectional area
Ap
5.00
Surface footing properties
Mass
Mf 3.84 · 104
Rolling moment of inertia
Jf,x 1.032 · 105
Pitch moment of inertia
Jf,y 1.032 · 105
Yaw moment of inertia
Jf,z 2.048 · 105
Surface footing length
Lf
5.6
Surface footing width
Wf
5.6
Pa
Pa
kg/m3
m4
m4
m4
m4
m2
kg
kg·m2
kg·m2
kg·m2
m
m
Table 4: Properties of the pylons and footings
Two different types of soil are used in the simulations: clay and sand. Their parameters are given in Table 5. Four different soil stratifications are analysed:
Case 0 : Bedrock, i.e. bridge supports are fixed,
Case 1 : Half-space of clay,
Case 2 : 5m of clay over half-space of sand,
Case 3 : 10m of clay over a half-space of sand.
22
Poisson’s ratio
ν
Mass density
ρ
Hysteretic damping η
Young’s modulus
E
P-wave speed
cP
S-wave speed
cS
Rayleigh wave speed cR
Clay
0.3
2 · 103
0.045
20 · 106
116.0
62.0
57.5
Sand
0.4
2 · 103
0.040
160 · 106
414.0
169.0
159.0
kg/m3
Pa
m/s
m/s
m/s
Table 5: Properties of different soil types
6.2
Soil effects to the computational model
To show the effects of the soil to the dynamic behaviour of the structure, FRFs are
compared for four different soil cases as described at Subsection 6.1. The excitation
point is excited and the response of the structure is observed at response point 1 (on the
rail) and response point 2 (on the deck directly under the rail) as shown in Figure 11.
The results are presented in Figure 12.
−8
x 10
FR [m/N]
Response point 1
Case 0
Case 1
Case 2
Case 3
1
0.5
0
0
5
10
15
Frequency [Hz]
20
25
30
20
25
30
−8
x 10
FR [m/N]
Response point 2
1
0.5
0
0
5
10
15
Frequency [Hz]
Figure 12: FRFs for four different soil stratifications: Response at Response point 1
(top) and Response point 2 (bottom)
23
For Case 0 the first eigenfrequency is above 6Hz, which means that any critical
speed of the vehicle has to be around 150m/s. This is beyond the scope of this paper.
When a soil body is added to the system, in Cases 1, 2 and 3, the first eigenfrequency
decreases significantly. Furthermore, the first eigenmode changes, from the deck deflecting in a U-shape between pylons, to a whole pylon deflecting downwards. It can
be seen that the width of the peaks increases but the magnitudes decrease because
there is more damping in the system. This illustrates that the soil changes the dynamic
structural behaviour of the system completely, the lower values of the first eigenfrequency makes the critical speeds of the travelling vehicle easier to achieve.
To determine critical speeds of the vehicle, soil stratification Cases 1, 2 and 3, are
checked for speeds from 30m/s to 90m/s. The response is checked at the 6th pylon
from the left, c.f. Figure 11, at the node connecting the pylon and the footing. The
obtained results are shown in Figure 13.
Case 1
Case 2
Case 3
1.05
w/w
ref
[−]
1.1
1
0.95
30
40
50
60
Speed [m/s]
70
80
90
Figure 13: Relative displacement for different vehicle speeds
Maximum displacements over time are checked for each speed and normalized by
the maximum displacement for the lowest speed. That way, all the graphs begin at a
value of one on the ordinate. For Case 1, a critical speed of approximately 56m/s is
determined, which is very close to the Rayleigh wave speed in clay, described by the
parameters given in Table 5. However, the maximum displacements for the critical
speed are not very high, around 6% higher than the reference displacement. For Cases
2 and 3 there is no critical speed, the maximum displacements for Case 2 stay approximately the same, while for Case 3 they are slowly increasing with the increasing speed.
In general, the effect of different speeds on the maximum footing displacements is not
high.
6.3
Effects of including the vehicle in the model
Vehicle and track unevenness are important for the structural behaviour of the system, as the wheel–rail interaction produces high-frequency vibrations. For this pur24
pose, a model that includes both vehicle and track unevenness is compared with a
model where the effects from the vehicle are modelled as constant moving forces.
The displacements in the z direction at Response point 2, c.f. Figure 11, are analysed.
Firstly, the bridge model with soil stratification Case 0, where the footings are fixed
to bedrock, is investigated for three different speeds. The results are presented in Figure 14. It can be seen that for the speed 30m/s both ways of modelling the vehicle
produce nearly identical results. With increasing speed, high-frequency vibrations become more pronounced. Finally, for a speed of 70m/s the high-frequency vibrations
on the overall dynamic behaviour of the structure becomes crucial. For high speeds,
proper modelling of the vehicle is very important.
−4
−5
2
2
Displacement [m]
Displacement [m]
x 10
0
−2
Force
Vehicle
−4
0
2
4
Time [s]
6
0.5
2
2
Displacement [m]
Displacement [m]
10
20
Frequency [Hz]
30
−5
0
−2
2
4
Time [s]
6
x 10
Speed: 50m/s
1.5
1
0.5
0
0
8
−4
10
20
Frequency [Hz]
30
−5
x 10
2
2
Displacement [m]
Displacement [m]
Speed: 30m/s
1
−4
0
−2
−4
0
Force
Vehicle
1.5
0
0
8
x 10
−4
0
x 10
2
4
Time [s]
6
8
x 10
Speed: 70m/s
1.5
1
0.5
0
0
10
20
Frequency [Hz]
30
Figure 14: Time (left) and frequencies (right) series of the displacements of the model
excited by a passing vehicle for soil stratification Case 0
25
Further, the same simulation is performed, just this time, soil stratification Case 1 is
introduced in the model. The results are presented in Figure 15. It can be seen that with
increasing vehicle speed, the displacements at the high frequencies are increasing,
however the excitation at high frequency is not as significant as in the previous case.
Therefore, it can be concluded that a proper vehicle modelling is still important for
such cases as structural fatigue life estimation, but for calculations of the maximum
displacements, the vehicle modelled as a set of constant forces predicts the response
quite well.
−4
x 10
0
Displacement [m]
Displacement [m]
x 10
−5
−10
Force
Vehicle
−15
0
2
4
Time [s]
6
8
1
Displacement [m]
Displacement [m]
x 10
−5
−10
−15
2
4
Time [s]
6
8
Displacement [m]
Displacement [m]
−10
−15
2
4
Time [s]
6
8
−4
Speed: 50m/s
0
0
x 10
−5
30
0.5
−4
0
10
20
Frequency [Hz]
1
x 10
0
Speed: 30m/s
0
0
−4
0
Force
Vehicle
0.5
x 10
0
−4
1
10
20
Frequency [Hz]
30
−4
Speed: 70m/s
0.5
0
0
10
20
Frequency [Hz]
30
Figure 15: Time (left) and frequencies (right) series of the displacements of the model
excited by a passing vehicle for the soil stratification Case 1
26
7
Conclusions
Dynamic analysis of a multi-span bridge is performed. The proposed model simultaneously accounts for wheel–rail interaction as well as structure–soil interaction. A
finite element model for the bridge structure and a semi-analytical solution for the soil
body are utilized. The results show that proper modelling of the soil body is very
important for the overall dynamic behaviour of the structure. Introducing the soil, the
first eigenfrequency of the structure decreases, and the shape of the first eigenmode
changes. The effects from the vehicle, more specifically wheel–rail interaction, cause
high frequency vibrations. This phenomenon becomes more important for higher vehicle travelling speeds, especially for the case where the bridge structure is on a hard
subsoil. By including soft soil in the model, more damping is introduced in the whole
system, which damps out the vibrations at high frequencies induced by the vehicle.
The work can be expanded by introducing a three-dimensional vehicle model instead of the two-dimensional model used in this paper and decreasing the number of
approximation used in rail track modelling. An improved model might show effects
which cannot be seen in the present computational model. Further, the vehicle can be
implemented as a full train model, i.e. a locomotive with multiple coaches.
To conclude, the work done in this paper shows that sophisticated models, that
are able to model different phenomena, are important in understanding the behaviour
of high-speed railway structures. The proposed model offers a simplified solution for
preliminary calculations but accounting for most significant contributions to vibrations
and deformations of the bridge–subsoil system.
References
[1] L. Meirovitch, “Analytical methods in vibrations”, MacMillan, London, United
Kingdom, 1967.
[2] L. Fryba, “Vibration of solids and structures under moving load”, Noodhoff International, Groningen, The Netherlands, 1972.
[3] D.J. Thompson, “Wheel-rail noise generation, Part III: rail vibration”, Journal of
Sound and Vibration 161, 421–446, 1993.
[4] B. Ripke and K. Knothe, “Die unendlich lange Schiene auf diskreten Schwellen
bei harmonisher Einzellasterregung”, Fortschritt-Berichte VDI Reihe 11, No.
155, 1991.
[5] L. Gry, “Dynamic modelling of railway track based on wave propagation”, Journal of Sound and Vibration 195, 477–505, 1996.
[6] T.X. Wu and D.J. Thompson, “A double Timoshenko beam model for vertical
vibration analysis of railway track at high frequencies”, Journal of Sound and
Vibration 224, 329–348, 1999.
[7] T.X. Wu and D.J. Thompson, “Analysis of lateral vibration behaviour of railway
track at high frequencies using a continuously supported multiple beam model”,
Journal of the Acoustical Society of America 106, 1369–1376.
27
[8] M.A. Heckl, “Railway noise-can random sleeper spacing help?”, Acustica 81,
559–564, 1995.
[9] P.J. Remington, “Wheel/rail rolling noise I: theoretical analysis”, Journal of the
Acoustical Society of America 81, 1805–1823, 1987.
[10] S.L. Grassie, R.W. Gregory, D. Harrison, K.L. Johnson, “The dynamic response
of railway track to high frequency vertical excitation”, Journal of Mechanical
Engineering Science 24, 77–90, 1982.
[11] X. Lei, N.A. Noda, “Analyses of dynamic response of vehicle and track coupling
system with random irregularity of track vertical profile”, Journal of sound and
vibration 258(1), 147–165, 2002.
[12] R.U.A. Uzzal, W. Ahmed, S. Rakheja, “Dynamic analysis of railway vehicletrack interactions due to wheel flat with a pitch-plane vehicle model”, Journal of
mechanical engineering Vol. ME39, No. 2, 2008.
[13] W. Zhai, Z. Cai, “Dynamic interaction between a lumped mass vehicle and a
discretely supported continuous rail track”, Vol. 63, No. 5, 987–997, 1995.
[14] A.V. Metrikine, K. Popp, “Instability of vibrations of an oscillator moving along
a beam on an elastic half-space”, European Journal of Mechanics, A/Solids 18
(2), 331–349, 1999.
