MEng Dissertation. Ju

A DYNAMIC ANALYSIS OF AN
ISOSTATIC FOUR SPAN HIGH
SPEED RAILWAY BRIDGE
Author: Juan Bernal Sanchez
Supervisor: Richard Llewellyn
March 2015
MEng Civil Engineering
School of Engineering & the Built Environment
Edinburgh Napier University

iii
A dynamic analysis of an isostatic four span High-Speed
Railway bridge
Bernal Sanchez, Juan
MEng in Civil Engineering. Project, Edinburgh Napier University,
Edinburgh.
17th March 2015
Keywords
Dynamic analysis High-speed Railway Bridge
Isostatic short span bridge Modal method
Fluid Viscous Dampers (FVD) HSLM-A
Serviceability Limit State (SLS) Structural damping
Finite Element Method (FEM) Resonance phenomenon
Abstract
Railways are continuously being developed with the introduction of high-speed lines
which bring numerous benefits due to the reduction of journey times. Increasing the speed
of this mode of transport over 200 km/h, combined with the resonance phenomenon is
causing difficulties to some bridge infrastructure. In the case of isostatic short span
bridges this results in a superstructure change due to the excess of the vertical acceleration
within the Serviceability Limit State (SLS). This research has introduced the dynamic
analysis based on a Finite Element Method (FEM) with software to study the load models
affecting severely on the bridge and the structure’s parameters. These include the mass,
stiffness and damping which influence the vertical acceleration and the resonance speed.
Modifying the structural mass and stiffness is obtained to result in a variation of the speed
at which the resonance phenomenon occurs. Furthermore, the structural mass and the
damping properties appear to influence directly the vertical acceleration of the deck slab,
whereas the stiffness does not generate any variation. Hence, a material called Fluid
Viscous Damper (FVD) has been introduced as an additional external damping property.
The generalised reduction of the vertical acceleration with the FVD converts it into the
most efficient and practical solution instead of changing the structural mass and/or
stiffness of the isostatic bridge model.

iv
Table of contents
Abstract............................................................................................................................iii
Table of contents ............................................................................................................. iv
List of tables and figures ................................................................................................ vii
List of equations ............................................................................................................... x
Acknowledgements ......................................................................................................... xi
List of abbreviations and symbols.................................................................................xiii
1. INTRODUCTION ....................................................................................................... 1
1.1. Case of study ...................................................................................................... 3
1.2. Project aims........................................................................................................ 4
2. LITERATURE REVIEW ............................................................................................ 5
2.1 Structural bridge design concepts....................................................................... 5
2.1.1 Isostatic bridges.................................................................................... 5
2.1.2 Structural bridge parameters ................................................................ 6
Damping of the structure ............................................................ 6
Mass of the structure................................................................... 7
Stiffness of the structure ............................................................. 7
2.2 High-Speed train design concepts...................................................................... 8
2.2.1 Initial requirements to undertake the dynamic analysis....................... 8
2.2.2 High-Speed Load Models. Eurocode Specifications ......................... 10
2.2.3 High-Speed train considerations ........................................................ 12
Speed considerations ................................................................ 12
Verification of the limit states .................................................. 13
Criteria for traffic safety. Serviceability Limit State................ 13
2.3 Dynamic analysis ............................................................................................. 15
2.3.1 Basic concepts.................................................................................... 15
The Single Degree of Freedom (SDOF)................................... 15

v
Multidegree of freedom (MDOF)............................................. 16
Damping in MDOF systems ..................................................... 16
2.3.2 Resonance phenomenon..................................................................... 17
2.3.3 Mode analysis..................................................................................... 19
Dynamic analysis. Finite Element Method analysis................. 19
Modal method. Mass participation factor................................. 20
Modal method. Model of a train of moving loads.................... 21
2.4 Fluid Viscous Dampers (FVD)......................................................................... 22
3. METHODOLOGY .................................................................................................... 25
3.1 Bridge model characteristics ............................................................................ 26
3.1.1 Materials’ properties .......................................................................... 26
3.1.2 Frame’s features................................................................................. 26
3.1.3 Deck slab............................................................................................ 27
3.1.4 Bearings.............................................................................................. 28
3.1.5 Foundation springs............................................................................. 29
3.1.6 Abutments .......................................................................................... 29
3.1.7 Prestressed beams............................................................................... 29
3.1.8 Bents................................................................................................... 29
3.2 Loads configurations........................................................................................ 30
3.2.1 Dead load ........................................................................................... 30
Linear loads. Rail tracks ........................................................... 30
Area loads ................................................................................. 31
3.2.2 Moving loads...................................................................................... 33
3.2.3 Load combinations. Dead and moving load....................................... 34
3.3 Dynamic analysis ............................................................................................. 35
3.3.1 Time-history analysis. Time step size................................................ 35
3.3.2 Structural bridge model. Main dynamic analyses.............................. 35

vi
4. ANALYSIS & RESULTS ......................................................................................... 37
4.1 Modal method. Mass participation factor......................................................... 37
4.2 Serviceability Limit State. Vertical Acceleration............................................. 39
4.2.1 Load models. HSLM-A1 to HSLM-A5 ............................................. 39
4.2.2 Load models. HSLM-A6 to HSLMA10............................................. 42
4.2.3 Structural mass................................................................................... 45
4.2.4 Structural stiffness.............................................................................. 47
4.2.5 Structural damping............................................................................. 49
4.2.6 Introduction of FVD. Damping Coefficient measurement ................ 51
4.2.7 Introduction of FVD. HSLM-A1 ....................................................... 54
4.2.8 Introduction of FVD. HSLM-A10 ..................................................... 57
4.3 Serviceability Limit State. Vertical deformation of the deck slab ................... 61
4.3.1 Structural mass................................................................................... 62
4.3.2 Structural stiffness.............................................................................. 63
4.3.3 Structural damping............................................................................. 64
5. CONCLUSIONS ....................................................................................................... 66
6. RECOMMENDATIONS FOR FUTURE WORK .................................................... 70
7. REFERENCES .......................................................................................................... 72
Appendix A. Serviceability Limit State. Vertical acceleration of the deck slab
Appendix B. Serviceability Limit State. Vertical deformation of the deck slab
Appendix C. Technical data. Design project planes. Isostatic four short span High-
Speed Railway bridge

vii
List of tables and figures
List of tables
Table 4.1 Theoretical resonance phenomenon. Load models HSLM-A1 to HSLM-A10
Table 4.3 Vertical velocity. Structural bridge model. HSLM-A1. First, second, third and
fourth span
List of figures
Figure 2.1 Values of damping to be assumed for design purposes.
Figure 2.2 Flow chart for determining whether a dynamic analysis is required
Figure 2.3 Limits of bridge natural frequency η0 (Hz) as a function of L (m)
Figure 2.4 HSLM-A
Figure 2.5 HSLM-A table
Figure 2.6 Application of HSLM-A and HSLM-B
Figure 2.7 Definition of angular rotations at the end of decks
Figure 2.8 Multi degree of freedom scheme
Figure 2.9 The peak value of the dynamic response occurs due to resonance. The value
is strongly dependent on the damping coefficient.
Figure 2.10 Displacement in a point in the centre of the bridge
Figure 2.11Regular train bogie’s distance
Figure 2.12 Simple moving train loads
Figure 2.13 Step force with ramp
Figure 2.14 A viscous damper as installed
Figure 2.15 The Ok-Yeo Bridge Dampers
Figure 2.16 Equivalent linear and non-linear FVDs
Figure 2.17 Force-Velocity relationships of Viscous Dampers
Figure 3.1 Structural bridge model. Deck slab
Figure 3.2 Structural bridge model. Bridge section design
Figure 3.3 Structural bridge model. Deck slab with prestressed beams
Figure 3.4 Technical data. Cross-section deck slab
Figure 3.5 Rail tracks. UIC60
Figure 3.6 Linear loads. Rail track load

viii
Figure 3.7 Mono-bloc sleeper
Figure 3.8 Area load. Ballast’s load and sleeper’s load
Figure 3.9 Moving load. Train lanes
Figure 3.10 Moving load. First wagons load model HSLM-A1
Figure 3.11 Moving load. Last wagons load model HSLM-A1
Figure 3.12 Bridge model. View of the whole structural bridge
Figure 4.1 Structural bridge model analysis. Total modal mass participation
Figure 4.2 Structural bridge model. Modal method. First mode shape (eigenmode)
Figure 4.3 Structural bridge model. Modal method. Second mode shape (eigenmode)
Figure 4.4 Structural bridge model. Modal method. Third mode shape (eigenmode)
Figure 4.5 Vertical acceleration. Load models. HSLM-A1 to HSLM-A5. First bridge
model’s span
Figure 4.6 Vertical acceleration. Load models. HSLM-A1 to HSLM-5. Second bridge
model’s span
Figure 4.7 Vertical acceleration. Load models. HSLM-A1 to HSLM-5. Third bridge
model’s span
Figure 4.8 Vertical acceleration. Load models. HSLM-A6 to HSLM-10. First bridge
model’s span
Figure 4.9 Vertical acceleration. Load models. HSLM-A6 to HSLM-10. Second bridge
model’s span
Figure 4.10 Vertical acceleration. Load models. HSLM-A6 to HSLM-10. Third bridge
model’s span
Figure 4.11 Vertical acceleration. Structural mass modification. Load model HSLM-
A1. First span
Figure 4.12 Vertical acceleration. Structural mass modification. Load model HSLM-
A1. Second span
Figure 4.13 Vertical acceleration. Structural stiffness modification. Load model
HSLM-A1. First span
Figure 4.14 Vertical acceleration. Structural stiffness modification. Load model
HSLM-A1. Second span
Figure 4.15 Vertical acceleration. Structural damping modification. Load model
HSLM-A4. First span
Figure 4.16 Vertical acceleration. Structural damping modification. Load model
HSLM-A4. Fourth span
Figure 4.17 Vertical shear forces. Structural bridge model. Load model HSLM-A1
Figure 4.18 Vertical velocity. Load model HSLM-A1. First bridge model’s span

