Stability of Beam Bridges under Bridge-Vehicle Interaction
AYOUB EL AMRANI1, HAFID MATAICH1, BOUCHTA EL AMRANI1,*
1Laboratory of Mathematics, Modeling and Applied Physics,
High Normal School, Sidi Mohamed Ben Abdellah University,
PB 5206 Bensouda 30040 Fez,
MOROCCO
*Corresponding Author
Abstract: -In this paper, we provide an accurate and reliable formulation for simulating the interactions of both
train/bridge subsystems and suitable for high-speed railway lines as well as for existing lines worldwide that are
being renewed or modernized. We model the train as a series of suspended masses, taking into account the
energy dissipation and the suspension system for each train vehicle. On the other hand, the bridge supporting
the rails with irregular elevations will be modeled as an Euler-Bernoulli beam. The mathematical formulation
of the interaction problem between the two subsystems requires the writing of two sets of equations, which
interact with each other through contact forces. Using a one-dimensional finite element formulation, a series of
equations are constructed by Modeling the beam structure. In addition, the suspended mass equations are first
discretized using Newmark's finite difference formulas, which then reduce the degrees of freedom (DOF) of the
vehicle to those of the bridge element. This solves the coupling problem between the two subsystems. The
derived component is known as the vehicle/bridge interaction (VBI) element. On the other hand, an iterative
procedure will be used subsequently to solve the non-linearity problem of the resulting system of differential
equations. MATLAB programs provide results that identify the critical parameters influencing the bridge's
dynamic stability.
Key-Words: - Beam bridge; Railway bridge stability; High-speed train; Suspended mobile mass; Bridge-vehicle
interaction, beam vibration, iterative procedure; Newmark.
Received: July 2, 2023. Revised: February 24, 2024. Accepted: March 21, 2024. Published: May 13, 2024.
NOMENCLATURE
The flat rate mass of the bodywork
The mass per unit length of the beam
The mass of a train wheel
The moment of inertia of the beam
The stiffness of the suspension unit
The overall length of the beam
The damping of the suspension unit
The abscissa of the action of contact force
Displacements of wheel and bodywork nodes
The stiffness of the bridge ballast
Velocities of displacement of wheel and
bodywork nodes
Rail irregularity at abscissa x
Accelerations of displacement of wheel and
bodywork nodes
The contact force between the two
subsystems
v
The constant speed of the train
Transversal displacement of the beam
p
The total weight of the two mass units and
The velocity of transversal displacement of
the beam
The Young's modulus of the beam structure
Acceleration of transversal displacement of
the beam
The Poisson's coefficient
Beam shape functions
Increment of wheel displacement
Increment of bridge displacement
Increment of bodywork displacement
Coefficients of the β-Newmark method
1 Introduction
In recent years, the construction of high-speed
railway tracks (TGV) and suspension bridges in
various countries around the world has seen
significant progress. This has given rise to several
phenomena, among which the vibrational effects on
bridges caused by the passage of these high-speed
trains have become a subject of growing interest.
Studies conducted by the authors [1], [2], [3], [4],
[5], [6], [7] show that vehicle speed is one of the
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factors, among others, affecting the transverse
dynamic instability of the bridge. Dynamic bridge-
vehicle interactions have been examined by several
researchers in the literature, [8], [9], [10], [11],
addressing the effects influencing the dynamic
behavior of bridges when interacting with moving
vehicles. These studies model moving vehicles as
mobile loads, moving masses, or suspended masses
in motion, taking into account suspension
mechanisms and energy dissipation associated with
vehicles. More sophisticated models that consider
various dynamic properties of vehicles or train cars
have also been implemented in the study of
vehicle-bridge interactions, [12]. In this book, and
more specifically in chapters 9 and 10, the authors
delve into the Modeling of 2D and 3D interaction
problems.