[15] G. Lombaert, G. Degrande, J. Kogut, S. Franc-ois, “The experimental validation
of a numerical model for the prediction of railway induced vibrations”, Journal
of Sound and Vibration 297, 512–535, 2006.
[16] X. Sheng, C.J.C. Jones, and M. Petyt, “Ground vibration generated by a load
moving along a railway track”, Journal of Sound and Vibration, Vol. 228, No. 1,
129–156, 1999.
[17] X. Sheng, C.J.C. Jones, M. Petyt, “Ground vibration generated by a harmonic
load acting on a railway track”, Journal of Sound and Vibration 225 (1), 3–28,
1999.
[18] L. Andersen, J. Clausen “Impedance of surface footings on layered ground”,
Computers and Structures 86 (1-2), 72–87, 2008.
28
Appendix B
Paper II
Paper II
109
Experimental validation of a numerical model for
three-dimensional railway bridge analysis by comparison with a
small–scale model
Paulius Bucinskas1 , Liuba Agapii1 , Jonas Sneideris1 , and
Lars Vabbersgaard Andersen2
1
Master student
professor
2 Associate
Department of Civil Engineering, Aalborg University, Denmark
Abstract
The aim of the paper is to perform dynamic analysis of a multi-span railway bridge
interacting with the underlying soil. A small-scale model of a bridge structure is constructed for experimental testing and the results are compared with a computational
model. The computational model in this paper is based on finite-element (FE) analysis for the bridge structure and a semi-analytical solution for the subsoil. The bridge
deck and columns are modelled using three-dimensional beam elements. The foundations are implemented as rigid footings placed on the ground surface. The vehicle
is modelled as a two dimensional 10-degrees-of-freedom system. The subsoil model
utilizes the Green’s function for a horizontally layered half-space. The small-scale experimental model consists of bridge deck, columns and footings which are made from
Plexiglas. An electric vehicle travels along the bridge deck on a track to simulate a
passing train. Mattress foam is used to substitute the subsoil. The model is equipped
with a number of accelerometers, strategically placed in certain positions to analyse
the dynamic structural response. Finally, the results obtained from experimental tests
are used to calibrate and validate the numerical model which is found to reproduce the
structural response of the experimental model fairly well.
Keywords: railway, multi-span bridge, dynamics, vehicle, small-scale experimental
model, structure-soil interaction, computational model validation.
1
Introduction
In this paper, a railway bridge structure with a passing vehicle is analysed. Firstly, it
is done by creating a numerical model capable of evaluating dynamic effects caused
by train traffic on the bridge–soil system. A sophisticated approach including different phenomena can be beneficial for obtaining more reliable results, thus allowing
1
optimization of the final structural design. Laboratory testing is an integral part of
the paper since it is used to validate the computational model. Thus, a small-scale
model is constructed and tested in the laboratory at Aalborg University, Denmark.
The dynamic behaviour of the structure found by the computational and experimental
approaches are compared.
Most analytical studies of structural response due to dynamic loading have been
based on Euler-Bernoulli beam theory, where the bridge structure has been considered
as a simply supported beam. Complexity of the problem solution depends on implementation of a dynamic load in a model. For the case of a simple moving constant
force, a closed-form solution was formulated and published by Meirovitch [1] and
Fryba [2]. Rieker [3] investigated capabilities of a numerical model based on finiteelement method (FEM) formulation to predict dynamic response of a beam under
moving continuous load. To assess the computational model, an experimental model
of a single-span aluminium beam subjected to a moving mass was used.
The ground vibration problem was studied by different authors, proposing approaches that analyse soil induced vibrations due to moving loads by using an FEM
model or, alternatively, Green’s function—sometimes combined with a boundaryelement method (BEM) model. Metrikine and Popp [4] studied ground motion by
using a model in which the vehicle was implemented as a two-degrees-of-freedom
system coupled with a beam of infinite length representing the track and a homogeneous half-space model representing the subsoil. Lombaert [5] applied previously
mentioned methodology in a BEM model to study railway traffic induced vibrations.
Sheng [6, 7] studied the ground vibration generated by a harmonic load that moves
along a railway track on a layered half-space.
A number of field tests has been performed by other researchers, analysing fullscale structural dynamic response of railway bridges due to real passing vehicles [8,
9, 10, 11]. However, full-scale tests are costly in terms of time and equipment. Thus,
for development and validation of computational models, as intended in this paper,
small-scale tests are preferable, since measurements are easier and faster to perform
in small scale—especially when large numbers of experimental tests are required.
A real structure can be scaled down to a small-scale model and vice versa using
similitude laws. Harris and Sabnis [12] provide tools for proper scaling of different
physical problems for static as well as dynamic cases. Previous work on small-scale
experimental models done by Bilello [13] includes an investigation of dynamic response of a single-span aluminium bridge model under a moving load. The study
concludes that the measured response of a beam is basically due to the first structural
mode of vibration. In a subsequent paper, written by Stancioiu [14], a four-span slender bridge model subjected to a load moving at constant speed is analysed. Here an
experimental model is used to validate a theoretical approach. The conclusion states
that the results obtained by the two different methods show good compatibility at low
speeds. However, results start to differ when the vehicle speed is increased.
In the next section, the experimental set-up is described. In Section 3 the computational model is presented, followed by its validation in Section 4. Results and
2
comparisons of the values obtained from a series of experiments and computational
model are presented in Section 5. Especially, a parameter study is carried out to compare the two models for different passage speed. Finally, the overall conclusions are
presented in Section 6.
2
Experimental model and set-up
A small-scale experimental model was built of Plexiglas in the laboratory as shown in
Figure 1. To recreate a realistic structural behaviour, a vehicle with four wheel sets and
travelling along the bridge on a railway track was used. To model subsoil, mattress
foam material was employed. It was determined that mattress foam provides a good
substitution for soil, since the ratio between real structural bending stiffness and soil
stiffness was maintained using Plexiglas for the bridge and mattress foam for the soil.
The bridge structure is described in Subsection 2.1, the passing vehicle is presented
in Subsection 2.2, and a description of the underlying soil is given in Subsection 2.3.
Finally, the measurement equipment set-up is described in Subsection 2.4.
Acceleration track
Gauge
Vehicle
Deck
Rail
Current
control
Foam
Gauge
z
Footing
y
Column
x
Deceleration track
Figure 1: Small-scale experimental model
2.1
Bridge structure
The small-scale bridge model consists of bridge deck, multiple columns and surface
footings. The material used for all parts was Plexiglas as it was easy to work with, and
its relatively small Young’s modulus made the collection of data from experiments
easier. The deck was made from one continuous piece of Plexiglas to which columns
3
were connected using screws. The columns and the surface footings were screwed
together in the same manner to ensure stiff connections between different parts of the
structure. The columns and footings have much higher bending stiffness than the deck
and are therefore assumed to be rigid.
The track for the vehicle was bolted to the deck (see futher description in the next
subsection) and the two ends of the bridge were fixed, i.e. clamped, by putting them on
concrete tiles resting directly on the floor in the laboratory and ballasting the Plexiglas
beam with heavy steel blocks. Thin rubber pads where installed under the Plexiglas
beam within the supported regions to prevent sliding and ensure good contact over the
full area. On the other hand, the footings were simply placed on the surface of the
underlying foam, relying on gravity to provide the necessary contact throughout the
dynamic tests. Detailed sections of the bridge are shown in Figure 2.
Rubber pad
Deck
Track
Column
120
5
119
86
119
Screw Bolt
Footing
z
5
20
Screw
Soil
20
x
89
z
20
y
120
Figure 2: Longitudinal section (left) and cross-section (right) of the bridge model
(measures in mm)
2.2
Vehicle and track
R
To reproduce the effect of a passing vehicle, a toy train and track made by LEGO
was used. The track was made of plastic with metal rails and constituting of 127.5mm
long segments interlocking with each other. Each track segment was bolted to the
Plexiglas deck with a single bolt placed along the centre line. Rail pads were placed
under each sleeper made of 1mm thin rubber tape to ensure good contact between
track and deck, thus avoiding slipping and sliding during vibrations. An illustration of
the set-up is given in Figure 3. It is noted that the track contributes significantly to the
overall stiffness of the experimental bridge model. Hence, the reduced stiffness in the
joints between adjacent track segments must be accounted for in the computational
model, and likewise distance between bolts (only one bolt per segment) can cause a
reduced shear capacity in the connection leading to further loss of bending stiffness.
Furthermore, the joints lead to surface irregularities repeated periodically along the
track. To obtain realistic interaction forces between the track and the vehicle, these
irregularities must be represented in the computation model.
4
Bridge deck
127.5
Bolt
2.7
42 64
y
x
Connection between two railway segments
Figure 3: Set-up of bridge deck with bolted railway track (measures in mm)
The main bridge had a length of 2500mm. At either end of the bridge, an extra
length of track was added for acceleration and deceleration of the vehicle in order to
reach a constant travelling speed across the bridge, cf. Figure 1. The vehicle is powered by electric current travelling through the metallic rails. It is modified by adding
extra mass to make the response of the structure easier to distinguish from the noise
in the signal and to avoid loss of contact between the wheels and the rails. Further, the
added mass leads to lower eigenfrequencies that more realistically represent those of
a vehicle in full scale. The travelling speed of the vehicle can be changed by changing
the electric current. Vehicle speeds up to 1.3m/s could be reached with this equipment.
An illustration of the vehicle is shown in Figure 4.
Added mass
y
Moving direction
Engine
Wheel set
Deck
x
Track
24
74
48
Figure 4: Sketch of the vehicle (measures in mm)
2.3
Soil
A foam mattress size of 2500mm×1000mm was chosen so that the boundaries would
have minimal impact on the overall behaviour (this was the maximum standard size
of foam mattrass available from the producer). Two layers of 120mm thickness each
were used and tested in two cases: one case with a single layer (120mm) of foam and
a second case with two layers (240mm) of foam. The mattresses were placed within
a box of plywood to fix the boundaries, cf. Figure 1. Furthermore, to allow future
analysis of saturated foam, the box was made water tight using silicon paste and the
bottom layer of foam was glued to the base of the box to counteract buoyancy during
water saturation and prevent slipping.
5
2.4
Measurement equipment set-up
All components of the equipment and software utilized in the tests are manufactured
by Brüel & Kjær (B&K). To evaluate the dynamic behaviour of the small-scale model,
a number of accelerometers (type 4507 Bx) were used. They were placed at the positions shown in Figure 5.