ix
Figure 4.19 Vertical acceleration. Introduction FVD. Load model HSLM-A1. First span
Figure 4.20 Vertical acceleration. Introduction FVD. Load model HSLM-A1. Second
span
Figure 4.21 Vertical acceleration. Introduction FVD. Load model HSLM-A1. Fourth
span
Figure 4.22 Vertical shear forces. Structural bridge model. Load model HSLM-A10
Figure 4.23 Vertical acceleration. Introduction FVD. Load model HSLM-A10. First
span
Figure 4.24 Vertical acceleration. Introduction FVD. Load model HSLM-A10. Second
span
Figure 4.25 Vertical acceleration. Introduction FVD. Load model HSLM-A10. Third
span
Figure 4.26 Vertical acceleration. Introduction FVD. Load model HSLM-A10. Fourth
span
Figure 4.27 Vertical deformation. Initial analysis. Load model HSLM-A1. First to
fourth bridge’s span
Figure 4.28 Vertical deformation. Mass modification. Load model HSLM-A1. First to
fourth bridge’s span
Figure 4.29 Vertical deformation. Stiffness modification. Load model HSLM-A1. First
to fourth bridge’s span
Figure 4.30 Vertical deformation. Damping modification. Load model HSLM-A1. First
to fourth bridge’s span

x
List of equations
Equation 2.1 Natural frequency for singly supported bridge subjected to bending only
Equation 2.2 Modal rigidity for an isostaic beam
Equation 2.3 D’Alembert’s Principle for equation of Motion for SDOF
Equation 2.4 D’Alembert’s Principle for equation of Motion for MDOF
Equation 2.5 Correlation between the structure’s frequency and the load’s frequency
Equation 2.6 Correlation between the structural natural frequency and the load’s
frequency
Equation 2.7 Resonance phenomenon as a function of the train bogies and train’s speed
Equation 2.8 Critical speed at which the resonance phenomenon occurs
Equation 2.9 D’Alembert’s Principle for equation of Motion for MDOF
Equation 2.10 Total mass participation factor
Equation 2.11 Typical damper characteristic law
Equation 3.1 Linear load. Rail track
Equation 3.2 Area load. Sleeper’s load
Equation 3.3 Area load. Ballast’s load
Equation 3.4 Area load. Total
Equation 4.1 Typical damper characteristic law
Equation 4.2 Calculation of the damping coefficient. Load model HSLM-A1
Equation 4.3 Calculation of the damping coefficient. Load model HSLM-A10

xi
Acknowledgements
This is, without any doubt, the most ambitious and challenging project I have ever carried
out in my entire life. The success implies such amount of hard work that reaching it is
only the final step of a very long way. Becoming a Civil Engineer is a complex career
that only some privileged can complete. This is the destiny I have decided to choose, and
I do not regret about my decision.
First of all, I want to thank Victor and Carlos because they have been my diary back up
throughout this entire year and they have become in my family at Edinburgh.
I also want to mention some people: Jose Maria, Chema, and Maria Del Carmen - the
Bernal Sanchez family. Actually, I can say that I form part of this incredible family and I
cannot imagine my life without their continuous support. I will never be able to
compensate the opportunity they have given to me of being studying at Edinburgh, but I
will try to make them feel proud of their little son every day. Thank you very much family.
My supervisor, Richard Llewellyn, has been the main guide I have got during my studies
at Napier University and I want to make him participate of all the achievements I have
obtained along the MEng. I have been gradually developing all my skills and he has
realised of my tireless effort to undertake this investigation. I am very fortunate of having
him as my supervisor.
I cannot forget the most important contribution to this project from Juan Jose Jorquera
Lucerga. He is the professor from the Polytechnic University of Cartagena who has been
in contact with the Spanish organization in order to get the structural bridge planes. I
really appreciate the assistance he has provided to me. This project could never have been
undertaken without his contribution.
I would also like to recognize the support provided from part of my lecturer Aamir
Khokhar who has greatly helped me in order to understand and achieve this challenging
investigation.
Finally, I want to thank the institution Napier University which has provided me the
opportunity to undertake this project and helped me every day in order to achieve the
success.

xiii
List of abbreviations and symbols
List of abbreviations
ADIF Adminitrador de Infraestructuras Ferroviarias
AVE Alta Velocidad Española
BSI British Standards Institution
ERTMS Europen Rail Traffic Management System
FEM Finite Element Method
FVD Fluid Viscous Damper
HSLM High Speed Load Modal
SLS Serviceability Limit State
UBC Uniform Building Codes
USA United Stated of America
UK United Kingdom
List of symbols
Capital Letter
C Damping
E Young’s modulus
F Force
FD Damping coefficient
Hz Hertz
I Moment of inertia
K Modal rigidity, Stiffness
L Length
MPa MegaPascals

xiv
N Newton
U1 First Principal Direction. X Direction
U2 Second Principal Direction. Y Direction
U3 Third Principal Direction. Z Direction
R1 First Principal Direction. X Rotation
R2 Second Principal Direction. Y Rotation
R3 Third Principal Direction. Z Rotation
V Velocity
Small Letter
c Damping
d Distance
f Natural frequency
h Hours
i Multiplier
k Stiffness
kg Kilograms
kg/m3
Kilograms per cubic meter
km Kilometres
km/h Kilometres per hour
kN KiloNewton
kNsec/m Kilonewtons per second per meter
m Meters
m/s2
Meters per squared second
m Mass
mm Millimetres
mm/s Millimetres per second

xv
N/m2
Newton per squared meter
s Seconds
𝑥̈ Acceleration
𝑥̇ Velocity
x, u Displacement
γbt Maximum vertical acceleration for ballasted tracks
γdf Maximum vertical acceleration for fastened tracks

A dynamic analysis of an isostatic four span High Speed Railway bridge Juan Bernal Sanchez
1
1. INTRODUCTION
Nowadays, the railway is one of the most important modes of transport which is
increasing its productivity in a significant way due to it offers an alternative from the
other types of transport, which are more congested, such as road transport. Moreover, the
railway transport is considered one of the safest mode of transport, specifically second
behind the air transport, which has been decreasing progressively the number of accidents
and fatalities since 1980’s based on the introduction of a new safety system; the ERTMS.
The European Rail Traffic Management System (ERTMS) is the safety system which
establishes a continuous control of the train speed in order not to exceed the maximum
allowable design speed. This system was originally introduced into the innovative mode
of transport within the railway engineering, the high-speed train, which has been able to
reach speeds over 200km/h (Bernal Sanchez, 2014).
The diary commercial businesses and the growing social necessities have assisted the
railway transport to develop continuously its technology during the past fifty years in
order to provide better services for the population and a greater freight transport for the
worldwide companies. Hence, the high-speed train has become in one of the most modern
and innovative overland mode of transport within the railway transport which is
continuously seeking to increase its speed in order to reduce progressively journey times
and transport costs (Moliner Cabedo, 2012). Japan, France and Spain have been some of
the initial countries which have been investing a high amount of money in the
development of the high-speed trains during the past thirty years. After these countries,
many other countries within the European Union such as Germany and Italy as well as
some others in the rest of the world such as China or the United States of America (USA)
have also been investing in the high-speed train technology (Dominguez Barbero, 2001).
Therefore, due to the fact that the high-speed transport has been steadily growing during
the past decades, the majority of the railway infrastructure such as the railway bridges
have been also developing in order to support the progressive increment of the train’s
speed and wagons length. Thus, it has occurred that some of the original high-speed
bridges which were designed for a specific speed limit have suffered unexpected
deficiencies. These have been provoked mainly by the resonance phenomenon which has
caused the demolition and replacement of many of these bridges (Alarcon Alvarez &
Museros Romero, 2002).

2
This is the case of the isostatic short and medium span bridges (between 10 and 25m)
which have been constructed in order to save short span distances. Moreover, within the
isostatic railway bridges, there are some specific typologies of railway bridges which have
been greatly affected by the resonance phenomenon such as the concrete deck slab over
prestressed beams or the reduced-weight deck slab bridges. These types of railway
bridges are more affected by the resonance effect due to a combination of a reduced-
weight and a low stiffness of the deck slab in comparison with the new type of high-speed
bridges such as the box deck slab or the trough beam bridge (Moliner Cabedo, 2012).
The resonance is the dynamic phenomenon caused by the combination of the repetitive
actions of the train’s bogies, and its regular distances, with the natural frequency of the
structural bridge producing the excessive acceleration of the deck slab. This phenomenon
has been lately appearing at the isostatic high-speed bridges, provoked by the progressive
increment of the train’s speed, and it has caused the progressive loosening of the ballast
layers. Moreover, it has also caused the damage of the ballast properties which has
simultaneously provoked the loosening of the buckling resistance of the rail track with
the reduction in the comfort of the passengers. Finally, the loosening of the ballast
compaction has incurred in some occasions the deterioration in the alignment of the
beneath rail tracks. Hence, all these consequences generate the necessity of a constant
reparation of the ballast layer conditions, the replacement of some of these layers, and
thus a huge increment in the maintenance and repair costs (Alarcon Alvarez & Museros
Romero, 2002).
The appearance of the resonance phenomenon has compromised the requirements
established by the European commission and the national annexes for railway bridges
concerning the Serviceability Limit State. This is due to the fact that the maximum
allowable vertical acceleration and deformation of the deck slab is exceed in many of the
isostatic short and medium span bridges which have been investigated (Mellier, 2010).
Therefore, the combination of the consequences in short and medium isostatic span
railway bridges have forced to modify the structural parameters such as the mass and/or
stiffness of the deck slab. This process has been carried out in order to modify the
resonance phenomenon conditions incrementing the design speed’s range of the train over
the structural bridge. One of the most used techniques to change the bridge parameters
has been the replacement of the deck slab, resulting in the demolition of the original deck