The treatment of dynamic interactions in a
vehicle-bridge system requires the determination of
two sets of motion equations, one for the bridge
structure (the stationary subsystem) and the other to
model the moving vehicle structure (the moving
subsystem) as presented in this work, [13]. The
interaction between the two subsystems is achieved
through contact forces existing at the contact points
of the two subsystems. Due to these forces, both
sets of equations will be coupled and nonlinear.
However, the contact position varies in time and
space, so the mass, damping, and stiffness matrices,
which are functions of the contact forces, must be
reformulated at each time step in a time-domain
analysis. Solving this system of coupled differential
equations requires us to adopt an iterative
procedure based on time integration using β-
Newmark finite difference formulas (which are
classical methods with average acceleration and
unconditionally stable associated with specific
values of γ=0.5 and β=0.25; for more details on this
algorithm, [14]).
In this study, we employ a dynamic
condensation approach to solve vehicle-bridge
interaction problems. This method has been used in
the literature, as described in reference, [12], where
the reduction scheme was used to condense the
vehicle's degrees of freedom to the associated
degrees of freedom of the bridge. However, if we
need to obtain the response of the vehicles, which
serves as a reference for assessing passenger
comfort, we cannot rely on the two approaches
mentioned above to obtain accurate solutions
because approximations have been made to connect
the vehicle's degrees of freedom (slave) to those of
the bridge (master).
2 Mathematical Modeling of the
Study Problem
Physical Modeling of the problem
The passage of a high-speed TGV train over a
beam bridge produces mutual effects between the
two subsystems, namely the vehicles on one side
and the bridge on the other side. These effects are
known as Vehicle Bridge Interaction (VBI)
dynamics, [15]. In this paper, we examine the
impact of these interactions on the dynamic
response of the bridge. The proposed physical
model for the bridge-vehicle interaction problem in
this study is as follows: the bridge is represented by
an Euler-Bernoulli beam structure (small
deformations and negligible shear effects) of length
L, and the train in motion at a constant speed v is
approximated by a sequence of suspended mass
units (N wheels), distributed in pairs for each front
and rear bogie of the wagon, as shown in Figure 1.
In this study, we propose not to consider the
effect of the variation in the distance between the
wheels on the dynamic response of the bridge.
Therefore, we assume that all the wheels of the
train are separated by regular distances , as
shown in the diagram in Figure 2. Consequently,
each suspended mass unit represents each train
carriage's front or rear half.
Fig. 1: A beam of length L under the loading of a succession of train cars
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Fig. 2: Idealization of the load on the beam by mobile loads separated by a uniform distance
2.1 Vehicle Dynamics
We consider a suspended mass model consisting of
two nodes, with one node associated with each of
the two concentrated masses. Similarly, the vertical
displacements of the two masses are denoted by the
coordinates . The equations of
motion for this model can be expressed according
to the fundamental principles of dynamics, as
described in [7].
(1)
is the contact force between the two models
representing the vehicle and that of the bridge,
which is given by the following expression:
(2)
The Hermitian shape function evaluated at the
point of contact force application, , is
, and the nodal displacement vector
of the active element
of the beam. For more details regarding the beam
finite element, [16]. Finally,
represents
the irregularity of the rail which will be dealt with
in section 6.3 and its expression is given later by
eq. (34). Let {∆y} be the increment of transverse
displacement of the suspended mass unit, then:
(3)
By substituting equation (3) into equation (1),
we obtain a temporal recurrence relation of the
following form:
(4)
Based on the β-Newmark finite difference scheme
[17], as follows:
(5)
With
(6)
We manipulate the equations of the system Eq.