64
64
3
482
64
64
5
4
384
64
64
7
6
529
239 145
1A
12
290
1
2
384
384
Impact hammer
Accelerometer
34567
10 384
2500
239
1B
8
11 384
482
13
9
384
290
Figure 5: The testing structure of the bridge model. Measures in millimetres
The data from the accelerometers were read by a front-end (type 3560 D) which fed
the data to a computer running PULSE LabShop Fast Track Version 18.1.1.9. Further,
data analysis is performed using Matlab. The data read from the accelerometers were
filtered using a high-pass filter at 7Hz. To determine the frequency response functions
(FRF) for input in two different positions (1A and 1B) and output in a number of other
points (1–11), an impact hammer (model 8202) was used to introduce an impulse. A
rubber tip was put on the hammer head to introduce a soft impact, thus reducing the
high-frequency contents in the input signal. The data from the hammer was amplified
using a NEXUS Conditioning Amplifier Type 2692-C before sending the signal to the
front-end and then to the computer. PULSE LabShop Fast Track software read the
data from the hammer and from the accelerometers and calculated directly the FRF.
The scheme of the experimental data collection set-up is shown in Figure 6. With
reference to Figure 5, accelerometers no. 12 and 13 were placed on two thin wires
that were triggered by the vehicle when it entered and left the main stretch of the
bridge deck. These “gauges”, utilized to time the vehicle passage and calculate the
vehicle speed, are visualized in Figure 1.
Computer with
Pulse LabShop
Accelerometers
Front-end
Amplifier Impact hammer
Figure 6: Measurement equipment set-up
6
3
Computational model of the bridge and subsoil
The paper aims to validate the computational model, described in this section. The
model consists of three main parts: The bridge structure described in Subsection 3.1,
the vehicle presented in Subsection 3.2, and the underlying soil described in Subsection 3.3. The bridge is modelled utilizing the FEM and the vehicle is implemented as
a two-dimensional 10-degrees-of-freedom system, while the soil is modelled using a
semi-analytical approach based on Green’s function for a layered half-space.
3.1
Computational model of the bridge
To model the dynamic behaviour of a bridge structure, three-dimensional (3D) beam
elements are used, c.f. Figure 7. The 3D beam elements have two nodes with six
degrees of freedom (d.o.f.) at either node (three translational and three rotational
d.o.f.). Extra mass, such as screws, accelerometers and wires, added on the smallscale experimental model is accounted for by point masses Mj , added at distances xa,j
from the nodes.
The beam element has the following properties in the local coordinate system:
Cross-sectional area A, mass density ρ, Young’s modulus E, shear modulus G, moment of inertia around the local y-axis Iy , moment of inertia around the local z-axis
Iz , polar moment of inertia I0 and torsional constant J. Displacements are denoted as
u, v and w in the x, y and z directions, respectively, and θ is rotation around x-axis.
x w,1
z
x w,i
y
Pw,1
u
xa,1 M1
Pw,i
M2
z
0
w
x
Mj
xa,j
y
z
v
w
Mj
ya,j
Figure 7: Three-dimensional beam element shown in the local x, z-plane (left) and
local y, z-plane (right)
It is assumed that displacements are small (i.e. linear) and that axial deformation,
bending and torsion are decoupled. Based on Bernoulli-Euler beam theory, the equation of motion for bending in the x,z-plane reads:
∂ 2 w(x, t)
∂ 4 w(x, t)
EIy
+ ρA
∂x4
∂t2
N
M
X
∂ 2 w(x, t) X
+
=
δ(x − xa,j )Mj
δ(x − xw,i (t))Pw,i (t), (1)
2
∂t
j=1
i=1
where the influence of the added masses Mj , j = 1, 2, ..., N , is accounted for via the
Dirac delta function δ(x). Further Pw,i (t), i = 1, 2, ..., M , is a number of point loads
7
applied to the bridge, in this case originating from the wheels of the passing vehicle
which will change position xw,i (t) with time t; hence, the subscript ‘w’.
Torsion around the x-axis is considered free (Saint-Venant torsion) and can be expressed as
GJ
N
X
∂ 2 θ(x, t)
∂ 2 θ(x, t)
∂ 2 θ(x, t)
2
−
ρI
+
δ(x
−
x
)y
M
= 0.
0
a,j a,j
j
2
∂x2
∂t2
∂t
j=1
(2)
Here it is taken into consideration that extra mass is placed at a distance ya,j from
the centre line of the beam, and it is assumed that no external moments are applied.
Further, no external forces are acting in the longitudinal direction, and therefore the
axial deformation in x-direction is governed by
N
X
∂ 2 u(x, t)
∂ 2 u(x, t)
∂ 2 u(x, t)
− ρA
+
= 0.
EA
δ(x − xa,j )Mj
∂x2
∂t2
∂t2
j=1
(3)
Finally, bending in the x,y-plane is described similarly to bending in the x,z-plane by
using Bernoulli-Euler beam theory with extra point masses:
N
X
∂ 2 v(x, t)
∂ 2 v(x, t)
∂ 4 v(x, t)
+
ρA
+
δ(x
−
x
)M
= 0.
EIz
a,j
j
∂x4
∂t2
∂t2
j=1
(4)
As it was mentioned previously, the experimental model has a railway track which
creates higher-stiffness areas where the track is bolted to the deck. This influences the
stiffness of the system and hence its eigenfrequencies and eigenmodes. To account
for this within the computational model, elements with higher stiffness are introduced
at those positions by using a different set of material and geometrical constants. The
values of these constants, as well as the length over which they are applied within each
track segment, are variables used in the model calibration.
The governing equations for the bridge structure are all discretized, using the standard Galerkin approach with cubic interpolation functions for transverse displacements and linear interpolation for longitudinal displacement and torsion. Damping
is introduced as Rayleigh damping to match the response of the experimental model.
3.2
Computational model of the vehicle
The vehicle is modelled as a two-dimensional 10-degrees-of-freedom system as shown
in Figure 8. The system has primary suspension described by stiffness Ksu,1 and damping Csu,1 as well as secondary suspension with parameters Ksu,2 and Csu,2 . The car
body has a mass Mc and pitch moment of inertia Jc , and it is described by the vertical
translation wc and the pitch rotation φc . Bogie 1 and bogie 2 have the masses Mt,1 and
Mt,2 , the pitch moments of inertia Jt,1 and Jt,2 , the vertical translations wt,1 and wt,2 ,
and the pitch rotations φt,1 and φt,2 . The four wheel sets have the masses Mw,i and the
vertical translations ww,i , i = 1, 2, 3, 4.
8
Moving direction
Vehicle body
Mc Jc
K su,2
wt,2
t,2
K su,1
z
x
c
C su,2
Bogie 2
C su,1 K su,1
Mw,4
Wheel 4
wc
K su,2
Mt,2 Jt,2
C su,1
Wheel 2
KH
Mw,3
Wheel 3
KH
P4
P3
2L1
K su,1
C su,2
wt,1
Mt,1 Jt,1
Bogie 1
C su,1 K su,1
C su,1
Mw,2
Mw,1
t,1
ww,i
Wheel 1
r
wr,i
KH
KH
P2
P1
2L1
2L 2
Figure 8: Multi-degree-of-freedom vehicle
The interaction forces between the vehicle and the track are modelled as
3
KH (| ww,i − wr,i − r(xw,i ) |) 2 for ww,i − wr,i − r(xw,i ) < 0
Pi =
0
for ww,i − wr,i − r(xw,i ) > 0
(5)
where KH is the Hertzian spring constant for the circular wheels and r(xw,i (t)) denotes the vertical track irregularities at the positions of the wheels, i = 1, 2, 3, 4. Note
that ww,i , wr,i and xw,i (and thus also r(xw,i )) are time dependent, since the positions of
the four wheel sets change with time as xw,i = xw,i 0 + V0 t assuming a constant speed
V0 of the vehicle. Obviously, the contact forces Pi obtained by Equation (5) must be
counter balanced by the forces Pw,i , i = 1, 2, 3, 4, in Equation (1), which provides the
necessary vehicle–structure interaction model.
The track irregularities are introduced as periodically repeating half-circles with a
radius of 0.5mm for every 127.6mm (corresponding to the length of track segments in
the physical model), c.f. Figure 8, to account for the track joints in the experiment.
It is noted that, appart from the periodically occuring bumps in the track segment
connection points, no track irregularities are accounted for in the present analyses.
3.3
Computational model of structure–soil–structure interaction
A semi-analytical approach is used to model the underlying soil. This subsection gives
a brief introduction, while a more detailed description can be found in [15] and the full
derivation of the method is given in [16]. The method utilizes the Green’s function in
9
frequency and wave-number domain. With ω representing the angular frequency and
kx and ky denoting the horizontal wave numbers,
Ūi (kx , ky , ω) = Ḡij (kx , ky , ω)P̄j (kx , ky , ω),
(6)
where Ūi (kx , ky , ω) and P̄j (kx , ky , ω) are the displacement and traction, respectively,
while Ḡij (kx , ky , ω) is the Green’s function. The solution used is valid for homogeneous linear viscoelastic material with horizontal boundaries between layers. The
equations of motion for the soil stratum are described by Navier equations in the frequency and wave-number domain, and by solving them the Green’s function is obtained. However, the Green’s function provides the displacement field for a known
stress distribution. To obtain an impedance matrix for a set of footings on the ground
surface, the following approach is taken, assuming that the footings are rigid and each
footing has six degrees of freedom, as shown in Figure 9:
1. The displacement corresponding to each rigid body mode is prescribed at N
points distributed uniformly at the interface between the footing and the ground.
2. The Green’s function matrix is evaluated in the wavenumber domain.
3. The wavenumber spectrum for a simple distributed load with unit magnitude
and rotational symmetry around a point on the ground surface is computed.
4. The response at point n to a load centered at point m is calculated for all combinations of n, m = 1, 2, ..., N . This provides a flexibility matrix for the footing.
5. The unknown magnitudes of the loads applied around each of the points are
computed. Integration over the contact areas provides the impedance [Zs (ω)].
p,1
z y
p,2
wp,1
vp,1
wp,2
p,1
vp,2
p,2
x
up,1
Soil surface
p,1
Connecting node
up,2
p,2
Rigid foundation
Figure 9: Coupling between structure and soil—model with two sqare footings
To couple the bridge structure with the underlying soil, firstly the structural stiffness
matrix [K], structural damping matrix [C] and structural mass matrix [M] are converted
to a complex dynamic stiffness (or impedance) matrix [Z(ω)]:
[Z(ω)] = [K] + iω[C] − ω 2 [M].
10
(7)
Further, the impedance matrix obtained for the soil, i.e. [Zs (ω)], is added to the dynamic stiffness matrix for the structure at the rows and columns corresponding to
connecting nodes at the base of all bridge columns:
[Z11 (ω)]
[Z12 (ω)]
[Z(ω)] =
.