3
slab, the reconstruction of the new deck slab and consequently the temporal closure of
the structural bridge (Moliner Cabedo, 2012).
The dynamic analysis is the typology of structural analysis introduced in the
investigations in order to study the structural response of the railway bridge under the
passage of a moving load, in this case the high-speed train. Hence, many analytical
theories of the dynamic analysis concerning the passage of the high-speed train over short
and medium isostatic span bridges have previously been mentioned such as the
superposition method or the inverse iteration (Karlsson & Nilsson, 2007). However, the
Finite Element Method (FEM), such as the modal method, is an innovative analysis that
has been progressively introduced in the dynamic analysis investigations of the high-
speed bridges. This analysis is undertaken in order to develop the 3D dynamic analysis
of structural bridges based on the introduction of a computational software. Thus, the
FEM has allowed extend the investigation of the dynamic analysis of the structural
bridges with the introduction of the multi-degree of freedom theories, reducing the
complex calculations concerning the design and modelling of high-speed bridges
(Dominguez Barbero, 2001).
Hence, the dynamic analysis of the high-speed railway bridge is an investigation that is
growing steadily with the apparition of the last case scenarios concerning the resonance
phenomenon and the modification of the structural bridge parameters. Thus, some
alternative solutions to the deck slab replacement have been investigated during the past
years as the introduction of additional external damping properties into the structural
bridge with Fluid Viscous Dampers (FVD). This material is presented as an innovative
application which is able to add external damping to the structural bridge in order to
mitigate the energy applied to the structure due to the moving load and the repetitive
action of wagon’s bogies (Martinez-Rodrigo, Lavado, & Museros, 2010).
1.1. Case of study
This research has undertaken the design project of an isostatic four short span high-speed
railway bridge provided by the Spanish organisation “Administrador de Infraestructuras
Ferroviarias” (ADIF). This railway bridge is located at one of the new high-speed lines
which are currently being designed and constructed in the Spanish railway network within
the Alta Velocidad Española (AVE). This high-speed line has been constructed for a
maximum design speed limit equal to 350km/h.

4
1.2. Project aims
The main aim of the research is to study the behaviour of the isostatic short span bridge
based on the dynamics analysis of the structural bridge under the passage of a high-speed
train. There are various project objectives which have been set to determine the final aim
of the investigation:
 To design and model the isostatic four short span bridge mentioned previously as the
case of study in order to carry out the dynamic analysis of the structural bridge.
 Introducing a Finite Element Method analysis in the investigation in order to undertake
the dynamic analysis of the isostatic bridge. The FEM will be undertaken introducing a
specialised bridge modeller software, CSiBridge2015.
 Using the Eurocode 1 specifications regarding the Serviceability Limit State (SLS) of
railway bridges. Establishing the maximum allowable values vertical with respect to the
acceleration and the deformation of the deck slab during the dynamic analysis of the
structural bridge, based on the Eurocode requirements.
 Using the Eurocode 1 to set the moving loads which simulate the passage of different
types of trains in order to comply with the European Interoperability requirements for
railway bridges. Thus, it will be essential to analyse the structural response of the
isostatic bridge to the passage of each one of the load models established by the
European commission.
 Studying the structural parameters that affect the behaviour of the isostatic railway
bridge under the passage of a high-speed train. This research will be mainly focused on
the structural response modifying specific structural parameters such as the structural
mass, stiffness and damping.
 This research seeks to determine the most relevant structural parameter in the mitigation
of the maximum vertical acceleration peak values obtained in the deck slab.
 Studying the introduction of additional external damping properties into the structural
bridge with the material Fluid Viscous Damper (FVD). This simulation is going to be
undertake in order to improve the dynamic behaviour of the isostatic bridge, mitigating
the energy applied to the structural bridge and reducing the vertical acceleration of the
deck slab.
 Finally, this investigation seeks to decide if the FVD is presented as a real and practical
solution in order to improve the dynamic performance of the isostatic short span bridge
designed and modelled for this analysis.

5
2. LITERATURE REVIEW
This statement has decided to divide the literature review of the dissertation in three main
sections: the definition and the features of the structural bridge, the high-speed design
upon the Eurocode considerations and the basic dynamic concepts introduced in the
analysis of the railway bridge. These three areas have been required to undertake the
entire analysis and development of the dissertation due to the fact that the dynamic
analysis of the structure is occurred because of the interaction between the high-speed
train and the structural bridge.
2.1 Structural bridge design concepts
The structural bridge that has been analysed by this statement possess some particular
features that difference it from other types of railway bridges. It is defined as an isostatic
bridge formed of four different short/medium spans (between 10 and 25 metres) and
constructed with two prestressed concrete beams per span length of the bridge as the
beams girders of the overall structure. The deck slab of this particular bridge is composed
by reinforced concrete. All these features will generate different results of the dynamic
behaviour depending on the span where the analysis is being undertaken (Alarcon Alvarez
& Museros Romero, 2002).
2.1.1 Isostatic bridges
This is an established typology of structural bridges together among many others such as
the hyperstatic or continuous bridge structures. These two typologies are classified within
the group of the bridges divided by its structural system. The isostatic bridge is also
considered as a non-continuous structural bridge where the spans do not present a
continuity and the girders which are located above them are simply supported. There is
an independence between the deck of each one of the spans and also between the deck
span and the supports. On the other hand, the hyperstatic bridge is the structural system
where there is a dependence between the deck of the different spans in which the bridge
is divided, and it could also exist a dependence with the supports of the structure
(Carnerero Ruiz, 2007).
This statement has decided to analyse an isostatic bridge because this structural bridge
become in one of the most used bridge at high-speed lines during the 80’s and 90’s, mostly
in France and Spain, where it has recently occurred numerous difficulties related to the

6
vertical acceleration and deformation of the deck slab at the spans of this type of bridge
(Dominguez Barbero, 2001). This is an emerging inconvenient that is being occurring at
low and medium span isostatic bridges. Specifically, these structural defects have
appeared in isostatic bridges where the concrete deck slab has been directly constructed
over the prestressed concrete beams, the isostatic bridges which have been constructed
with solid concrete slabs or the lightweight concrete slab bridges. These types of isostatic
bridges were preferably constructed in the past in order to take advantage of the short and
medium spans. However, these types of bridges are characterised by reduced mass deck
slabs and also reduced stiffness structures which result in a higher sensitivity with regard
to the resonance effect. Consequently, this produces excessive vertical acceleration and
vertical displacement peak values of the deck slab (Moliner Cabedo, 2012).
2.1.2 Structural bridge parameters
The dissertation is focused on the analysis of the isostatic four span bridge in order to
obtain some results concerning the dynamic behaviour of the entire structure. The
Eurocode 1 establishes the main parameters which will generate a different dynamic
behaviour of the structure under the passage of a high-speed train (British Standards
Institution, 2003). A variation of the different bridge parameters and its analysis under
the passage of a high-speed train has already been undertaken by various authors and
articles. However, the results of these statements have been obtained mainly with the
employment of numerical analysis and using some 2D software to analyse these
structures. Most of these articles recommend the introduction of 3D software in order to
analyse the dynamic behaviour of structural bridges and introduce the effect of some
variations such as the eccentricity of the loads applied on the load as well as the transverse
deformation and acceleration of the deck slab bridge (Alarcon Alvarez & Museros
Romero, 2002; Mellier, 2010; Goicolea & Antolin, 2012; Moliner Cabedo, 2012).
Damping of the structure
The Eurocode 1 mentions at the section 6.4.6.3.1 (BSI, 2003) numerous principles which
must be taken into account about the structural damping of the railway bridge:
 The first principle, and one of the most relevant, set that the peak values obtained in
the structure as a cause of the traffic vehicles and the resonant phenomenon conditions
are highly dependent of the damping conditions.

7
 The second principle establishes that the worst case scenario with respect to the
damping conditions should be undertaken in the design of the structural bridge.
Additionally, the Eurocode 1 provides a table (figure 2.1) where the lower limit of the
damping percentages is determined depending upon the type of material which is
introduced at the structure and also the span length of the structural bridge.
Mass of the structure
Additionally, the Eurocode 1 (BSI, 2003) establishes some principles regarding the mass
of the whole structure which should be taken into account before carrying out the analysis:
 The first principle concerning the structural mass sets that the maximum dynamic loads
are probably to occur at resonance peaks, and that this phenomenon is occurred when
the frequency of the loading or a multiplier of this value coincide with the natural
frequency of the structure. Moreover, it mentions that the underestimation of the
structural mass will incur in an overestimation of the natural frequency of the structural
bridge. Thus, it provokes an overestimation of the speed at which the rail traffic
reaches the resonance conditions.
 The Eurocode 1 also mentions that the maximum acceleration of the structure is
inversely proportional to the mass of the structure. This is a known concept which has
been previously measured by many other authors who have analysed the dynamic
behaviour of the structural bridge varying the mass of the structure. These
investigations have been mainly analysed in a one degree of freedom structures or 2D
bridge models (Dominguez Barbero, 2001; Gonzalez, 2008; Moliner Cabedo, 2012).
Stiffness of the structure
The Eurocode 1 mentions the same principle for the stiffness of the structure than in the
case of the mass of the structure. Thus, the maximum dynamic load effects will occur
when a multiple of the frequency loading and the natural frequency of the structure
coincide. In the case of the structural stiffness, the overestimation of the stiffness will
Figure 2.1 Values of damping to be assumed for design purposes
Source: BSI (2003)

8
overestimate the natural frequency of the structure and overestimate the traffic speeds at
which the resonance occurs. Hence, this would be the opposite case in comparison with
the variation of the structural mass.
There is an important principle concerning the stiffness parameter at the structural bridge
where it is mentioned that “owing to the large number of parameters which can affect the
Young’s modulus, it is not possible to predict enhanced Young’s modulus values with
sufficient accuracy for predicting the dynamic response of a bridge” (BSI, 2003).
2.2 High-Speed train design concepts
The British Standards Institution (BSI, 2003) is the main official European document
which has been introduced at this statement in order to justify the dynamic analysis of the
structural bridge under the passage of a high-speed train. This document corresponds to
the official Eurocode 1, which is the document assigned to the actions on structures, and
the statement is specifically focused on the second part of the main document about the
traffic loads on bridges. There are numerous sections within the document about the static
and dynamic analysis but there are only a few of them which are going to be mentioned
at this statement.
2.2.1 Initial requirements to undertake the dynamic analysis
The section 6.4 of the document specifies the numerous conditions which must be
required for the dynamic analysis of a structural bride under the passage of a train. Within
these documents, the section 6.4.2 mentions the different factors that will influence the
dynamic behaviour of a structural bridge:
 The overall mass of the structure.
 The speed of the moving load that is passing over the structure.
 The span length of the structural bridge considered.
 The foreseeable vertical irregularities of the rail track.
 The overall damping of the structural bridge.
 The number of axles, the axle load and the spacing between the axles.
 The natural frequency of the entire structure and the natural frequency of the elements
which form part of the structure.
 The mode shapes (eigenmodes) associated to the structure.