(4) and substituting the expressions of the equation
Eq. (5), we obtain the following equivalent stiffness
equations:
(7)
In order to simplify the writing of the expressions,
we set:
-
(8)
And also:
(9)
+
(10)
Finally, the vertical displacement increments of
the suspended mass unit can be
written in terms of the bridge displacement at
time , as follows:
(11)
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To simplify, we always assume:
(12)
2.2 Bridge Dynamics
We adopt a classical Euler-Bernoulli approach in
the plane which assumes small deformations
therefore we neglect all transverse shear
deformations. So, the beam element (Figure 3) will
be characterized by two degrees of freedom DOF
per node, a deflection in the x-y plane and a
rotation
around the z-axis. The beam
dynamics is approximated using Hermite
interpolation functions (the functions are listed
in Appendix as A-1), which are able to give an
exact nodal solution in the finite element (super-
convergent element).
Fig. 3: The proposed vehicle-bridge interaction
elementary element is a mass suspended from a
beam
Using Hamilton's energy principle, [18], the
bridge element that reacts with the suspended mass
unit is governed by the following motion equations:
(13)
Where, are respectively
the mass, damping and stiffness matrices of the
bridge element, and represents the external
nodal load applied to the element. In the case where
the bridge element behaves as a two-dimensional
rigid beam element, we assign four DOFs for each
element (one translation and one rotation for each
node). The beam damping in this case is assumed
to be of Rayleigh type, so the matrix can be
expressed as a linear combination of the mass and
stiffness matrices, [19]. The matrix , and
the damping , are presented in Appendix A-2,
A-3 and A-4.
Let be the transverse displacement
increment of the beam element, then after the time
increment :
(14)
By substituting equation Eq. (14) into the motion
equation Eq. (13), the motion's equations of bridge
can be expressed in an incremental form.
(15)
3 The Condensed Equation of the
Bridge (Equation of the VBI
System)
Equation Eq. (15) shows the coupling between the
two subsystems via the displacement of the
suspended mass on the right side of the
equal sign. However, the other terms are known,
such as the external nodal load , which is
known at the last time step, just like the
displacement of the bridge element . In order
to decouple the equations of the two sub-systems,
we assumed that , using the
expression of from equation Eq. (11). In this
way, we obtain, from equation Eq. (15), the
condensed equations of motion for the beam
element at time , considering the effect of
the suspended mass unit interaction. Therefore, the
degrees of freedom (DOF) of the vehicle are
condensed to those of the bridge elements in
contact, [20].
(16)
With, is the stiffness matrix of the condensed
system.
(17)
The load forces induced by the wheels,
including contributions from rail irregularities and
ballast stiffness.
(18)
The resistance forces associated with the suspended
mass unit:
(19)
The assembly of the interaction problem
between the two subsystems requires a repetitive
loop over all the elements interacting with the
suspended mass (the elements denoted as VBI Eq.
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(16)), as well as over the elements that do not
interact with the vehicle. Therefore, the overall
system of equations of motion for the two
interacting subsystems is:
(20)
Where denotes all the displacements
of all the nodes of the beam representing the bridge
at time , and represents the
displacement increments of the bridge from time t
to , with:
(21)
The external nodal loads and the
resisting forces on the right side of equation
Eq. (20) are constructed as follows:
(22)
The damping matrix [Cb] is constructed using
the procedure based on the assumption described in
Appendix A.
1. Solving the resulting equation
(equivalent stiffness equation).
First, the acceleration
and the velocity
of the bridge can be related to the displacement
increments between the time instants
and t using type finite difference
formulas, as follows:
(23)
By substituting the expressions Eq. (23) into
the equation of the system Eq. (20), the final
equation can be transformed into the following
equivalent stiffness equations:
(24)
With,
(25)
(26)
The matrix
and the vector
respectively representing the effective stiffness and
the global external load are treated as constants
during each time step.
4 Non-linearity Treatment (Iterative
Procedure)
When a vehicle passes over a beam, there are
mutual interactions between the two subsystems,
the vehicle and the beam, through the contact force.