(8)
[Z21 (ω)] [Z22 (ω)] + [Zs (ω)]
Here [Z11 (ω)] is the impedance submatrix for nodes that are not connected to the soil
and [Z22 (ω)] is the impedance submatrix at the connecting nodes. Further, symmetry
of the FE system matrices implies that [Z12 (ω)] = [Z21 (ω)]T .
4
Calibration of the model
To properly evaluate the dynamic response of the system, the computational model
was calibrated to fit the dynamic properties of the small-scale experimental bridge
model, focussing on the first eigenmodes. This was achieved by comparing the frequency response functions obtained from the numerical model with those from the
experimental model, c.f. Figure 10. The experimental model was excited using the
impact hammer. Points with accelerometers 5 and 7 were checked for an excitation at
point 1A (see Figure 5).
The calibration was done for four different cases:
• Case A: The bridge model is fixed on a solid surface (the laboratory floor) and
no rail is present,
• Case B: The bridge model is fixed on a solid surface (the laboratory floor) and
the rail is bolted to the bridge deck,
• Case C: The whole structure is placed on one layer of mattress foam,
• Case D: The structure is placed on two layers of mattress foam.
For the Case A, the results obtained from the two different models follow the same
path: the first eigenmode is approximately at 46Hz, while the second one appears
at around 128Hz. The results for Case B show that the computational model is able
to predict the experimental behaviour quite well. Compared to Case A it is seen that
bolting the railway track to the bridge deck makes the structure more stiff and therefore
the eigenfrequencies are higher—for the first eigenmode it is around 61Hz and for the
second one approximately 168Hz. Another effect of the track is an increased damping
in the system which can be seen by the increased width of the peaks in the frequency
response functions.
Cases C and D show that the difference between the measured and computed results increases compared to the previous cases, but the results still match well. The
difference may be caused by the boundary conditions of the experimental model that
are not correctly accounted for in the computational model. Thus, the horizontal extent of the soil (i.e. the foam mattresses) is finite in the experimental set-up; but it is
assumed to be infinite in the computational model.
11
100
Numerical
Experimental
Case A
80
FR [(m/s 2)/N]
FR [(m/s 2)/N]
100
60
40
20
0
0
50
100
150
80
60
40
20
0
0
200
25
50
100
150
200
50
100
150
200
50
100
150
200
50
100
150
Frequency [Hz]
200
25
20
FR [(m/s 2)/N]
FR [(m/s 2)/N]
Case B
15
10
5
0
0
50
100
150
20
15
10
5
0
0
200
15
15
5
0
0
15
FR [(m/s 2)/N]
FR [(m/s 2)/N]
10
50
100
150
5
50
100
150
Frequency [Hz]
5
15
Case D
10
0
0
10
0
0
200
FR [(m/s 2)/N]
FR [(m/s 2)/N]
Case C
200
10
5
0
0
Figure 10: FRF comparison between experimental and numerical models for Case A
(bridge without railway), Case B (bridge with railway), Case C (bridge on one layer
of foam), and Case D (bridge on two layers of foam). Graphs on the left present data
from accelerometer 5, while graphs on the right are from accelerometer 7.
12
In either case, the first eigenfrequency decreases—for Case C to 24Hz and for Case
D 22Hz. Overall, the behaviour is different from Cases A and B, but the difference
between having one layer and two layers of mattress foam is relatively small. The
FRF analyses indicate that the computational model is able to reproduce the measured
response in the low-frequency range with adequate accuracy.
As a main result of the FRF analysis, calibrated values of the material properties are
obtained. Values for the bridge model parameters are given in Table 1. It is recalled
that the moment of inertia for the deck around the local y-axis includes the effect of
the track. As a consequence of this, it varies along the length of the bridge to account
for reduced stiffness around track segment joints and increased stiffness around bolts
connecting the track to the Plexiglas beam. With reference to Table 1, Iy = Ic,y is
used for the stiffer parts where track is bolted to the deck, while Iy = Id,y is applied to
the rest of the deck. The properties for the vehicle are listed in Table 2. The mass of
each bogie given in the Table 2 is idealised in order to make the computational model
work properly. Finally, the values of parameters used for the soil (mattress foam) are
provided in Table 3.
Deck, Id,y
Moment of inertia around y-axis
Deck, Ic,y
Column, Ip,y
Deck, Id,z
Moment of inertia around z-axis
Column, Ip,z
Deck, Jd
Torsional constant
Column, Jp
Deck, Id,0
Polar inertia moment around x-axis
Column, Ip,0
Young’s modulus
E
Density
ρ
Poisson’s ratio
ν
2.06 · 10−9
2.5 · 10−9
5.73 · 10−8
7.2 · 10−7
1.06 · 10−6
4.8 · 10−9
3.55 · 10−8
7.21 · 10−7
1.11 · 10−6
2.5 · 109
1182
0.37
m4
m4
m4
m4
m4
m4
m4
m4
m4
N/m2
kg/m3
-
Table 1: Calibrated cross-sectional properties and material parameters of the bridge
Primary suspension stiffness
Ks,1
Secondary suspension stiffness
Ks,2
Primary suspension damping
Cs,1
Secondary suspension damping
Cs,2
Hertzian spring constant
KH
Mass of car body
Mc
Mass of bogie
Mt
Mass of wheel set
Mw
Pitch moment of inertia for car body Jc
Pitch moment of inertia for bogie
Jt
785
320
4
1
10000
0.5
0.1
0.06
0.0015
1.62 · 10−5
Table 2: Calibrated properties of the vehicle
13
N/m
N/m
N· s/m
N· s/m
N/m3/2
kg
kg
kg
kg·m2
m4
55000 N/m2
27200 N/m2
37.50 kg/m3
0.01
-
Young’s modulus E
Shear modulus
G
Density
ρ
Poisson ratio
ν
Table 3: Calibrated material properties of mattress foam
5
Results and comparison
For Cases B, C and D described in Section 4, the magnitudes of acceleration in the
frequency domain were measured for a vehicle passing with three different speeds:
V0 = 0.53m/s, 0.97m/s and 1.31m/s. The same speeds were analyse with the calibrated computational models. The response at the middle of the bridge, i.e. at accelerometer 5 (cf. Figure 5) was analysed. The experimental data was averaged from
ten tests. From the computational as well as experimental data, the root-mean-square
values of acceleration hAiRMS are determined in each 1/3 octave frequency band:
r
hAiRMS =
1
(A1 A∗1 + A2 A∗2 + · · · + An A∗n ),
n
(9)
where A1 , A2 , etc., denote the Fourier coefficients of the accelerations at the n frequencies in a given 1/3 octave band, and A∗1 , A∗2 , etc., denote their complex conjugates.
The values are converted into decibel using the reference acceleration value 1µm/s2 :
2
hAiRMS [dB re 1µm/s ] = 20 log10
hAiRMS
A0
,
A0 = 1µm/s2 .
(10)
Figure 11 illustrates the results for Case B. Computational and experimental acceleration response data provide the highest acceleration in the 63Hz 1/3 octave band,
where the first eigenfrequency of the structure lies. The tested speeds do not provide significant difference. In general, at the lower frequencies the magnitudes of
accelerations are lower than recorded at higher frequencies, and overall the numerical model underestimates the magnitudes of accelerations. The differences are up to
about 20dB, corresponding to a factor 10. This is judged to be a result of missing
track surface irregularities in the computational model, and the vehicle model may
need better calibration. For comparison it should be noted that a calculation based
on a computational model with no vehicle, but with a set of constant travelling forces
applied to the track, leads to far worse results. Thus, without explicitely modelling the
vehicle, differences of more than two orders of magnitude were observed between the
measured and calculated results.
14
<A>RMS [dB re 1µm/s 2]
100
80
60
40
Experimental 0.53m/s
Numerical
0.53m/s
Experimental 0.97m/s
Numerical
0.97m/s
Experimental 1.31m/s
Numerical
1.31m/s
20
0
12.5
16
20
25 31.5 40
50
63
80 100
1/3 octave band centre frequency [Hz]
125
160
Figure 11: Accelerations in frequency domain; Case B (bridge with railway)
<A>RMS [dB re 1µm/s 2]
100
80
60
40
Experimental 0.53m/s
Numerical
0.53m/s
Experimental 0.97m/s
Numerical
0.97m/s
Experimental 1.31m/s
Numerical
1.31m/s
20
0
12.5
16
20
25 31.5 40
50
63
80 100
1/3 octave band centre frequency [Hz]
125
160
Figure 12: Accelerations in frequency domain; Case C (bridge on one layer of foam)
Cases C and D, shown in Figure 12 and Figure 13, illustrate that numerical and
experimental results follow the same trend, especially at higher frequencies peaking in
15
<A>RMS [dB re 1µm/s 2]
100
80
60
40
Experimental 0.53m/s
Numerical
0.53m/s
Experimental 0.97m/s
Numerical
0.97m/s
Experimental 1.31m/s
Numerical
1.31m/s
20
0
12.5
16
20
25 31.5 40
50
63
80 100
1/3 octave band centre frequency [Hz]
125
160
Figure 13: Accelerations in frequency domain; Case D (bridge on two layers of foam)
the 80Hz 1/3 octave band. However, as in Case B the accelerations are underestimated
by the computational models. For Case D, data from the two models match better
which is slightly surprising since the frequency response is not modelled with higher
accuracy, cf. Figure 10.
Generally, it can be seen that adding the mattress foam in the experimental setup decreases the first eigenfrequency, thus making the response within the 20Hz and
25Hz 1/3 octave bands higher. However, the acceleration response for all three cases
is similar at higher frequencies. While the FRFs for computational and experimental
models match well, the acceleration response for the passing vehicle is constantly underestimated in the numerical model. It can be concluded that differences arise from
the vehicle and track irregularities used in the numerical model which does not properly recreate all the effects caused by a small-scale vehicle used in the experiments.
6
Conclusions
Dynamic analysis of a multi-span railway bridge was performed. For that purpose a
small-scale experimental model was built and compared with a computational model.
The effects caused by an underlying soil, a passing vehicle and track irregularities
were taken into account. It was shown that the underlying soil has a significant effect
on the overall behaviour of the structure, changing the eigenfrequencies and adding
more damping to the system as would be expected. The frequency response functions
show a good agreement between the two models after the material and cross-sectional
16
properties of the compuational model had been calibrated. When the structure was
excited by a passing vehicle, the two models behave in a similar manner. However, the
results start to differ more. This illustrates that a proper modelling of the vehicle and
track irregularities is important for realistic dynamic structural response evaluation.