9
Furthermore, the section 6.4.4 specifies the previous analysis which must be undertaken
before deciding the introduction of a static or dynamic analysis of the structural bridge.
The flow chart of the process which determines if the structural bridge should carry out a
dynamic analysis can be visualised in the figure 2.2. In order to obtain the value no to
establish if the structural bridge should carry out a dynamic analysis, some previous
calculations should be obtained from the figure 2.3. This figure represents the upper and
lower limit where the dynamic behaviour is not required depending upon the natural
frequency of the structure and the span length in which the structural bridge is divided.
For a simply supported bridge the Eurocode provides and standard equation depending
on the maximum deflection at the mid span of the structure. This equations is shown as:
𝑛0(𝐻𝑧) = 17.75/√𝛿0 (2.1)
Figure 2.2 Flow chart for determining whether a dynamic analysis is required
Source: BSI (2003)

10
Figure 2.3 Limits of bridge natural frequency η0 (Hz) as a function of L (m)
Source: BSI (2003)
2.2.2 High-Speed Load Models. Eurocode Specifications
The section 6.4.6 mentions the requirements for a dynamic analysis. The first point within
this section corresponds to the loading and load combinations. There are various
principles which must be adopted for the dynamic analysis of a structural bridge under
the passage of a high-speed train (BSI, 2003):
 The second principle, one of the most relevant at this investigation, establishes that the
dynamic analysis has to be undertaken introducing the load model HSLM on bridges
which have to follow the interoperability criteria. This is applicable to the international
high-speed lines which can be used by the European countries.
 The third principle mentions that it exist two types of HLSM load model; HSLM-A
and HSLM-B, which are characterised by the length of the coaches.
 It is also mentioned that the load model HSLM-A and HSLM-B represent the effects
from part of different types of high-speed trains, such as the articulated, the
conventional or the regular high-speed train. This principle is established to comply
with the interoperability requirements within the European countries. The HSLM load
models would be the representation of the European trains which should be considered
in the national railway lines.

11
Figure 2.5 HSLM-A table
Source: BSI (2003)
Figure 2.4 shows the shape of a standard Load Model HSLM-A, where the different
number of coaches and the position of the axle loads which are applied are specified
graphically. The Load Model HSLM-A shows three different types of coaches: the first
one is the power car (leading and trailing power cars identical), the second one is the end
coach (leading and trailing end coaches identical) and the third coach type is the most
numerous which is the intermediate coach with a constant length of the coach and the
bogies.
Figure 2.5 shows the values of the length coach, the number of coaches, the bogie axle
spacing and the point force P. Each one of those values change depending upon the type
of load model that is introduced in the model analysis. These values represent the
standards load models for the entire European Union from the HSLM-A1 to the train load
model HSLM-A10. This is an essential data in the analysis of the dynamic behaviour of
the structural bridge where the distance of the bogies and the length of the coach will
affect the consequent behaviour of the force induced into the bridge.
Figure 2.4 HSLM-A
Source: BSI (2003)

12
Figure 2.6 Application of HSLM-A and HSLM-B
Source: BSI (2003)
Eurocode 1 establishes that the HSLM-A and HSLM-B load models should be introduced
in the structural bridge depending on the features of the main structure (BSI, 2003). Figure
2.6 shows the conditions which must be complied in order to introduce the load model A
or the load model B. This statement has decided to introduce the load model A with all
the ten standard load models due to the fact that the span length is 21 metres and the
structural bridge is considered a continuous structure even though this bridge is an
isotactic structural bridge.
2.2.3 High-Speed train considerations
Speed considerations
The section 6.4.6.2 of the Eurocode 1 mentions the minimum considerations which
should be introduced in the dynamic analysis of the structural bridge concerning the
different speeds of the passage of a high-speed train over the structure. Eurocode 1
establishes some principles which have been undertaken by this research:
 The first principle mentions that each one of the trains and the load model HSLM
which are passing over the structural bridge must be designed considering a maximum
speed over the maximum line speed established. The Eurocode 1 establishes an initial
value of the design speed of the line equal to 1.2 times the maximum line speed.

13
 The second principle within this section mentions that the series of speed of the load
models HSLM which are introduced at the structural bridge have to start from 40m/s
to the maximum design speed determined previously based on the maximum line speed
of the structural bridge.
Verification of the limit states
The section 6.4.6.5 shows the different statements to ensure the traffic safety related to
the passage of the train over the structural bridge. It is also mentioned some statements
about the dynamic impact factor, concept applied within the statistic analysis.
Nevertheless, this project has decided not to introduce this concept which is assigned to
the static analysis but not to the dynamic analysis of the structure. Hence, the Eurocode 1
establishes that:
 The first principle within this section, and one of the main objectives of this research,
establishes that a verification of the vertical acceleration peak values has to be carried
out as part of the Serviceability Limit State (SLS) and the traffic safety. This is due to
the fact that the Eurocode 1 establishes that an excessive value regarding the peaks of
the vertical acceleration can incur in a track instability and the consequent derailment
of the train.
 The second principle mentions that the maximum allowable peak values of the vertical
acceleration through the rail track are specified at the national annex document A2 of
EN 1990.
Criteria for traffic safety. Serviceability Limit State
The AnnexA2 (European Committee for Standardization, 2003) is the specific normative
of railway bridges and the limits states of the same under the passage of a high-speed
train. First of all is required to mention that the loads introduced at the SLS have to be
applied with characteristic values based on the criteria appointed by the Eurocode 1.
The section A2.4.4.2 “Criteria for traffic safety” (European Committee for
Standardization, 2003) is divided in five main sections: the vertical acceleration of the
deck, the deck twist, the vertical deformation of the deck, the transverse deformation and
vibration of the deck, and the longitudinal displacement of the deck. This statement is
mainly focused on the two first serviceability limit states.

14
A. Vertical acceleration of the deck:
The national annex A2 mentions that the maximum peaks values of the vertical
acceleration of the deck slab should not exceed the next design values:
a) γbt = 3.5m/s2
for ballast track.
b) γdf = 5.0m/s2
for direct fastened decks.
Concerning the frequencies which must be introduced at the analysis of the structural
bridge, the Eurocode 1 establishes that it should be considered one of the next criteria:
a) 30Hz;
b) 1.5 times the frequency of the first mode of vibration of the element being considered
including at least the first three modes.
B. Vertical deformation of the deck:
The first principle, and the most significant concerning the deformation of the railway
deck slab, sets that for all the structures which are loaded with characteristic vertical
values the maximum vertical deformation of the deck slab due to the rail loads should not
exceed the value L/600.
Dominguez Barbero (2001) establishes an equation for the case of the isostatic railway
bridges (equation 2.1) where it establishes the correlation between the vertical
deformation of the deck slab and the structural parameters. It states that the variation of
the modal rigidity of the isostatic bridge depends uniquely upon the span length, and the
structural stiffness. Thus, it can be established previously to the analysis of this
investigation that the results expected correspond to a modification of the vertical deck
deflection with the increment of the structural stiffness. However, the modification of the
structural dead load mass of the structure shall not provoke any increment regarding the
vertical deformation of the deck slab.
𝐾𝑖 = 𝑖4
∗ 𝜋4
∗
𝐸𝐼
𝐿4
∗ 𝜇( 𝐿) = 𝐾𝑖(𝐿3
, 𝐸𝐼, 𝑖4
) (2.2)
Figure 2.7 Definition of angular rotations at the end of decks

15
2.3 Dynamic analysis
2.3.1 Basic concepts
The dynamic analysis is one of the most complex analysis that can be found at the
structural mechanics and this is due to the fact that the typical analysis of a structure is
based uniquely upon the statistic analysis. The dynamic forces are characterised by
changing its position and/or magnitude in the time and it can result in a different response
of the structure in comparison with the statistic analysis (Gustaffson, 2008).
The dynamic analysis has become in one of the most developed analysis during the last
decades in railways being caused by the apparition of a known mode of transport such
the high-speed railway. Moreover, the dynamic analysis is required in the majority of
railway structures and it conditions the project in most of the cases. Thus, the employment
of simplified methodology concerning the dynamics analysis such as the experimental
enveloping methods is obsoleted. This is due to the apparition of the high-speed train
which reaches speed over the 200km/h and with a variety of railway infrastructures such
as the response of the isostatic bridges or some kind of hyperstatic bridges; the continuous
viacducts (Dominguez Barbero, 2001).
The Single Degree of Freedom (SDOF)
The Single degree of freedom (SDOF) is a system in which there is only one independent
displacement coordinate. SDOF systems allows to express the dynamic behaviour of
many structures in terms of single coordinate and that SDOF approach can apply directly
on those. The response of complex linear structures can be expressed as the sum of the
responses of a series of SDOF systems so that this same treatment once again applies to
each system in the series. Thus, the SDOF analysis techniques provide the basis for
treating the vast majority of structural dynamic problems (Khokhar, 2014a).
There are some basic parameter regarding the SDOF and the D’Alembert’s Principle
(equation 2.3) for equation of Motion for SDOF such as the mass element, the spring
element, the damping element and the exciting force (Gonzalez, 2008):
𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝐹(𝑡) (2.3)