This type of phenomenon is nonlinear, and its
resolution requires an iterative procedure (such as
the modified Newton-Raphson method) to
eliminate the unbalanced force between the two
subsystems. This iteration procedure is presented as
the equivalent stiffness equation system of the
interacting VBI system described by Eq. (24) must
be modified, that is:
(27)
The exponent «i» indicates the current iteration
number. The resistant force vector in Eq (26) must
be structured for iterations as follows:
(28)
The initial conditions (i=1) are:
(29)
In each time increment, the displacement
increments of the bridge, for all iterations
performed, can be cumulated as:
(30)
Taking into account the loop of iterations, the
acceleration and speed of the bridge can be
obtained by:
(31)
In the current incremental step iteration « i »
the total vehicle response is calculated by:
(32)
5 Numerical Results and
Interpretations
The two subsystems interact with each other, as
presented in Figure 1, and will be roughly
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characterized by the coefficients and parameters
listed in Table 1.
5.1 Validation and Limitations
First, let's examine the parameters presented in
Table 1, by setting the vehicle parameters
to zero, in this situation, all loads
passing through the beam are considered mobile
loads. On the other hand, in the second case, by
assigning zero values to the damping cv and the
mass of all train wheels, as well as very high
values to the stiffness of the vehicle and the
ballast , the system obtained is a moving mass.
The third case takes into account all the real
properties of the bridge and the vehicle described
by a suspended mass. In all three cases, we
consider only the contribution of the first vibration
mode of the beam.
Let's start with a numerical evaluation of the
theoretical formulation proposed above, validating
the importance of treating non-linearity. We
consider a situation corresponding to the first case
(mobile load) with two loading scenarios defined
by the ratio of the vehicle's mass to the beam's
mass, denoted
, where the first
scenario corresponds to
being equal to
0.1%, and the second scenario to
being
equal to 10%. The static deflection of the beam,
denoted is equal to 9.3200 mm.
In Table 2, we execute the iterative algorithm for
handling non-linearity to determine the impact
factor IF, defined as
, over eight
iterations. The current calculations are performed
with a train speed of v equal to and a
fundamental frequency of the beam, denoted , of
. The speed parameter is defined as
.
Table 1. Parameters of the interaction problem, bridge structure, and suspended mass
Vehicle-related
settings
The flat rate mass of the bodywork
The mass of a train wheel
0.00
The rigidity of the suspension unit
The damping of the suspension unit
0.00
v
The constant speed of the train
p
The total weight of the two mass units and
The Young's modulus of the beam structure
The Poisson's ratio
m/s
Characteristics of
the bridge
structure
The mass per unit length of the beam
The moment of inertia of the beam
The overall length of the beam
The abscissa of the point of contact force
V
The rigidity of the bridge ballast
1595
Rail irregularity
Table 2. Digital recording of the « IF » impact factor for
with
Iteration
« i »
Proposed model
[16]
[17]
1
9,2174
0,989
16,2168
1,7401
2
10,1494
1,089
20,8768
2,2411
3
11,1933
1,201
23,8592
2,5600
4
12,2185
1,311
26,6552
2,8601
5
13,1505
1,411
28,8920
3,121
6
14,1757
1,521
30,8492
3,3112
7
14,4553
1,551
32,7132
3,5114
8
14,6883
1,576
33,6452
3,6101
IF =
1.598
IF =
3.730
According to the results from the previous
table, the impact factor observed at mid-span of the
beam under the loading of the first situation,
and is a result very
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close to the reference, [21]. However, in the second
loading situation with
, the
obtained factor requires more
iterations for proper convergence, according to the
results from [22]. From this observation, we notice
a good correlation between the results obtained and
those from the literature. Consequently, we can
continue with the numerical manipulations using
the theoretical approach adopted previously.
A graphical representation of processing
algorithm convergence is shown in Figure 4. In this
figure, the impact factor "IF" and the
deviation
are
drawn in a histogram for each iteration in the case
of
. Processing equation Eq. (27)
returns a portion of the deflection used to correct
the beam displacement incrementwith an
accumulated deviation according to equation Eq.