An existing experimental model could be improved by implementing a better defined vehicle and track system, thus making the results more reliable. Further, experiments could be carried out with different combinations of parameters. Adding extra
mass on the bridge or changing the span length would change the eigenfrequencies
of the structure, and soaking the mattress foam with water would decrease the wave
speeds in the material, thus making the critical speeds easier to reach experimentally.
To summarize, the structure–soil–structure and vehicle–structure–vehicle interaction are important phenomena. Disregarding either phenomenon severely diminishes
the accuracy of the model. Calibration of a computational model is higly important.
The model must account for all physical aspects related to the dynamic interaction
problem in a proper manner—qualitatively as well as quantitatively. The present study
indicates that such model development and calibration can be achieved, though some
improvement, validation and calibration are still necessary.
Acknowledgement
The authors would like to show their gratitude to Anders Vabbersgaard Pedersen for
R
generously making the LEGO
model train available. Further, the PULSE LabShop
softwave and the front-end hardware was put at the disposition of Aalborg University
by Brüel & Kjær Sound & Vibration Measurement A/S at no cost. This support is
highly appreciated.
References
[1] L. Meirovitch, “Analytical methods in vibrations”, MacMillan, London, United
Kingdom, 1967.
[2] L. Fryba, “Vibration of solids and structures under moving load”, Noodhoff International, Groningen, The Netherlands, 1972.
[3] J.R. Rieker, M.W. Trethewey, “Experimental evaluation of beam structures subjected to moving mass trains”, in “Proceedings of the IMAC-XII International
Modal Analysis Conference”, Honolulu, Hawaii, USA, Vol. 2, 968–974, 1994.
[4] A.V. Metrikine, K. Popp, “Instability of vibrations of an oscillator moving along
a beam on an elastic half-space”, European Journal of Mechanics, A/Solids 18
(2), 331—349, 1999.
[5] G. Lombaert, G. Degrande, J. Kogut, S. Franc-ois, “The experimental validation
of a numerical model for the prediction of railway induced vibrations”, Journal
of Sound and Vibration 297, 512-–535, 2006.
17
[6] X. Sheng, C.J.C. Jones, and M. Petyt, “Ground vibration generated by a load
moving along a railway track”, Journal of Sound and Vibration, Vol. 228, No. 1,
129-–156, 1999.
[7] X. Sheng, C.J.C. Jones, M. Petyt, “Ground vibration generated by a harmonic
load acting on a railway track”, Journal of Sound and Vibration 225 (1), 3-–28,
1999.
[8] J.W. Kwark, B.S. Kim, Y.J. Kim, E.S. Choi, “Dynamic behavior of the bridge
for KHSR due to high-speed train”, Structural Dynamics, EURODYN2002, 205–
209, 2002.
[9] R. Karoumi, J. Wiberg, A. Liljencrantz, “Monitoring traffic loads and dynamic
effects using an instrumented railway bridge”, Engineering Structures 27, 1813–
1819, 2005.
[10] H. Xia, N. Zhang, “Dynamic analysis of railway bridge under high-speed trains”,
Computer and structures, Vol. 83, 1891–1901, 2005.
[11] H. Nassif, P. Lou, Y. Wang, “Dynamic Modeling and Field Testing of Railroad
Bridges”, in “Proceedings of the 31st IMAC, A Conference in Structural Dynamics”, Topics in Dynamics of Bridges, Vol. 3, 2013.
[12] H.G. Harris, G.M. Sabnis, “Structural modeling and experimental techniques”,
CRC Press, New York, USA, 2000.
[13] C. Bilello, L.A. Bergman, D. Kuchma, “Experimental investigation of a smallscale bridge model under a moving mass”, Journal of Structural Engineering 130
(5), 799–804, 2004.
[14] D. Stancioiu, S. James, H. Ouyang, J.E. Mottershead, “Vibration of a continuous beam excited by a moving mass and experimental validation”, Journal of
Physics: Conference Series 181, 2009.
[15] P. Bucinskas, L. Agapii, J. Sneideris, L.V. Andersen “Numerical modelling of
dynamic response of high-speed railway bridges considering vehicle–structure
and structure–soil–structure interaction”, 2015.
[16] L. Andersen, J. Clausen “Impedance of surface footings on layered ground”,
Computers and Structures 86 (1–2), 72–87, 2008.
18
Appendix C
Numerical code
The calculations described in this thesis are performed by numerical codes written in Matlab
software. The main code that includes 3-D bridge structure, modelled with rail, rail-pads, sleepers, ballast connected with the deck, non-linear wheel–rail interaction, structure–soil–structure
interaction (SSSI) etc. is called NUM_FULL.m, it is included in the USB flash drive attached.
A detailed flow chart illustrating the principles of the code is given in the next pages. The
flow chart is separated in to number of areas “blocks” (separated by dashed lines), which each
represent different parts of the code. Each area has a number and a name (written in bold text,
e.g. 1.1 Input. General settings), the same system is used in the code as well, thus it is easy to
cross-reference the flow chart and the code itself. The main variables used in the codes are also
shown in the flow chart (written in italic text, e.g. coord_fix).
Main “blocks” shown in the flow chart are :
1.1 Input. General settings. General settings for the code. It can be chosen (run_multi) if a
new soil impedance matrix should be calculated, or a previously calculated one loaded; if a new
vertical track profile should be generated (new_unevenness), or an old one loaded; if soil surface
displacements should be found (run_soil_dis); what plot type should be displayed (plot_type).
The identifier for the calculations is set, which is later used in the naming of output files. Finally,
local and global iteration settings are defined including the tolerances (glob_iter_tolerance and
iter_tolerance).
1.2 Input. Time. The model uses a number of different time series: so called local (t_loc),
global (t) and calculation (t_cal) time. Local time series are used for the vehicle calculations,
including local iteration, the time steps are very small in order for the local iteration to work
properly and the time series ends, when the vehicle leaves the bridge. The global time is used for
the bridge–soil system calculations, the time steps are significantly bigger and more time is left
after the vehicle leaves the bridge to let the vibrations to dissipate, since time steps are relatively
big, a smaller number of frequencies need to be computed. The frequencies obtained from the
global time series are used for the soil impedance matrix calculations. Finally calculations time
is introduced for the final results of bridge-soil system as well as soil displacements, it allows
to calculate a chosen time window from global time with bigger accuracy. Multiple time series
are needed in order to optimize the performance and reduce the computation time of the code.
Parameters time_skip and time_skip_loc describe when the first wheel of the vehicle reaches the
deck.
1.3 Input. MDOF. Input for the vehicle. Primary and secondary suspension system (K_s1,
K_s2, C_s1 and C_s2), the force acting on the wheel (move_F and wheel Hertzian spring constant (G_wheel) need to be defined.
129
Track unevenness is also defined here, firstly the length of the track that has unevenness is
defined (uneven_L), the uneven track is only defined on top of the deck, the track before and
after the bridge is considered to be level. Then the number of wave numbers used for track
unevenness generation is defined (uneven_points), as well as the constant factors (k_c, k_r and
A_tr).
1.4 Input. Bridge properties. Material and cross-sectional parameters of the bridge, that
includes rail, rail-pads, sleepers, ballast, deck and pylons. The exact parameters needed can be
seen in the flow chart. The moments of inertia are given in local coordinate system.
1.5 Input. Bridge geometry. The FEM mesh for the model is generated automatically from the
parameters defined here. Firstly the numbers of spans and pylons are defined (num_span and
num_support). Later variables dx_span, dx_support, dx_track. The spacing between sleepers
spacing_sleeper and z- coordinate in level_sleeper. The level_sleeper is only used for definition
of coordinates for the nodes and plotting, thus it has no effect on the system stiffness.
The starting nodes for the spans are defined in coord_span, there should always be a node
where deck begins and ends, as well as at points connecting the deck and pylons. Variable
coord_support defines the top node (lines 1,3,5 etc.) and bottom node (lines 2, 4, 6 etc.). There
should be a shared connecting node in coord_span and coord_support. To define the track
before and after the bridge, variable coord_track is used, whereas the track over the bridge deck
has nodes obtained by offsetting the nodes of the deck by a distance in z-direction.
Variables xy_span and xy_support define the local x,y-plane, used for correct orientation of
every element. Nodes coupled with soil are defined in soil_coord and fixed nodes in coord_fix.
These nodes have to be already defined in the geometry.
The surface footings are defined by half width r0, height h0, scale and orientation orien.
1.6 Input. Soil. The soil parameters are defined by Young’s modulus (E_soil), Poisson’s ratio
(nu_soil), mass density (rho_soil), damping ratio (eta_soil) and layer height (h_soil). Each soil
layer is defined by a separate value. To define a half-space, the layer height is set to 0.
The points on the soil surface where the displacements should be calculated are defined in x_soil
and y_soil. The origin for this coordinate system is the centre of the first footing.
Further settings for footings shape, discretization settings etc. need to be defined as well, as
shown in the flow chart.
2.1 MDOF. Vehicle FEM matrices. Stiffness, mass and damping matrices for the vehicle are
set.
2.2 MDOF. Track unevenness. If a new track profile is generated- a PSD function is calculated
and by assigning random phase angles the Fourier coefficients for track unevenness are found.
If new track unevenness is not generated, an old file is loaded.
3.1 Bridge. Mesh generation. A mesh for the FEM model is created, by defining the coordinates, topology etc. Topology (Topo_bridge, Topo_deck, Topo_support) defines which nodes
are connected and node numbers are defined by line number in Coord matrix.
130
C. Numerical code
3.2 Bridge. FEM matrices FEM matrices are generated for the full bridge structure- firstly for
the deck and pylons, later rail and sleepers are added.
4.1. Soil. Impedance matrix for soil Impedance matrix for surface footings is calculated by
calling an external file Multipod_fix.exe. The .exe code is built using a code written in Fortran
which was provided by thesis supervisor Lars Vabbersgaard Andersen.
5.1 Iteration. Setup Each vehicle wheel positions in local (t_loc) and global time (t), as well
as unevenness values for each position (un_position) are found.
5.2 Iteration. MDOF Start of the global iteration. Vehicle movements are calculated in time
domain, using local iteration scheme in local time. Results are transformed to global time to
use in the next section.
5.3 Iteration. Bridge. Axial forces acting on the wheels are applied to the rail. The impedance
matrix for the bridge structure in frequency domain is established and the footing impedance
matrix is added. The obtained results are inverse Fourier transformed to time domain and later
transformed to local time. If convergence of global iteration is reached, the code continues
forward; if not section 5.2 Iteration. MDOF is repeated.
5.4 Iteration. Conversion to “calculations” time. Bridge displacements from iteration are
converted to calculations time series (t_cal).