16
Multidegree of freedom (MDOF)
Even though the single degree of freedom let analyse the dynamic behaviour of some
kind of simple structural systems, the majority of the mechanical structures are based on
more complex multidegree of freedom models (Karlsson & Nilsson, 2007).
Figure 2.8 Multi degree of freedom scheme
Source: Karlsson & Nilsson (2007)
Karlsson and Nilsson (2007) states an expression which is the generalised fundamental
of motion using matrix equation. The terms M, C and U represent the system’s mass,
damping and stiffness matrices and the term p(t) represent the external vector load
dependent of the time (equation 2.4):
𝑀𝑢̈ + 𝐶𝑢̇ + 𝐾𝑢 = 𝑝( 𝑡) (2.4)
Damping in MDOF systems
a. Viscous damping
The viscous damping is considered by different authors as the main damping factor within
a material and specifically introduced for the system modelling since this parameter is
linear. It acts dissipating the energy in a vibrating system creating a force proportional to
the velocity of the particle which has been considered but opposite to the direction where
the particle is moving to (Khokhar, 2014a).
b. Modal damping
The type of damping most frequently used in the structural dynamic computation is the
modal damping. Modal damping is assumed to satisfy orthogonality. By utilising this
diagonalised form of the modal damping, the set of coupled equations of motion are
transformed into N un-coupled equations of motion in modal coordinates. Unlike the
Rayleigh damping, the modal damping allows all N damping factors to be assigned
individual values. Typical values are in the range between 1% and 10% (Karlsson &
Nilsson, 2007).

17
Figure 2.9 The peak value of the dynamic response occurs due to resonance. The
value is strongly dependent on the damping coefficient.
2.3.2 Resonance phenomenon
The resonance phenomenon is the conditional phenomenon of the structures under the
passage of trains and especially under the passage of high-speed trains which are able to
reach speeds over 200-220 Km/h, when the resonance phenomenon is more evident
(Carnerero Ruiz, 2007). The resonance phenomenon in the railway bridge occurs because
of the repetitive action of the bogie loads of the train acting over the structural bridge,
with a cadence coincident with the eigenmodes of vibration from the structure. This event
coinciding with the movement caused by the natural frequency of the structure generates
a progressive increment of the energy transmitted on the bridge, with the consequent
apparition of excessive displacements and accelerations (Carnerero Ruiz, 2007; Karlsson
& Nilsson, 2007; Moliner Cabedo, 2012).
The natural frequencies and eigenmodes of the structure
Gillet (2010) comments that the natural frequency of the structural bridge is one of the
most important dynamic characteristics of the bridge. It is also said that “they characterise
the extent to which the bridge is sensitive to dynamic loads”. This parameter is measured
by the number of vibrations per second of the structure or Hertz (Hz). Gillet (2010)
highlights an important concept concerning the natural frequency of the structure which
is that a mechanical systems with a continuous distributed mass have an infinite number
of natural frequencies. Natural frequencies are calculated by the equation:
𝑓𝑗 = 𝜆𝑗 (2.5)

18
Figure 2.10 Displacement in a point in the centre of the bridge
Source: Gustafsson (2008)
The mathematic condition that indicates us the speed at which the resonance phenomenon
can occurs, due to the action of a train, is that one at which the period that two successive
bogies of a train acting (te) on an specific point of the bridge is coincident with a
eigenmode of vibration of the structure (Tn) or one of its numerous multiples (i)
(Gustaffson, 2008):
𝑡𝑒 = 𝑇𝑛 ∗ 𝑖 (2.6)
This expression can also be expressed as a function of train’s speed (v), the regular
distance between the bogies of the train (d), the first natural frequency of the railway
bridge (f) and the eigenmodes of vibration of the structure considered (i) (Carnerero Ruiz,
2007):
𝑑
𝑣
=
1
𝑓
∗ 𝑖 (2.7)
Rewarding the equation, the critical speed (v) at which the resonance phenomenon
appears is calculated as:
𝑣 =
𝑑∗𝑓
𝑖
(2.8)
Figure 2.11 Regular train bogie’s distance

19
2.3.3 Mode analysis
There are numerous mode analysis which can be used to solve the dynamic analysis of a
structure for both SDOF and MDOF models. Depending on the complexity of the model
and the structure, there are several analysis which can be employed such as the numerical
analysis or the analysis based on the Finite Element Methods. Within the numerical and
analytical analysis, there are numerous types of analyses which can be applied to a
structure with some limitations which can restrict their use. Some of the most known
analytical analyses are the superposition method, the numerical integration or the inverse
iteration (Eriksson & Trolin, 2010).
However, this statement is focused on the analysis of the structural bridge based on a
Finite Element Method software which in its turn provides different Finite Element
analyses of the structure depending on the grade of accuracy in which the structure is
analysed.
Dynamic analysis. Finite Element Method analysis
The versatility in the application of these types of analysis in complex structures as well
as the facility for creating these types of models has established the Finite Element
Method analysis an excellence tool within the Civil Engineering. Within the generality of
methods within this type of FE analyses, there are two ways to undertake the calculation
of the dynamic solicitations caused by the passage of a points load train which are the
direct integration method and the modal method (Dominguez Barbero, 2001). The use of
this type of methodology is available for both the linear and nonlinear dynamic behaviour
of complex structures. “This analysis is based on the direct time integration of the
dynamic equilibrium equations of the structure, under the actions of a train of loads” (Da
Silva Dias, 2007). Hence, the structural model can be analysed with the complete
integration of the structure taking a discrete system with N degrees of freedom. However,
the FE analysis can be also undertaken reducing the number of degree of freedom by the
previous modal method of the structure (Da Silva Dias, 2007).
The software used for the analysis of this dissertation has available the two types of
methods based on a Finite Element model which are the direct integration and the modal
method:

20
a) Direct time integration methods: The N degree of freedom which characterise the
structure have to be defined and they are solved by each instant of the integration
(equation 2.7). Taking into account that the equation are generally coupled, these are
solved simultaneously (Dominguez Barbero, 2001; Da Silva Dias, 2007):
𝑀𝑢̈ + 𝐶𝑢̇ + 𝐾𝑢 = 𝑝( 𝑡) (2.9)
b) Modal method: This method is only applicable for a linear behaviour of the main
structure. First of all, the auto values are extracted and the n more significant eigenmodes
of vibration are selected (N >> n). In a second stage, the eigenmodes of vibration are
integrated in the time. The equation obtained from each one of the eigenmodes of vibraton
is uncoupled from the rest, thus in the last term, the system is reduced to a system of an
only degree of freedom (Dominguez Barbero, 2001). As Da Silva Dias (2007) states, the
number of eigenmodes of vibration of the main structure must be representative of the
bridge dynamic response. The modal method is the one which has been chosen for this
research as the Finite Element analysis of this investigation using the software
CSIBridge2015.
Modal method. Mass participation factor
The mass participation factor is an often studied parameter when trying to determine
which modes must be included in the modal method. The mass participation factor is an
indicator of the percentage of the total mass which is being introduce actively at the model
and the direction in which the model is moving to (Karlsson & Nilsson, 2007). It is proven
that modes with a large mass participation factor have a greater influence on describing
the response of the structure to a dynamic load. Unreliable solution will be obtained if
modes of vibration with significant mass contribution are missed (Da Silva Dias, 2007).
Numerous national building codes such as the Eurocode or the Uniform Building Codes
(UBC) require that the sum of the all significant modes must be greater than the 90% of
the total mass included in the analysis (Karlsson & Nilsson, 2007).
𝑆𝑀𝑃𝐹𝐽 = ∑ 𝑀𝑃𝐹𝑖𝑗𝑚
𝑖=1 ≥ 0.9; 𝑗 = 𝑥, 𝑦, 𝑧 (2.10)
The equation 2.8 is stating that the total translational mass of the considered modes must
be greater than 90%.

21
Figure 2.13 Step force with ramp
Source: Khokhar (2014b)
Modal method. Model of a train of moving loads
A. Simple moving train loads
The simplest way to define a train of moving loads in a Finite Element model is applying
a load history in each one of the nodes that form part of the structure as it can be shown
in the figure 2.12. For a time step ti and an axle force, F, a nodal load Fj is assigned to the
node j if the axle is above an element that contains that node.
B. Response to Arbitrary Dynamic Excitation. Step Force with Ramp
This step force with ramp is one of the numerous response to Arbitrary Dynamic
Excitation which can be found in the fundamental concepts of the Response to Periodic
and Arbitrary Dynamic Excitation (Khokhar, 2014b). The point A explains the analysis
of a train of moving loads but with simple moving loads where there is not interaction
between the loads in each one of the nodes where they are passing. However, this is not
a real behaviour of the Finite Element model where the increment of the speed of the train
over a structure will generate an ascendant force applied to each one of the infinite nodes
in which the model is divided (CSI, 2011). That is why a step force with ramp model
represents more accurately the behaviour of the moving train loads passing over the
structural bridge. Figure 2.13 represents the increment of the load at the node while the
time is varying and how the maximum load is maintained when the train is passing over
the bridge.
Figure 2.12 Simple moving train loads
Source: Dominguez Barbero (2001)