(30)
. In fact, from
this illustration we see small corrections gained in
each iteration (red bars) and an almost uniform
distribution of "IF" values from iteration 1 to 8. So,
incorporating the non-linearity treatment algorithm
into the numerical analysis of the problem is not
necessary.
Therefore, in the case of low ratios of
(which indicates small
deformations according to Euler-Bernoulli
approximations), the incorporation of the non-
linearity processing algorithm in the numerical
analysis of the problem is not necessary.
Once again, the graphical representation of the
convergence of the non-linearity processing
algorithm is illustrated in Figure 5. In this figure,
the impact factor « IF » and the difference
are plotted for
8 iterations.
Fig. 4: Graph shows the impact factor of the excited bridge in the case of
Fig. 5: Graph shows the impact factor of the excited bridge in the case
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In contrast to the interpretation made for
illustrations in the case of
. If the
mass of the vehicle increases relative to that of the
bridge, then the non-linear treatment of equation
Eq. (27) is necessary, as the large deviations
shown in Figure 5.
5.2 Results Related to a Single Suspended
Mass
In this section, we investigate the dynamic response
(lateral deflection) at the mid-point of the beam
under the action of a suspended mass in uniform
motion (constant train speed), considering only the
contribution of the first mode of vibration of the
beam. This situation corresponds to the second
specific case of the suspended mass described in
Section 6.1, complemented by the parameters from
Table 1. This response is represented in Figure 6.
As can be seen, the resulting dynamic response
based on the VBI element Eq. (24) corresponds
well to that of the first mode of the closed-form
solution presented in [23].
(a)
(b)
Fig. 6: The evolution of the impact factor of the
beam (a) the special distribution (b) the planar
distribution
The illustration of the relative displacement at
mid-span
of the beam in a 3D
space consists of three directions (time parameter
, the velocity parameter
, and
another upward direction for the impact factor
(relative displacement)
in Figure
6(a). Furthermore, Figure 6(b) presents a top-down
view of Figure 6(a). In addition, Figure 6 provides
a general illustration of the effect of time and
vehicle velocity parameters on the dynamic
behaviour of the bridge.
Figure 7 shows the intersection of planes
perpendicular to the velocity axis and their cross-
section for values S_1=0,0.3,0.5,1 and 2, which
intersect the graphical evolution of relative
displacements according to curves representing the
temporal evolution of the beam displacement.
Fig. 7: The impact factor at mid-span of the bridge
for different speeds of the suspended mass
From Figure 7, we can see that at very high
speed the maximum dynamic displacement occurs
when the suspended mass leaves the bridge. As the
speed of the load decreases, the peak (the
maximum relative displacement) appears when the
suspended mass is close to the centre of the beam.
By progressively increasing the speed, the position
of the load producing the maximum dynamic
displacement at mid-span moves towards the end of
the bridge.
5.3 Dynamics of the Bridge under a Train
Loading
During the dynamic analysis of the interactions
between the two subsystems, the train can be
modeled as a sequence of (N=10 wheels)
suspended masses (moving at a constant velocity v)
separated from each other by uniform distances
equal to the length of each railcar (other
properties and parameters are presented in Table 1),
as shown in Figure 3.
The curves shown in Figure 8 are obtained after
a numerical study of the mathematical formulation
of Section 2. Using the small deformation
approximation, the displacement of the beam will
be obtained by the principle of superposition of all
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the responses induced by individual suspended
masses, such as:
(33)
(a)
(b)
Fig. 8: The impact factor of the mid-span bridge as
a function of time for a series of 10 loads (a)
damping factor ξ_1=0.0 (b) damping factor
ξ_1=0.05
By manipulating the method described above,
the transverse deflection at mid-span of a beam
with a span of , simply supported at its
edges, is obtained. With a fundamental frequency
(for the first mode), (Table 3)
, the
study is conducted in two damping situations
. The time evolution of relative
displacements illustrated in Figure 8(a) and Figure
8(b) shows several maximum values depending on
the combinations of the effects of the exciting loads
on the beam. Damping has the effect of stabilizing
the vibrations of the bridge.