6.1 Soil displacements. Calculations. An optional section. Soil surface displacements calculations are performed. Load the .txt files created by Multipod_fix.exe, if the files were loaded
before the same data is read from a .mat file. This is done because the .txt files takes a long time
to load, loading .mat files is considerably shorter. The soil displacements are found by multiplying soil displacements from unit footing displacements by the real footings displacements,
in frequency domain. Then the results are transformed to time domain using calculations time
(t_cal).
7.1 Plotting Plots a chosen figure. Possible to plot figures:
1- Displacement animation, bridge displacements excluding soil, e.g. Figure C.1.
2- Geometry with node numbers. Plots all the node numbers in the structure, e.g. Figure C.2
and Figure C.3.
3- Topology check, used to verify that the geometry of the bridge is generated correctly. If
topology is generated wrong there would be missing links between the nodes, e.g. Figure C.4.
5- Soil displacements, animation of just soil surface displacements, e.g. Figure C.5.
6- Structure and soil combined, animation of bridge as well as soil surface together, e.g. Figure C.6.
7- GIF file, creates .gif file of previous plot.
8- Displacement animation of the vehicle, including axial forces and track profile, e.g. Figure C.7.
9- Track vertical profile, e.g. Figure C.8.
10- Axial forces acting on a wheel, plotted as a function of x-coordinate, e.g. Figure C.9.
C. Numerical code
131
z coordinate [m]
Displacements. u X 2000.
Time 1.98 s.
4
2
0
−2
−4
−6
0
50
100
150
x coordinate [m]
200
−50 5
y coordinate [m]
z coordinate [m]
Figure C.1: A frame from bridge displacements animation
0
−2
−4
−6
1
3
22
78
4
28
9
30
35
2
33
36
4
35
3
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x coordinate [m]
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y coordinate [m]
Figure C.2: Geometry with nodes numbers
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Figure C.3: Geometry with nodes numbers (view zoomed to the deck–column joint
132
C. Numerical code
z coordinate [m]
0
−2
−4
−6
0
50
100
150
200
x coordinate [m]
y coordinate [m]
z coordinate [m]
Figure C.4: Topology checking plot
Displacements. u X 1000.
Time 1.98 s.
5
0
−5
0
50
100
150
x coordinate [m]
−50 5
−10
y coordinate [m]
z coordinate [m]
Figure C.5: A frame from soil displacements animation
Displacements. u X 1000.
Time 1.98 s.
2
0
−2
−4
−6
−8
0
50
100
150
x coordinate [m]
−50 5
−10
y coordinate [m]
Figure C.6: A frame from soil-structure system displacements animation
C. Numerical code
133
6
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Track profile
x 10
Unevenness
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Acting forces
150
200
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5
Axial force
x 10
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0
Figure C.7: A frame from vehicle displacements animation
−3
Y coordinate, [m]
5
Track profile
x 10
0
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0
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120
X coordinate, [m]
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Figure C.8: Track vertical profile
5
Axial force
4
Acting forces
x 10
3
2
1
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140
Figure C.9: Axial forces acting on the wheels
134
C. Numerical code
1.5 Input. Bridge geometry
num_span - number of spans
num_support - number of supports
Starting and ending coordinates for
structure sections:
coord_span - span sections
coord_support - pylons
coord_track - rail sections
Defining local x,y- plane::
xy_span - deck
xy_support - pylons
Sleepers distribution:
level_sleeper - z coordinate for all
spacing_sleeper - spacing in between
3.1 Bridge. Mesh generation
3.2 Bridge. FEM matrices
Nodes defined in the input:
1.4 Input. Bridge properties
Material properties of bridge:
E - Young's modulus
G - Shear modulus
rho - deck and pylons mass density
rho_found - foundation mass density
zeta - damping ratios for damping
FEM for deck and pylons:
coord_track
coord_span
z
z y
coord_support
x
Creating nodes for sleepers:
z
Deck cross-sectional properties (local
coordinate system):
I_y_d -moment of inertia around y-axis
I_z_d -moment of inertia around z-axis
J_d - torsion constant (around x-axis)
I_0_d - polar moment of inertia
A_d - cross-sectional area
L_m - dist. between bridge/rail centres
x
FEM for rail:
spacing_sleeper
level_sleeper
Pylon cross-sectional properties (local
coordinate system):
I_y_c -moment of inertia around y-axis
I_z_c -moment of inertia around z-axis
J_c - torsion constant (around x-axis)
I_0_c - polar moment of inertia
A_c - cross-sectional area
z y
x
x
Creating extra nodes for deck, rail and pylons:
dx_track
z
1.1 Input. General settings
Calculate soil impedance matrix?
run_multi
Simulate new track unevenness?
new_unevenness
Calculate soil displacements?
run_soil_dis
Save full output?
save_full
Choose results to display:
plot_type
Global iteration settings:
glob_iter_tolerance - tolerance
max_loops - maximum loop number
Local iteration settings:
iter_tolerance - tolerance
betta, gamma- Newmark integration
constants
x
Railpad, sleeper, ballast properties:
K_p - railpad stiffness
C_p - railpad damping
K_w - ballast stiffness
C_w - ballast damping
m_s - sleeper mass
m_bl - ballast mass
Fixed and coupled with soil nodes:
Adding surface footings mass to mass matrix
z
FEM matrices for the whole bridge structure:
K_b - stiffness
C_b - damping
M_b - mass
- fixed
- coupled with soil
x
Adding surface footings:
1.3 Input. MDOF
Mass of vehicle parts:
vec_b_mass - bogie
vec_w_mass - wheel set
vec_c_mass - car body
2.1 MDOF. Vehicle FEM matrices
MDOF vehicle FEM matrices:
K_v - stiffness
C_v - damping
M_v - mass
z y
Distances:
vec_b_lenght - wheel base
vec_c_lenght - dist. between bogies
centre lines
z
x
x
Full system variables:
Node_Dof - stores numbers of dof for each node
Coord - coordinates of every node (line number node number)
Q - boundary conditions
Load file Unevenness.mat, that contains:
A_comp-Fourier coefficients for unevenness
un_k - wave-numbers for unevenness
Variables describing deck/pylons geometry:
Topo_bridge - deck and pylons topology (node
numbers for each element)
Topo_deck - deck topology
Topo_support - pylons topology
Variables describing rail geometry:
Topo_rail - rail topology
Variables describing sleepers geometry:
Topo_sleepers - sleepers topology
Nodes_sl - rail/deck nodes that sleepers couple to
Suspension system:
K_s1 - primary suspension stiffness
K_s2 - secondary suspension stiffness
C_s1 - primary suspension damping
C_s2 - secondary suspension damping
2.2 MDOF. Track unevenness
Unevenness
Surface footings settings:
r0 - half width of each
h0 - height of each
scale - ratio between length and width
orien - orientation
z y
dx_support
Vehicle
Nodes coupled with soil:
soil_coord
Rail properties:
E_rail - Young's modulus
rho_rail - mass density
I_y_rail - moment of inertia (y-axis)
A_rail - cross-sectional area
dx_span
x
Fixed nodes:
coord_fix
Connecting rail and deck through sleepers:
dx_track
Bridge
Number of FEM elements for each:
dx_span - span section
dx_support - pylon
dx_track - rail section
Generate unevenness PSD:
un_S
Random phase angles:
phi
Generate unevenness for specified settings:
A_comp-Fourier coefficients for unevenness
un_k - wave-numbers for unevenness
No
Yes
Wheel Hertzian constant:
G_wheel
Force acting on a wheel:
move_F
Simulate new track unevenness?
new_unevenness
Track unevenness:
uneven_L - length for unevenness
uneven_points - no. of wave-numbers
k_c, k_r, A_tr - constant factors
Local time properties:
dt_loc - size of time step
time_steps_loc - number of time steps
time_skip_loc - time offset
Global time properties:
dt - size of time step
time_steps - number of time steps
time_skip - time offset
1.6 Input. Soil
Number of footings:
nPod
Points on soil surface for output:
x_soil - x-coordinate
y_soil - y-coordinate
Soil properties (value for each layer):
E_soil - Young's modulus
nu_soil - Poisson's ratio
rho_soil - mass density
eta_soil - damping ratio
h_soil - height of layer
Soil code settings (Multipod_fix.exe):
intRes - find results by interpolation
allOut - produce full output
Footing settings:
lShape - load type
damTyp - damping type
fShape - shape of footing
Discretization settings:
nWo -no. of wave-numbers in Fourier space
nRad - no. of radii in spacial discretization
nPts - no. of points in radial direction
a0max- max, dimensionless frequency
1.2 Input. Time
Calculations time properties:
dt_cal - size of time step
time_steps_cal - number of time steps
time_skip_cal - time offset
5.1 Iteration. Setup
Unevenness value for every wheel position
for t_loc:
un_position
Vehicle position for every global time step:
position
Global time series:
t
5.3 Iteration. Bridge
4.1 Soil. Impedance matrix for soil
Calculate soil impedance matrix?
5.2 Iteration. MDOF
Apply F_wheel_global to the bridge deck:
F - force matrix for the bridge structure
(column- time step, line- dof)
Yes Call file Multipod_fix.m:
the input is structurized for .exe file to read
Start global iteration:
ddd - global iteration step
Next global iteration step:
ddd=ddd+1
Fourier transform of matrix F:
F_fft
No
Load earlier calculated files
Input is written to file Multipod.txt
Call Multipod_fix.exe:
reads input from Multipod.txt
Start bridge/soil system calculation for every
frequency, till Nyquist frequency:
ii - frequency step number
omega - circular frequency from global time
Next time step:
i=i+1
Estimates values for vehicle:
u_star - displacements, for time step
v_star - velocities, for time step
Calculate bridge impedance matrix for
omega(ii):
D_freq- impedance matrix
Output, footings impedance matrix:
ReZ .txt - real part of soil impedance matrix, for each frequency
ImZ .txt - imaginary part of soil impedance, for each frequency
Start local iteration for time step:
countr - local iteration counter
Next local iteration step:
countr=countr+1
Add footings impedance to connecting nodes
for omega(ii)
Read .txt file and create complex:
Z - footing impedance matrix
Output, soil surface displacements:
ReU .txt -real part of soil displacements from unit foundations disp.
ImU .txt - imag. part of soil displacements from unit foundations disp.