22
2.4 Fluid Viscous Dampers (FVD)
The Fluid Viscous Dampers (FVD) have been steadily used during the last twenty years
as an innovative solution created to work initially both as a seismic isolator and as a
passive energy dissipater technology (Castellano, Colato, & Infanti, 2004). The fluid
viscous dampers have been also recently introduced in some concrete and monumental
masonry buildings to protect them from the seismic conditions and wind loads due to the
high availability and the low cost on the market of these systems (Castellano, Colato, &
Infanti, 2004). Additionally, this technology has been introduced in some buildings in
order to attenuate the excessive floor vibrations provoked by the human activities
occurred because of the ignorance of the damping conditions of the materials employed
in the building construction (Saidi et. al., 2011). Castellano, Infanti and Kang (2004) have
an article concerning the introduction of the Fluid Viscous Dampers (FVD) as a retrofit
technology in some Korean bridges in order to protect them from the seismic vibrations
provoked by the area earthquakes. This article also highlights that the fluid viscous
dampers have resulted to be efficient, economical and reliable to protect the structures
which are located at earthquake areas.
Figure 2.14 and 2.15 shows the mechanisms used in the introduction of the fluid viscous
dampers at the bridge bearings of the Korean bridge appointed at the article written by
Castellano, Infanti, & Kang (2004) in order to mitigate the energy transmitted by the
earthquakes at the areas located.
Figure 2.14 A viscous damper as installed
Source: Castellano, Infanti, & Kang (2004)

23
Figure 2.15 The Ok-Yeo Bridge Dampers
Source: Castellano, Infanti, & Kang (2004)
An article written by Lavado, Domenech and Martinez-Rodrigo (2014) explains the
introduction of FVD introduced with clamped beams in the space that remains between
the deck slab and the abutments. The article remains as an essential reference of this
statement due to the fact that this article is one of the few articles found by the author
where the FVD technology has been introduced in a high-speed railway bridge. This
article undertakes the analysis about the effectiveness of this technology and the
consequent reduction of the excessive vertical acceleration, mainly during the resonance
phenomenon caused by the passage of a high-speed train.
Figure 2.16 represents the difference between the linear and non-linear fluid viscous
dampers and the mitigation of the energy with respect to the displacement experimented.
This statement has decided to introduce uniquely linear Fluid Viscous Dampers (FVD) in
order to mitigate the peaks of the excessive vertical acceleration of the deck slab.
Figure 2.16 Equivalent linear and non-linear FVDs
Source: Berton, Bolander, & Strandgaard (2004)

24
Figure 2.17 shows again the relation between the damper force (FD), and the relative
velocity (V) experimented by the FVD. It can be shown how the linear damper presents
a continuous linearity of the relation between these two variable, whereas the variation of
the exponent value will incur in an exponential behaviour of this correlation.
Figure 2.17 Force-Velocity relationships of Viscous Dampers
Source: Hwang, J.-S. (2005)
Hence, this research has decided to simplify the calculations concerning the introduction
of this innovative material and has established a simulation regarding the application of
linear FVD at the bearings of the bridge model. Thus, equation 2.11 is the typical damper
characteristic law and it represents the variables required to introduce the material linear
Fluid Viscous Damper into a structure. This equation establishes the correlation between
the force damping applied to the material (FD), the coefficient damping introduced into
the FVD and the relative velocity experimented by the FVD. This equation also presents
the exponent α concerning the velocity that is applied to the non-linear FVD, but for this
investigation the final value for this exponent will be 1, corresponding to the linear
correlation previously mentioned:
𝐹𝐷 = 𝐶 ∗ 𝑉 𝛼
(2.11)

25
3. METHODOLOGY
This dissertation has introduced a 3D software in order to carry out the dynamic analysis
of the isostatic railway bridge. CSiBridge®2015 is the computational software created by
the same company as SAP2000 and it is considered a Finite Element Model software
where the structure is completely analysed in infinitesimal sections. This software allows
to model, analyse and design bridge structures from many different typologies with the
employment of all the materials available within the field of the Civil Engineering. This
computational tool is able to model all the types of bridge geometries, boundary
conditions and load cases. Moreover, this software allows introduce, with a high degree
of accuracy, most part of the elements currently present at the bridge structures such as
the span conditions, abutments, the deck slab features, bearings, spring foundations,
bents, hinges and steel tensioning conditions. In fact, it allows you model the structure
depending upon the degree of accuracy of the structural design introduced at the software.
Thus, the structural bridge can be divided into beam elements, shell elements or the entire
structure as a unique solid object model.
This research has introduced the shell element as the main model conditions for various
reasons but the most important is the limitation of the computational conditions due to
the fact that this project has been analysed in a last generation laptop with some memory
running limitations. The shell element is a more complex model in comparison with the
beam element model where the model is divided in numerous shell elements with
different thickness, depending on the section of the structure, and a zero eccentricity.
Hence, this research has introduced the shell element model to design and analyse the
isostatic four short span bridge which has been obtained from the Spanish organisation
“Administracion de Infraestructuras Ferrovariarias” (ADIF). Unfortunately, the Spanish
organisation did not provide the exact location and the Spanish high-speed line that
contains the bridge structure in order to preserve this confidential information. However,
the organisation has provided all the technical information related to the structural
parameters/aspects as well as the totality of the bridge planes of the site construction
(Appendix C). Thus, this research has designed and modelled the isostatic bridge from
design planes and technical data which is currently employed to construct a new high-
speed line in the Spanish railway network. Therefore, the majority of the bridge sections
have been simulated according to the real isostatic four short span bridge.

26
3.1 Bridge model characteristics
3.1.1 Materials’ properties
The main materials introduced at the model are the various types of concrete of the
different bridge sections as well as the active and passive steel embedded into the
reinforced concrete. Actually, most of these materials are unique in the Spanish
construction due to the fact that it has also got its own construction normative:
HA-30: This is a type of concrete characterised by a ‘specified concrete compressive
strength’ (f´c) equal to 30MPa. Moreover, the Young’s Modulus (E) has got a value equal
to 33.6MPa and a poison’s ratio 0.2. The density of the material is 2549kg/m3
.
Additionally, a percentage of viscous damping equal to 1% has been introduced at the
material based on the BSI requirements.
HA-50: This is another type of concrete material with the only difference in comparison
to the HA-30 that the value of the ‘concrete compressive strength’ (f´c) is equal to 50MPa
and the Young’s modulus 38660MPa. A viscous damping equal to 1% has also been
introduced at this concrete material.
Rebar Y1860S7: This denomination corresponds to the steel material employed as the
active steel embedded into the reinforced concrete. The main material properties are:
density = 7849kg/m3
, E = 195000MPa, minimum yield stress (Fy) = 1767MPa, minimum
tensile stress (Fu) = 1860MPa and viscous damping equal to 0.5%.
Rebar S500D: This denomination corresponds to the steel material employed as the
passive steel embedded into the reinforced concrete. The main material properties are:
density = 8000kg/m3
, E = 200000MPa, minimum yield stress (Fy) = 500MPa, minimum
tensile stress (Fu) = 575MPa, and viscous damping equal to 0.5%.
3.1.2 Frame’s features
The main frame sections used in the bridge model have been divided into four groups:
the piers, the head piers, the prestressed beam section and the deck slab section (figure
3.1). The research has modelled all the technical data provided by ADIF and it has been
applied into the structural bridge model. Thus, the piers, the deck slab and the head piers
have been modelled with the concrete type HA-30, whereas the prestressed beams have
been modelled with the concrete type HA-50 established by the Spanish Organisation.

27
Figure 3.2 Structural bridge model. Bridge section design
Figure 3.1 Structural bridge model. Deck slab
3.1.3 Deck slab
The deck slab of the bridge model is considered as one of the essential parts of the
structure since it is the section of the bridge which is going to support the overall load
applied over the structure. Figure 3.2 and 3.3 represent the design of the deck slab and
the prestressed beams which have been installed above the piers and the head piers. The
methodology of the dynamic analysis has been focused on the mid points of the deck slab
spans in order to measure the maximum vertical acceleration and vertical deflection as
part of the SLS. The deck slab has been designed and modelled having into account the
prescriptions provided by ADIF concerning the type of concrete, the steel embedded into
the concrete and the location of these steel rebar.

28
Figure 3.3 Structural bridge model. Deck slab with prestressed beams
3.1.4 Bearings
The software introduces the bearings at the structural bridge in order to determine the
boundary conditions of the analysis. Six degrees of freedom are available to be modified
(U1, U2, U3, R1, R2 and R3) between the options of fixed, released and introducing an
specific stiffness. Hence, this research has distinguished between two types of bearings:
the bearings which are located at the abutments and the bearings located on the head piers.
With respect to the bearings introduced on the abutments, this research has decided to
model these bearings as completely fixed bearings based on the data provided by ADIF.
Secondly, the bearings introduced over the head piers have been modelled as fixed in the
three main directions and R3, with R1 and R2 released. Thus, this model let the rotation
of the bearings in the RX and RY directions simulating a pinned connection between the
deck slab and the head piers and consequently the isostatic conditions of the investigation.

29
3.1.5 Foundation springs
The foundation springs are the elements which represent the connection between the
ground and the structural bridge. As it has happened with the bearing properties, two
different foundation spring conditions can be found: the foundation spring between the
abutment and the ground, and the foundation spring between the bents of the piers and
the ground. Nevertheless, both types of foundation springs have been modelled with the
same boundary conditions; completely fixed in the three linear and rotational directions.
This allows the complete fixed conditions, avoiding the translation in any direction.
3.1.6 Abutments
The abutments are the structural elements which are located at the extremes of the bridge
which connect the deck slab and the girder support conditions with the foundation springs
and the ground. This research has decided to simplify this section within the structural
bridge connecting the prestressed beams integrally with the foundation springs and the
deck slab instead of doing it uniquely at the bottom of the beams. Thus, the fixed
connection between the deck slab, the prestressed beams and the foundation is complete,
even though this does not represent the real structural behaviour.
3.1.7 Prestressed beams
This research has designed the prestressed beams introducing active steel bars Y1860S7
which have been used according to the technical data provided. The prestressed forces of
the steel bars have not been completely reached by this research since it would incur an
excessive employment of computational memory that would not let the proper analysis
of the research. Moreover, this research has also simulated the passive steel bars that have
been appointed at the technical planes by the introduction of steel bars S500D, modelling
the most realistically possible the structural response of the prestressed beams.
3.1.8 Bents
The bents have been designed simulating an approximated realistic behaviour of the
technical data provided by ADIF. Thus, the connections between the piers and the ground
have been uniquely obtained by the support conditions mentioned at the spring foundation
properties. Furthermore, the head pier which is originally located at the bents of the
structural bridge has been located at the top of the pier columns due to the software
computational limitations.