In the last section of the proposed study, we
consider a bridge schematized as a pre-stressed
concrete beam with simple supports at the
boundaries, and its properties are as follows.
In Figure 9, we present the impact factor
for the displacement of the mid-
point of the bridge excited by the series of
suspended masses as a function of the first velocity
parameter
. The parameter is
defined as the ratio between the excitation
frequency of the moving train
and the first
(fundamental) frequency of the bridge. The
form of irregularity proposed in this study is the
one adopted by [24] and is expressed as follows.
(34)
Table 3. Beam Properties
Fréquences en
Liaisons aux frontières
34
8
10.3
32562
Appui simple
Crossed by a train characterized by the following
parameters (Table 4).
Table 4. Technical properties of the train
20
20000
0
1600
76
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Fig. 9: Maximum impact factor at mid-range as a function of the speed parameter for two damping ratios
We observe that the successive periodic effects
of the train wheels action induce multiple peaks of
relative displacements. These peaks indicate
resonant responses of the beam. Therefore, the
largest peak of the impact factor (dynamic relative
displacement) is produced at , which
corresponds to a train speed of
.
Furthermore, considering damping, the maximum
displacement will be reduced, and consequently,
the resonant response. This result is consistent with
the work under the passage of a single HS20-44
truck, [25].
6 Conclusion
It is important to note that dynamic loads only
cause minor damage to the bridge, such as the
resonance phenomenon (with damping, resonance
may have a negligible effect), but they result in
continuous deterioration of the bridge, increasing
maintenance needs. Examining how these factors
affect the bridge's response is crucial for operating
costs. To better understand the structural behavior,
the results of several parametric studies have been
presented. Furthermore, as the suspended mass
behaves in some ways like a mass applied to the
bridge, the inclusion of the inertial effect of moving
vehicles, represented by the suspended mass model,
has led to a slight reduction in the maximum bridge
response. Additionally, it is very clear that the
harder the ballast (with significant rigidity), the
higher the train speed at which the resonance
phenomenon is triggered. The rigidity of the
moving vehicle suspension system has a negligible
effect when determining bridge dynamics as the
objective. The results mentioned above pertain to
bridge safety but not to vehicle dynamics and,
therefore, passenger comfort.
Acknowledgements:
The authors would like to thank sidi Mohammed
ben Abdellah University for supporting this work.
The authors also thank the professors of the “ENS”
Fez for their assistance in manuscript correction.
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.6
Ayoub El Amrani,
Hafid Mataich, Bouchta El Amrani
E-ISSN: 2224-3429
65
Volume 19, 2024
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APPENDIX
The expressions of the Hermite shape functions are:
(A-1)
"Let the element « e » with length « l » be as
follows:
The mass and stiffness matrices of each element
are:
(A-2)
A beam with a square cross-sectional shape is
characterized by a moment of inertia
, where "a" is the side length of the section.
(A-3)
For most cases in civil engineering, it is not
cost-effective to consider all modes when
calculating the damping matrix .. In the case
where two modes are considered, known as
Rayleigh damping, the damping matrix is reduced
to
(A-4)
Where the first natural frequencies of the beam are
, let's assume that the damping ratio is
ξ1=ξ2=ξ.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Ayoub El Amrani and Hafid Mataich carried out
the simulation, the optimization and implemented
the Algorithm in C++. They discussed the results
of the work.
- Bouchta El Amrani supervised all the work.
This work is the work of the group supervised by
Professor Bouchta EL AMRANI laboratory of
mathematics, modelling and applied physics.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.e
n_US
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.6
Ayoub El Amrani,
Hafid Mataich, Bouchta El Amrani
E-ISSN: 2224-3429
66
Volume 19, 2024