Find force acting in each wheel using
u_wheel_loc and un_position:
F_MDOF- full vehicle force matrix
(column- time step, line- dof)
Solve system and obtain:
U_red(:,ii) - displacements in frequency
domain for omega(ii)
omega(ii+1)>Nyquest frequency
No
Determine for time step (column- time step,
line- dof):
a_MDOF - accelerations
v_MDOF - velocities
u_MDOF - displacements
Yes
Inverse Fourier transform to time domain:
u - bridge displacements in global time
6.1 Soil displacements. Calculations
Read .txt files and create complex:
unit_U - unit displacements of soil
Save variable unit_U to unitU.mat file
Find rail displacements under every wheel:
u_wheel - displacements in global time
Read unitU.mat file:
unit_U - unit displacements of soil
Transform u_wheel to local time:
u_wheel_loc - displacements in local time
Extract footings displacements:
load_U - displacements in frequency domain
Multiply:
Freq_U = load_U x unit_U
Start vehicle calculations with t_loc:
i - local time step number
Next frequency step:
ii=ii+1
Output:
frq .txt - frequencies and frequency numbers
Yes
Calculations time series:
t_cal
Unevenness
Vehicle position for every local time step:
position_loc
Vehicle
Local time series:
t_loc
Bridge
1.2 Input. Time
Vehicle travelling speed:
speed
No
Check for local convergence:
diff (local convergance)<iter_tolerance
Yes
No
t_loc(ii+1)>max(t_loc)
Check for global convergence:
glob diff (global convergence )<glob_iter_tolerance
And is maximum loop count reached:
ddd < max_loops
No
Yes
Transform obtained forces from local time to
global time:
F_wheel - forces in local time
F_wheel_global - forces in global time
Calculated before?
Yes
No
Yes
Inverse Fourier transform using t_cal:
Time_U - soil surface displacements
Calculate soil displacements ?
Transform u to calculations time:
u_cal - bridge displacements in t_cal
No
Save full output?
5.4 Iteration. Conversion to
"Calculations" time
Save full output to file:
FullOutput.mat
7.1 Plotting
Display results according to plot_type
Appendix D
Experimental testing of a small–scale
bridge model
In this appendix, more details about an experimental small–scale bridge model testing are given,
which are not mentioned in the paper "Experimental validation of a numerical model for threedimensional railway bridge analysis by comparison with a small–scale model". Firstly, construction solutions for particular parts of the structure itself are described, including pictures of
the real system. Also photographs of the vehicle and measuring equipment are presented. Later,
tests performed to obtain frequency response functions are described briefly, including response
graphs, for impact hammer and one of the accelerometers placed on the bridge. Subsequently,
follows an explanation about tests done with a passing vehicle, introducing a dynamic response
of the structure, obtained at one of the observation points, in frequency and time domains.
Construction solutions for particular parts of the system
The first step in building the experimental model was consideration of the overall size of the system. The larger the model, the more realistic dynamic behaviour of the bridge structure could
be recreated. However, it was decided to build a small–scale model due to space limitations
in the laboratory and since it is relatively cheap and easy to build. The next step was selection
of materials where the main factor was structure stiffness: the bridge should not be too slender
so it would deflect due to its own weight, but it should not be too stiff as well because later it
would be hard to excite it using a reasonable amount of mass. Finally, Plexiglas was chosen
for the bridge structure. To substitute the underlying soil, mattress foam is used since the ratio
between real structural bending stiffness and soil stiffness was maintained using Plexiglas for
the bridge and mattress foam for the soil. Also this material has open pores, so it can be soaked
with water, thus changing its parameters. The whole system for experimental testing is shown
in Figure D.1.
Before building a model, preliminary static tests were done using different kinds of Plexiglas
pieces and mattress foams in order to determine material properties and chose the most suitable
ones. Therefore, simple weighting and geometry measuring was done to obtain mass, dimensions and density. Also bending tests applying static load were performed to determine Young’s
modulus.
The next step, was assembling the model. Firstly, holes were drilled through the bridge deck
on the centre line for every 0.127m to bolt the railway pieces together with the deck. Also pairs
of holes, centred by centre line, were drilled through the deck every 0.12m to fix the columns.
139
They were made repeating the same distance periodically in purpose to change a span length.
Similarly pairs of holes were made in columns and footings so, finally, all separate pieces were
screwed together making a rigid connection as shown in Figure D.2.
Figure D.1: The whole system used for experimental tests
Further, when materials were selected, the separate parts of the model were made. The idea
was to make a model using as few pieces as possible to avoid unnecessary connections and
thus, contacts between separate pieces which could introduce unfavourable vibrations in the
system. On the other hand, it was considered to make the model light and comfortable enough
to transport it by hands. Therefore, the bridge deck was made as a solid piece of 3m length,
while columns and footings were cut of Plexiglas separately by producing 16 pieces of the
same size, to build 8 columns with footings. For the tests bridge model, 6 supports were used.
However, extra pieces were made in case tests with decreased span length were to be performed.
Railway pieces were bolted to the bridge deck strong enough just to hold them in their positions.
By bolting too hard, the piece deflects in the middle and introduces a U-shape, thus the whole
track becomes “wavy” which is an unfavourable form for a track. Moreover, small pieces of
rubber were glued under every sleeper and the whole railway piece was glued to the bridge deck
as shown in Figure D.2. In this way, the hard contact between track and deck was avoided and
therefore more realistic structural behaviour was achieved.
140
D. Experimental testing of a small–scale bridge model
Figure D.2: Frontal view of deck–column and column–footing connection (top left),
deck–column connection from the top (top right) and railway bolted to the
bridge deck (bottom)
At both ends of the bridge, ramps were made of concrete tiles, at the same level as a track, to
provide a space for extra track for train acceleration and deceleration as shown in Figure D.3.
The ramps were built stable and heavy so they could not introduce any motion during the tests
but they were also considered to be fairly light for transportation purpose. Due to that tiles were
chosen instead of a single solid block.
At each end of the bridge, 0.25m of the deck together with bolted railway track were glued to
rubber pads underneath and placed on the ramps. The rubber pads were used to prevent a hard
contact between the Plexiglas deck and the concrete tiles when the vehicle was entering and
leaving the bridge. Fixed supports at both bridge ends were installed by placing steel blocks on
the sides of the deck and aligning them to the sides of the concrete ramps, i.e. imitating clamps,
cf. Figure D.3. A strip of paper was taped between two steel blocks on the top and in the middle
of it an accelerometer was glued, as shown in Figure D.3. The accelerometer was connected
with a slim wire which was disturbed every time by a passing vehicle. Using this small set-up
at both ends of the bridge, a system to measure the vehicle speed was created.
D. Experimental testing of a small–scale bridge model
141
Figure D.3: Clamping and speed measuring system on acceleration ramp (top), clamping and speed measuring system on deceleration ramp, also a break at the
end of the track (bottom)
An extra track of 1.14m length was used for acceleration of the vehicle so the required speed for
the test could be achieved before the locomotive reached the bridge. Thus, the vehicle travelled
through the bridge at a constant speed. Similarly 0.76m of track was added after the bridge for
deceleration. The extra track at both ends was placed on multiple layers of polystyrene again to
prevent hard contact between railway track and concrete tiles. At the end of the track a break
was installed to stop the vehicle completely as shown in Figure D.3. A piece of foam with
additional mass on the top was used for a break. Therefore, stopping was soft and the vehicle
was not damaged.
As it is already known a LEGO® vehicle powered by an electric motor was used in the tests,
cf. Figure D.4. To reduce unfavourable vibrations in the vehicle itself it was considered to
strengthen or simply glue most of the separate pieces. Therefore, extra mass was glued to the
LEGO® platform and bogies were fixed together to car body using strips of adhesive coated
rubber. As a result, only the wheels of the vehicle could roll freely. The speed of the vehicle
was adjusted by a controller, cf. Figure D.4.
142
D. Experimental testing of a small–scale bridge model
Figure D.4: Vehicle (top), speed controller (bottom)
The subsoil was substituted by mattress foam which was placed in a box made of waterproof
plywood sheets by screwing them together on the sides and the bottom. The gaps in the connections were filled with silicon paste so, the box could keep water inside. This was done in
purpose to do tests with soaked mattress foam. However, firstly tests were performed using a
single layer of dry mattress foam. For that, it was cut just to fit in the box such that, it could
not slide or introduce any other motion. It was glued with the same silicon paste to the bottom
of the box through all bottom surface area, to ensure stationary contact. Finally, the whole box
with a mattress inside was placed on a layer of 0.02m rubber, cf. Figure D.1, to ensure good
contact with the concrete floor in the laboratory and to reduce noise in the system caused by
other working machines in the building. For the test case with two dry foam layers, the second
identical mattress foam piece was cut and simply placed on top of the first layer. The last tests
were performed using one layer of soaked foam. To prevent the foam from floating, concrete
tiles were placed on its top, distributing them over the whole area as shown in Figure D.5. Later
the box and foam were filled by water pouring it inside using buckets, where a capacity of one
bucket was 16l. The procedure of soaking foam was done slowly by stepping on every tile, in
this way squeezing air out of the pores. Finally, 264l of water was used to saturate the mattress,
the tiles were removed and the bridge model was simply placed on the top of the wet foam.
D. Experimental testing of a small–scale bridge model
143
Figure D.5: Tiles distributed over the surface of mattress foam during soaking procedure
144
D. Experimental testing of a small–scale bridge model
Measuring equipment specifications
The measuring equipment utilized during the tests, including computer software, is produced
by Brüel & Kjær (B&K).
An impact hammer, cf. Figure D.6, was used to excite the structure, in order to obtain frequency
response functions. The type 8202 hammer is designed to excite and measure impact forces on
structures with masses in the range of approximately 2kg to 3000kg. However, this range is
very much dependent on the structure mechanical properties and the mechanical background
noise imparted to it by ambient conditions. The impact hammer is supplied with three interchangeable impact tips of aluminium, plastic and rubber. The choice of impact tip determines
the impulse shape (amplitude and duration) and the bandwidth of the excitation. For increased
head mass, a 40g head extender is available.
The force was measured by the built-in force transducer type 8200, while the structural response
was measured by an accelerometers, type 4507 Bx, glued to the bridge structure using special
wax, as shown in Figure D.6. The calibration was done for every single accelerometer using
accelerometer calibrator, type 4294, before performing tests. The wires connecting accelerometers with a front-end were duct taped to a structure, close to the accelerometer to avoid any
disturbance by swinging.
Figure D.6: Impact hammer (top left), accelerometer (top right), glued accelerometers
(bottom)
D. Experimental testing of a small–scale bridge model
145
The signal from the impact hammer was sent to NEXUS Conditioning Amplifier Type 2692-C,
cf. Figure D.7, which is designed for applications where very high charge inputs can occur. It
contains four charge channels.
The signals from all accelerometers used in the measurements and amplifier were transmitted
to the front-end, type 3560 D, equipped with five 12-ch input modules, type 3038B, as shown
in Figure D.7, thus allowing to connect 60 accelerometers to a system in total. However, it
was decided that 13 accelerometers is a sufficient number, enough to obtain all necessary measurements. Type 3560-D front-end also contains DC power supply unit and controller module.