30
3.2 Loads configurations
3.2.1 Dead load
One of the most important aspects of this dissertation is the load combination and the
different loads introduced into the analysis. The structural bridge has got its own weight
load caused by the materials which have been employed to construct the bridge. However,
there are many other loads introduced at the design due to the fact that this is a railway
bridge. Hence, this mode of transport requires some different infrastructure elements such
as the ballast underneath the tracks, the rail tracks, and the sleepers which form part of
the rail tracks. All these elements have been designed according to the British Standards
requirements (BSI, 2003) which specify the conditions of all these loads related to the
railway bridge infrastructure. This research has decided to divide the railway
infrastructure loads within two groups: the line loads, which are the rail track, and the
area loads, which are the loads caused by the sleepers and the ballast.
Figure 3.4 Technical data. Cross-section deck slab
Linear loads. Rail tracks
The rail section introduced at the high-speed line is a UIC60 (figure 3.5) which has got a
higher weight in comparison to the conventional rail tracks due to it is required to increase
the inertia of the rail track, and thus the resistance to the flexion. Therefore, this rail track
must have a minimum weight equal to 60.21kg/m (figure 3.6). The bridge structure has
got two lanes, one in each direction, thus it has been introduced two rail tracks per lane:
𝐿𝑖𝑛𝑒𝑎𝑟 𝑙𝑜𝑎𝑑. 𝑅𝑎𝑖𝑙 𝑡𝑟𝑎𝑐𝑘 =
60.21𝑘𝑔
𝑚
=
590.458𝑁
𝑚
(3.1)

31
Figure 3.6 Linear loads. Rail track load
Figure 3.5 Rail tracks. UIC60
Source: Ruano Gomez (2007)
Area loads
A. Sleepers
With respect to the sleepers, the Spanish high-speed line has introduced an innovative
sleeper, specially prepared for the rail track UIC60, known as mono-bloc sleeper (figure
3.7). This sleeper has got a length equal to 250cm, a thickness equivalent to 30cm and a
weight of 350kg/sleeper. The mono-block sleepers are designed to be introduced
underneath the rail track with a consecutive distance between each other equal to 60cm.
However, this research has decided to generate the worst case scenario based on an area
load with an overall length equal to the bridge length (84m), the thickness of the sleepers
(0.3m) and the weight per sleeper. Thus, the area load is calculated as:
𝐴𝑟𝑒𝑎 𝑙𝑜𝑎𝑑. 𝑆𝑙𝑒𝑒𝑝𝑒𝑟 =
350𝑘𝑔
𝑚3 ∗ 0.3𝑚 =
105𝑘𝑔
𝑚2 = 1029.69825𝑁/𝑚2
(3.2)

32
Figure 3.8 Area load. Ballast’s load and sleeper’s load
Figure 3.7 Mono-bloc sleeper
Source: Ruano Gomez (2007)
B. Ballast track
The ballast introduced at the original high-speed line presents various variable thickness
all along the rail track, nevertheless this research has decided to present the worst case
scenario with the introduction of the area load caused by the ballast underneath the rail
tracks. The ballast used usually in high-speed lines has got a density equal to
2026.8969kg/m3
. Thus, the area load is obtained from the density of the ballast and the
constant thickness introduced by this investigation equal to 0.6 metres. Therefore, the
total area load is calculated as (figure 3.8):
𝐴𝑟𝑒𝑎 𝑙𝑜𝑎𝑑. 𝐵𝑎𝑙𝑙𝑎𝑠𝑡 =
2039.4037𝑘𝑔
𝑚3 ∗ 0.6𝑚 =
1223.6422𝑘𝑔
𝑚2 = 12000𝑁/𝑚2
(3.3)
𝐴𝑟𝑒𝑎 𝑙𝑜𝑎𝑑. 𝑇𝑜𝑡𝑎𝑙 =
1029.69825𝑁
𝑚2 + 12000𝑁/𝑚2
= 13029.69825𝑁/𝑚2
(3.4)

33
3.2.2 Moving loads
CSiBridge®2015 also allows design the vehicles which are going to pass over the bridge.
Thus, the research has designed the load models corresponding to high-speed trains
passing over the bridge model based on the Eurocode requirements specified in the
literature review. The point axle loads introduced due to the position of the bogies axles
and the length of the coaches have been designed and applied into the model analysis
depending upon the train load model (figure 2.5).
Figure 3.9 Moving load. Train lanes
Figure 3.9 represents the train lanes designed over the structural bridge model according
to the national annex specifications but also the technical data provided by ADIF. Each
one of the lanes have been represented by a colour. The green colour is the first lane and
it is the lane which has been chosen to be used as the train lane during the analysis of the
investigation. The blue colour represents the second train lane crossing the structural
bridge in the opposite direction, and it has not been used in this investigation.
All the load models have been analysed passing over the bridge model with a speed’s
range between 100km/h and 500km/h. This range of speed has been undertaken
deliberately in order to analyse higher speeds than the design high-speed line limit, which
is 350km/h. Thus, this research has carried out a dynamic analysis each 10km/h with
respect to each one of the load models, with a total of 40 measurements per load model,
obtaining ten different graphs in each one of the isostatic four span railway bridge. This
will allow compare the results of the structural response to the dynamic analysis,
depending on the load model that is passing over the bridge and the speed of the train.

34
3.2.3 Load combinations. Dead and moving load
Finally, the dynamic analysis has been undertaken having into account the load
combination of the dead load (self-weight bridge structure, sleeper, rail track and ballast)
and the moving load (from load model HSLM-A1 to HSLM-A10). BSI (BSI, 2003) points
out that the load combination when carrying out a Serviceability Limit State should be
undertaken introducing a factor 1.0 to the dead load and 1.0 to the variable load, which is
the case of the moving loads. Therefore, this research has applied the overall dead load
and each one of the load models (HSLM-A1 to A10) to the model depending upon the
final objective regarding the dynamic analysis undertaken (figure 3.10 and 3.11).
Figure 3.10 Moving load. First wagons load model HSLM-A1
Figure 3.11 Moving load. Last wagons load model HSLM-A1

35
3.3 Dynamic analysis
The dynamic analysis is the final step within the methodology. Once the entire structure
has been designed, the structural parameters have been modelled and the load
combinations have been introduced, the dynamic analysis has to be carried out. The
dynamic analysis is based on the theory which has been exposed in the literature review.
The Finite Element Method analysis is provided by the software with two different
options: the direct method integration and the modal method. This research has decided
to undertake the modal method due to the higher accuracy of this methodology and the
possibility to modify the eigenmodes values of the whole structure. This is going to allow
establish a comparison between the frequency of loads introduced and the natural
frequency of the entire structure.
3.3.1 Time-history analysis. Time step size
The time history analysis is a specific typology of dynamic analysis within the FEM and
it provides both linear and non-linear responses of the dynamic structural analysis under
the action of loads which are varying with the time. The time-history analysis works
introducing either the direct integration method or the modal method. This research has
introduced the time-history analysis with the modal method.
Hence, the time-history analysis is determined by a group of parameters where one of the
most relevant is the step size. The step size establishes the space of time between one
analysis and the following analysis. Thus, if an analysis introduces a step size of 0.2 sec
with a total number of 100 steps, it implies a total duration of the dynamic analysis equal
to 20 sec. This investigation has been limited by the step size due to the fact that a higher
computational memory is required when the step size is lower. Hence, this research has
estimated that the duration of each one of the time-history analysis shall be around 15
seconds. Therefore a step size of 0.1 seconds with a total number of 150 steps per analysis
have been undertaken in each one of the dynamic analysis carried out.
3.3.2 Structural bridge model. Main dynamic analyses
Once all the settings concerning the use of the computational software, the loads applied
into the structure and the materials as well as the structural conditions which have been
employed for the construction of the bridge design model have been established, the main
analyses which are going to be undertaken are divided into:

36
 In first place, analysing the number of shapes modes (eigenmodes) which are going to
be introduced in the dynamic analysis has to be undertaken.
 After that, the load models HSLM-A1 to HSLM-A10 are going to be introduced at the
structural bridge model in order to determine the structural response in terms of the
vertical acceleration of the deck slab.
 Modifying the structural mass of the bridge model based on the increment of the
density of the different types of concrete introduced. This will allow determine the
SLS concerning the vertical acceleration and the deflection of the deck slab.
 Varying the structural stiffness of the bridge model with the modification of the
Young’s modulus (E) of the concrete types. This will allow calculate both the vertical
acceleration and deformation of the deck slab as part of the SLS.
 Modifying the structural damping by introducing various values of additional modal
damping into the entire structure will be required to establish some results concerning
the vertical acceleration and the deformation of the deck slab (SLS).
 Introducing the Fluid Viscous Dampers (FVD) as a structural damping tool based on
the addition of external structural damping properties. It will be mainly introduced at
the bridge bearings as a practical application of the FVD. The SLS regarding the
vertical acceleration of the deck slab is going to be measured introducing two load
models; HSLM-A1 and HSLM-10.
Figure 3.12 Bridge model. View of the whole structural bridge

37
4. ANALYSIS & RESULTS
4.1 Modal method. Mass participation factor
Before calculating the Serviceability Limit State concerning the vertical acceleration and
deformation of the deck slab, this research has decided to measure the mass participation
factor as one of the previous calculations in order to carry out the dynamic analysis. The
modal method has been set to determine the eigenmodes values which should be required
at the bridge model. As it has been seen in the literature review (page 20), the mass
participation factor should be at least the 90% of the total mass in order to comply with
the Eurocode 1 specifications (BSI, 2003). Figure 4.1 shows that both the static and the
dynamic analysis present an overall value over the 90% of the total mas. This is an
essential requirement before carrying out any model analysis due to the fact that is the
manner to verify that the design is complying with the basis dynamic analysis conditions.
Hence, the research has determined that this bridge model needs to analyse a total number
of 20 eigenmodes in order to reach the 90% of the total mass.
Figure 4.1 Structural bridge model analysis. Total modal mass participation
Figure 4.2, 4.3 and 4.4 are representing the first three mode shapes (eigenmodes) which
have been required to calculate the analysis of this research. It can be observed that the
increment of the eigenmodes cause a greater number of oscillations in each one of the
isostatic spans in which the structural bridge is divided. Thus, the increment of the
eigenmodes let the investigation to generate a higher degree of accuracy when the analysis
of the vertical acceleration is being calculated.
This is essential for the analysis of the structural bridge model due to the fact that the
mass participation factor can change completely the results obtained during the analysis
of the SLS of the deck slab. Thus, the Eurocode 1 has established a minimum value for
both the static and dynamic analysis where the 90% of the total mass shall be reached.

38
Figure 4.2 Structural bridge model. Modal method. First mode shape (eigenmode)
Figure 4.3 Structural bridge model. Modal method. Second mode shape (eigenmode)
Figure 4.4 Structural bridge model. Modal method. Third mode shape (eigenmode)

39
4.2 Serviceability Limit State. Vertical Acceleration
4.2.1 Load models. HSLM-A1 to HSLM-A5
The first analysis of this research has been based on the introduction of the load models
HSLM-A1 to HSLM-A5 into the bridge structure model. Figure 4.5 shows the correlation
between the vertical acceleration at the deck slab of the structural bridge and the speed of
the train for the case of the first span of the bridge. It can be mainly observed that for each
one of the load models exists two vertical acceleration peaks. Moreover, the graph also
shows how the peaks of the vertical acceleration do not occur at the same speed for each
one of the load models. The HSLM-A1 presents a peak vertical acceleration equal to
3.88m/s2
at 400km/h, whereas the load model HSLM-A2 presents its peak vertical
acceleration of 3.657m/s2
at 420km/h. This is due to the resonance speed mentioned in
the literature review (page 17) which occurs as a combination of the regular distance of
the train bogies, the natural frequency of the structure and the eigenmode of vibration
considered. Thus, the research has produced a table (table 4.1) calculating the theoretical
resonance speed at the multiplier i=1 and i=1.5 of each one of the different load models.
Therefore, it can be demonstrated that the theoretical resonance speed occurs
approximately at the same resonance speed which has been calculated at the analysis.
Figure 4.5 Vertical acceleration. Load models. HSLM-A1 to HSLM-A5. First bridge model’s span

40
This is the first important finding of this section, due to the fact that these results assist in
validating the structural bridge model which has been designed for the research. The
concurrence of the theoretical resonance speed (table 4.1) with the resonance speed values
which have been obtained in this first analysis establish the validation of the bridge model.
However, the absolute vertical acceleration at the deck slab is the main aim of this
analysis. Figure 4.5 shows a horizontal line which represents the maximum allowable
vertical acceleration (3.5m/s2
) stipulated by the BSI. Figure 4.5 shows that the majority
of the load models, both at the critical speed and with a multiplier 1.5, exceeds the
allowable vertical acceleration. The analysis of the different load models helps to
represent the exact speed at which the resonance speed occurs and thus the train speed’s
that should be avoided at the structural bridge. Some of these load model produce greater
peak vertical acceleration values due to the weight of the point loads and the rest of the
train bogies distances.
Figure 4.6 represents the correlation between the absolute vertical acceleration and the
train’s speed for each one of the load models, but in this case, at the third span of the
bridge. This graph shows the same speed’s resonance as in the figure 4.5, however there
is a variation with respect to the vertical acceleration of the first span analysis. Figure 4.5
shows that there is a tendency of the peak vertical acceleration values to increase with the
multiplier i=1.5 in comparison with the critical speed (i=1). Figure 4.6 shows the same
tendency but with decreased acceleration values and with higher peaks values at critical
speed than the multiplier i=1.5 in the load models HSLM-A4 and HSLM-A5. This
difference in the structural response of the two spans to the dynamic analysis can be
attributed to the boundary conditions of the span 1 (completely fixed-pinned) and the span
3 (pinned-pinned).
Situacion f0 (Hz) Distance bogies (m) Critical speed i=1 (km/h) Critical speed i=1.5 (km/h)
HSLM-A1 4.01807 18 260.370936 390.556404
HSLM-A2 4.01807 19 274.835988 412.253982
HSLM-A3 4.01807 20 289.30104 433.95156
HSLM-A4 4.01807 21 303.766092 455.649138
HSLM-A5 4.01807 22 318.231144 477.346716
HSLM-A6 4.01807 23 332.696196 499.044294
HSLM-A7 4.01807 24 347.161248 520.741872
HSLM-A8 4.01807 25 361.6263 542.43945
HSLM-A9 4.01807 26 376.091352 564.137028
HLM-A10 4.01807 27 390.556404 585.834606

41
Figure 4.6 Vertical acceleration. Load models. HSLM-A1 to HSLM-5. Second bridge model’s span
Additionally, figure 4.7 shows the correlation between the vertical acceleration and the
train’s speed at the fourth span of the bridge model. Resonance speeds of the load models
are equal to the theoretical resonance speeds and the speeds at the third and fourth span.
Nevertheless, it can be observed that the structural response to the dynamic analysis at
the fourth span presents the opposite behaviour than in the case of the first span (figure
4.5). The peaks of vertical acceleration are greater at the critical speed than at the
multiplier i=1.5, with a tendency to decrease these values at higher speeds. This is due to
the boundary conditions of the first and the fourth span. While the first span has been
completely fixed at the abutment and fixed at the bottom of the prestressed beam, the
fourth span has a fixed support at the bottom of the prestressed beam and a complete fixed
support all along the girders.
Figure 4.7 Vertical acceleration. Load models. HSLM-A1 to HSLM-5. Third bridge model’s span

42
4.2.2 Load models. HSLM-A6 to HSLMA10
As undertaken previously with the load models HSLM-A1 to HSLM-A5, this research
analysed the rest of the load models; from the HSLM-A6 to HSLM-10. This analysis has
been carried out with the initial values of the structural bridge, without any modification
of the structural parameters. Thus, the analysis of the resonance speeds of these load
models should coincide with the theoretical resonance speeds recorded in table 4.1. This
lets the investigation validate the suitability regarding the design of the structural bridge
model as with the first five load models.
Figure 4.8 represents the values of the absolute vertical acceleration of the deck slab
depending upon the train’s speed at the first span of the bridge structure. First of all, due
to the fact that the speed of the multiplier of the resonance speed (i=1.5) in these models
(HSLM-A6 to HSLM-A10) is over 500km/h, a unique range of peak values can be
observed which coincides with the resonance speed of each one of the load models.
However, the analysis of the first five load models has shown two ranges of peak values
and this is due to the fact that the multiplier of the resonance speed was below 500km/h.
A vertical acceleration peak value of 12.01 m/s2
is shown in the load model HSLM-A10
at 370km/h, whereas the theoretical resonance speed has been calculated as 390km/h.
Thus, it can be observed that the theoretical and the Finite Element Method analysis
approximately coincide with the calculation of the resonance speeds and its multipliers in
the same way as the first five load models.
Figure 4.8 Vertical acceleration. Load models. HSLM-A6 to HSLM-10. First bridge model’s span

43
These slight variations between the theoretical and the analytical values of the resonance
speed are due to the fact that the theoretical calculation of the resonance speeds consider
a completely regular length of the train bogies. However, the literature review of this
research has shown that the specific distances of each one of the bogies in each load model
are established by the BSI are not exactly represented as regular bogie distances. Hence,
this research has set irregular bogie distances as part of the analysis and it has obtained
that the introduction of these irregular bogie distances results in lower resonance speeds
than the theoretical resonance speeds with fixed regular bogie distances.
Figure 4.8 shows that each one of the last five load models presents a range of speed
values where the allowable vertical acceleration is exceeded. The load model HSLM-6
shows a speed range between 270km/h and 330km/h where the vertical acceleration is
exceeded, whereas the load model HSLM-A9 represents a speed range between 300km/h
and 410km/h where the vertical acceleration is reaching values of 10.13m/s2
at 360km/h.
This is a relevant result due to the fact that this Finite Element analysis can determine the
range of speed values where the each type of train is exceeding the allowable vertical
acceleration and when to act under this technically unacceptable scenario.
Apart from this result, it is also observed that the values of the vertical acceleration of the
last five load models are much higher in comparison with the first five load models. Thus,
results over the 6m/s2
can be observed in most of the last load models, whereas the first
load models were less than 5-6m/s2
. Moreover, the graph of this load model is clearer in
terms of representation regarding the correlation between the vertical acceleration and the
train’s speed, obtaining consecutive higher values of the vertical acceleration when the
bogie distances of the load models are higher. Specifically, this is because higher
distances of the train bogies allow the software analyse a higher number of results per
second. However, a problem occurred during the first load models where the length of
the train wagons, combined with the lack of precision of the time step of the
computational tool, caused less accurate results in comparison with the last five models.
Figure 4.9 represents the correlation between the vertical acceleration and the train’s
speed at the second span of the structural bridge. This graph shows similar shapes than
the graph of the first span (figure 4.8). Large differences cannot be observed between the
first and second span due to the fact that there are not peaks values related to the multiplier
of the resonance speed as occurred with the first load models. However, the results of the
vertical acceleration show lower values in comparison with the first span, and it means

MEng Dissertation. Ju

Recommended

Recommended

More Related Content

What's hot

What's hot (8)

Similar to MEng Dissertation. Ju

Similar to MEng Dissertation. Ju (20)

MEng Dissertation. Ju