Finally, partly-proceeded data were transferred to a PC with Pulse LabShop software for further
analysis. Software features and testing procedure are described later in this section.
Figure D.7: Amplifier (top), front-end (bottom)
146
D. Experimental testing of a small–scale bridge model
Experimental tests applying impulse load in purpose to obtain frequency
response functions
The following steps describes how frequency response functions were obtained by exciting
structure with an impact hammer and processing data using Pulse LabShop software. For all
testing cases (Cases A, B, C, D and E), described in Appendix B, the tests and data analysis were
done in the same manner. The MTC Hammer template in Pulse LabShop computer software
was used for a modal analysis.
Step 1. Set up the hardware, place accelerometers on the structure and open Pulse Labshop
computer program. At the beginning, hardware has to be configured also in the program
as shown in Figure D.8. All connected transducers (accelerometers and impact hammer)
are recognized automatically by the program and indicated by a green light in Hardware
Setup window, telling us that hardware is working properly and ready to perform measurements. At this point the template is activated.
Figure D.8: Hardware set-up in Pulse LabShop software
Step 2. The bridge structure geometry is created straight in Pulse software in Geometry Basics
window, by drawing simple lines and connecting them at the nodes, cf. Figure D.9. Also a
mesh is implied introducing extra nodes which are used later to define degrees of freedom.
D. Experimental testing of a small–scale bridge model
147
Figure D.9: Meshed bridge geometry
Step 3. Measurement degrees of freedom are assigned by placing transducers in a created geometry, on the nodes as shown in Figure D.10. It is important to know that transducers
can be applied just at the nodes, thus sufficient amount of them has to be defined to assign
all transducers in exact locations like they are applied on a real structure.
Figure D.10: Transducers assigned at the nodes. Red arrows define accelerometers,
while black hammers denote excitation points
148
D. Experimental testing of a small–scale bridge model
Step 4. Set configurations in Analysis Setup window, according to tips given in Note for Analysis Setup window, cf. Figure D.11. In this case, a frequency span of 200Hz is used with
400 lines. It is important to keep in mind that all frequencies set in the span later have to
be excited during the tests, therefore too big spans can lead to failure. Obviously, higher
frequencies could be excited by using a harder tip of the hammer or hitting the structure
at stiffer parts, like almost rigid deck–column connections, but here a double-hit problem
arises, which does not allow performing tests successfully. A relatively small number of
lines was chosen to decrease time of measurements, since it does not have a big influence
on accuracy of the results.
Figure D.11: FFT analysis set-up
Figure D.12: Trigger set-up. Its parameters (left) and graph of impulse forces (right)
D. Experimental testing of a small–scale bridge model
149
Step 5. The trigger is set up at this point. The hammer has to be prepared for hitting the structure by choosing the right tip. In our case, the rubber tip is considered as the proper one,
since the structure is relatively small and stiff. The start button is clicked in the Hammer
Trigger Setup window, cf. Figure D.12, and the bridge is excited by hitting multiple times
one of the bolts, connecting deck and railway. The impact of the hammer should be seen
clearly at the graph as shown in Figure D.12. The level and the hysteresis lines were set
to an appropriate amount, so even weak hits would be accounted for. Figure D.13 shows
hammer reading in frequency and time domains, obtained by a force transducer mounted
in the hammer itself. Figure D.14 illustrates the response of accelerometer number 2 in
frequency domain.
Force [N]
0.04
0.03
0.02
0.01
0
0
50
100
Frequency [Hz]
0.5
1
Time [s]
150
200
1.5
2
Force [N]
20
10
0
−10
0
Acceleration [m/s2]
Figure D.13: Force transducer reading in frequency domain (top) and time domain
(bottom). Response for both domains is obtained for Case E
0.1
0.05
0
0
50
100
Frequency [Hz]
150
200
Figure D.14: Accelerometer, number 2, reading in frequency domain. Response is
obtained for Case E
150
D. Experimental testing of a small–scale bridge model
Step 6. After setting up the trigger, hammer weighting has to be configured. It is done by
performing a single proper hit, which means that the hit should trigger measurements, it
should not be too weak and all frequencies should be excited in the frequency span set
in Step 4, and it should not be a double hit. Later, appropriate values for leading, shift,
length and trailing parameters at the Hammer Weighting Setup window, cf. Figure D.15,
are selected. The main idea of doing that is to trim a signal from both sides, thus removing
noise, and leaving just one peak caused by an impulse load.
Figure D.15: Hammer weighting set-up. Its parameters (left), proper hit graph (right)
Figure D.16: Accelerometer weighting. Its parameters (left), proper hit graph (right)
D. Experimental testing of a small–scale bridge model
151
Step 7. Response weighting is done in a similar manner as hammer weighting described in the
previous step. Again the structure has to be excited by a single proper hit and fitting
values of leading, shift and tau parameters have to be chosen in the Response Weighting
Setup window, as shown in Figure D.16. This time, value selection is based on how much
damping has to be introduced into the system, i.e. unfavourable noise has to be damped
out of the system.
Step 8. At this step measurements are done by hitting the bridge structure by proper hits (as
many as set in “average” in Analysis Setup window in Step 4) at all excitation points
defined in Step 3. In our case, five averages are chosen and two excitation points are
defined, however data is presented just for one excitation point because of similarity, i.e.
structural response obtained by hitting two different excitation points is almost the same.
Figure D.17 shows all windows open in Measurement tab.
Figure D.17: Measurement tab with Measurement Control, Level Meter, Average
Counter, Analyzer, Double Hit Detector and Measurements Displays
windows
Step 9. Finally, FRFs are validated at the last step. At this point eigenfrequencies of the structure are determined after performing an FFT analysis and FRFs are plotted for every
single accelerometer according to the reference point, cf. Figure D.18. Also dynamic
behaviour of the structure can be inspected by checking an animation at a particular frequency. It is noted that nodes with no accelerometers will remain fixed–so the animation
152
D. Experimental testing of a small–scale bridge model
does not fully represent the actual motion of the model. Just the nodes with accelerometers and the ones excited by hammer will move. Further, data is exported into MATLAB
for a comparison with the numerical model.
Figure D.18: FRF Validation tab with frequency response functions (left) and deformed geometry animation (right)
D. Experimental testing of a small–scale bridge model
153
Experimental tests when the bridge is subjected to a passing vehicle
Tests with a passing vehicle are performed, adjusting three different speeds, in the same manner
for Cases B, C, D and E, which are described in Appendix B, by the following steps.
Step 1. Set up hardware, place accelerometers on the structure and open Pulse LabShop software. In this case, no existing template is used, thus it has to be created. Firstly, the
hardware has to be configured in the Pulse program as it is explained in the “Experimental tests applying impulse load in purpose to obtain frequency response functions” part,
Step 1. However, this time the program recognizes just accelerometers by itself. Thus, the
force transducer has to be assigned manually by choosing transducer type in the Hardware
Setup window as shown in Figure D.19. In our case it is 8202 30.6m force transducer.
Figure D.19: Hardware set-up. Yellow indicator next to hammer means that force
transducer is assigned but still not ready to use for measurements. Template has to be activated
Step 2. Settings in the FFT analyser are selected. Frequency span of 200Hz is chosen to have
the same frequency band as in impulse load tests but this time 1600 lines are used to
extend measurement time to 8 seconds, cf. Figure D.20.
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D. Experimental testing of a small–scale bridge model
Figure D.20: FFT Analyser window
Step 3 All transducers (force transducer and accelerometers) are added to a Signal Group manually in the Measurement Organiser window as shown in Figure D.21. Also group parameters are set in Signals window, Channel tab, cf. Figure D.21. Default values are used
for all parameters except maximum peak input. It is important to keep in mind that too
low value for maximum peak input parameter ‘cuts’ the highest peaks which exceed the
signal input threshold, thus part of data points is lost and measurements become unreliable. To ensure that full signal is received, the signal monitoring should be done before
performing tests. In our case, 7.071 V is chosen as an proper value for maximum peak
input. The template is activated at this point.
Figure D.21: Measurement Organizer window on the left and Signal group properties
on the right
Step 4. The tests are performed running vehicle 10 times for each speed (0.53m/s, 0.97m/s and
1.31m/s). Later data from 10 tests for one speed are averaged and compared with results
obtained from the computational model. In Figure D.22 a time series of accelerations of
the structure obtained by accelerometer number 7 is presented.
D. Experimental testing of a small–scale bridge model
155
15
Speed: 0.53m/s
Acceleration [m/s2]
10
5
0
−5
−10
−15
0
1
2
3
15
4
Time [s]
5
6
7
8
Speed: 0.97m/s
Acceleration [m/s2]
10
5
0
−5
−10
−15
0
1
2
3
15
4
Time [s]
5
6
7
8
Speed: 1.31m/s
Acceleration [m/s2]
10
5
0
−5
−10
−15
0
1
2
3
4
Time [s]
5
6
7
8
Figure D.22: Accelerometer number 7 readings in time domain for tests with a passing
vehicle for Case E. Data from three different tests, single test for each
speed
In Figure D.23 accelerations in frequency domain for accelerometer number 7 are presented, again for three different speeds.
156
D. Experimental testing of a small–scale bridge model
Acceleration [m/s2]
0.1
0.08
Speed: 0.53m/s
0.06
0.04
0.02
0
0
50
Acceleration [m/s2]
0.1
0.08
150
200
100
Frequency [Hz]
150
200
100
Frequency [Hz]
150
200
Speed: 0.97m/s
0.06
0.04
0.02
0
0
50
0.1
Acceleration [m/s2]
100
Frequency [Hz]
0.08
Speed: 1.31m/s
0.06
0.04
0.02
0
0
50
Figure D.23: Accelerometer 7 readings in frequency domain for passing vehicle tests
for Case E. Data from three different tests, single test for each speed
Averaged accelerations in frequency domain from 10 tests for each speed are presented
in Figure D.24.
D. Experimental testing of a small–scale bridge model
157
Acceleration [m/s2]
0.04
0.03
0.02
0.01
0
0
Acceleration [m/s2]
0.04
50
100
Frequency [Hz]
150
200
100
Frequency [Hz]
150
200
100
Frequency [Hz]
150
200
Speed: 0.97m/s
0.03
0.02
0.01
0
0
0.04
Acceleration [m/s2]
Speed: 0.53m/s
50
Speed: 1.31m/s
0.03
0.02
0.01
0
0
50
Figure D.24: Accelerometer number 7 readings in frequency domain for tests with a
passing vehicle for Case E. Data from three different tests, averaged test
for each speed
Further, data is compared with the numerical model.
158
D. Experimental testing of a small–scale bridge model
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