traffic induced bearing loads and movements of a …
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TRAFFIC INDUCED BEARING LOADS AND MOVEMENTS OF A STEEL PLATE-GIRDER BRIDGE
Analytical part of a research into relevant parameters for type testing related to wear and fatigue of structural bearings
A Delft University of Technology thesis by V. Bos
Master of Science ‘Civil Engineering’
January 31, 2011
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University Delft University of Technology (www.tudelft.nl) Faculty Civil Engineering and Geosciences (www.citg.tudelft.nl) Department Design and Construction Section Structural and Building Engineering Principal Ministerie van Infrastructuur en Milieu Rijkswaterstaat (www.rijkswaterstaat.nl) Dienst Infrastructuur Afdeling Staal-, Werktuigbouw en Installatietechniek Master of Science program Civil Engineering Structural Engineering Steel and Timber Construction Graduation committee Prof.ir. F.S.K. Bijlaard (Delft University of Technology) Dr. M.H. Kolstein (Delft University of Technology) Dr.ir. C. van der Veen (Delft University of Technology) Ir. L.J.M. Houben (Delft University of Technology) Dr.Ing. J.S. Leendertz (Rijkswaterstaat) Prof.Dr.Ing. U. Freundt (Bauhaus-Universität Weimar) Date January 31, 2011 Student (number) V. Bos (1380966)
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Summary
Bearings are structural devices which principally transmit vertical forces from a bridge’s
superstructure to its substructure and allow the superstructure to rotate locally about 1 or several
axes. Additionally, they either transfer horizontal forces or allow local horizontal translations of the
superstructure with little resistance. The combination of such movements and contact stresses
causes wear of some bearing components as PTFE-sheets, internal seals and guides. Because
of this, bearings have to be replaced or renovated many times in a bridge’s life, which is rather
expensive and hence not desired by governments, for example. Nowadays, wear sensitive
components are tested for durability or in order to determine the coefficient of friction after a
certain sliding distance. These tests prescribe very little variation in the test parameters (contact
stress, sliding distance and movement velocity) which is exactly what wear is related to. Also the
correctness of the total sliding distances in these tests is questioned. A research into actual
values for these parameters probably results in extended service lives for bearings.
This thesis reports the results of an investigation into bearing loads and movements of a steel
plate-girder bridge (2 side spans of 93.5m, main span 162m) in the Netherlands. The behaviour is
examined in a static analysis in which frictional resistance in the bearings itself is disregarded.
Subsequently, a linear dynamic analysis (disregarding friction as well) was performed in order to
analyse possible dynamic amplifications. A nonlinear analysis of the bridge appeared to be too
complicated (with this program) at the moment and is, therefore, not done and recommended for
further research.
The results show that the support displacements and vertical reactions can perfectly be analysed
statically. The resultant effect of dynamic amplification and frictional resistance appears to be
negligible. The input parameters for the long-term friction test type B (EN 1337-2) correspond to
the results of an average lorry on the bridge. The input parameters for the durability test for
internal seals (EN 1337-5) correspond to the heaviest lorry only. The accumulated displacements
in both tests are much smaller than those found in the analysis (e.g. 80km/year translation
instead of 10km in total). Although it is found that the longitudinal translation and rotation about
the lateral axis are reduced in case of multiple lorries on the bridge (a traffic simulation is needed
in order to obtain the real values, but 30-50% reduction is estimated), still the current values are
exceeded largely. The vertical reactions of intermediate supports increase maximally 10% due to
a single lorry while end support reactions increase up to 55%, however, maximally 20% applies to
70% of the lorries. Finally, the horizontal reactions are predominantly influenced by frictional
resistance of the (intermediate) supports. Since friction is not included in the finite element
calculation, cooperation between supports is disregarded, consequently the development of
horizontal reactions cannot be considered as reliable.
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Preface
January 2011
This thesis has been written in order to complete the Master of Science program ‘Civil
Engineering’ (track ‘Structural Engineering’ and specialisation ‘Steel and Timber Construction’) at
the Delft University of Technology.
I will greatly thank Rijkswaterstaat and the graduation committee for giving me guidance, support
and the opportunity to do this investigation. Additionally, I thank Arie Romeijn for introducing me
in this subject.
Furthermore, I really appreciated everybody’s enthusiasm and commitment in this project.
I enjoyed writing this thesis, hopefully you will enjoy reading!
Vincent Bos
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Definitions, symbols and abbreviations
Definitions
* Definition taken from EN 1337
Backing plate * Metallic component (of a sliding element) which supports sliding materials. Bearing * Bearings are elements allowing rotation between two members of a structure and
transmitting the loads defined in the relevant requirements as well as preventing displacements (fixed bearings), allowing displacements in only one direction (guided bearings) or in all directions of a plane (free bearings) as required.
Degree-Of-Freedom A parameter for displacement (translation or rotation) of a specific point in a
structure. Displacement In this thesis, a displacement is either a translation or rotation. Dynamic Nodal Load A time-dependent point load that can be applied to a node in a finite element
model. Elastomeric pad / disc * Component (of a pot bearing) which provides the rotational capability. Fatigue Load Model A model prescribing axle configurations, axle loads and number of lorries that
must be considered in a fatigue (life) assessment. Friction Pendulum System Isolator In this thesis, a finite element feature that may be used to model friction. Guide * Sliding element which restrains a sliding bearing from moving in one axis. Influence Generating Point Location where influence values are calculated in a MIDAS finite element
influence line analysis. Internal seal * Component (of a pot bearing) which prevents escape of the elastomer material
through the clearance between the recess walls and the piston when a compressive force is applied.
Lorry Influence Line (own definition) Graph that illustrates the magnitude of a parameter, due to a specific lorry as a
function of the longitudinal position of that lorry on the bridge deck. Lubricant * Special grease used to reduce the friction and wear in the sliding surfaces. Mating surface * Hard smooth metallic surface (of a sliding element) against which the PTFE or
composite materials slide. Piston * Component (of a pot bearing) which closes the open end of the recess in the pot
and bears on the elastomeric pad Pot * Component (of a pot bearing) with a machined recess which contains the
elastomeric pad, piston and internal seal. Polytetrafluoroethylene * A thermoplastic material used for its low coefficient of friction. Restraining force Force that is generated on a structural element when the deformation of that
element is prevented. Restraint A (theoretical or physical) limitation of movements. Slide path / sliding distance Relative displacement between 2 (sliding) surfaces. Sliding plane / surface * Combination of a pair of flat or curved surfaces of different materials which allow
relative displacements. Sliding plate See backing plate Spherical cap Partly-spherical steel component of a spherical bearing that lies upon a spherical
shaped PTFE-sheet and can either be directly connected to the superstructure or supporting a translational sliding plane.
Substructure Part of a bridge (piers, abutments, foundation) which transfers the loads, that act
on the superstructure, to the solid ground.
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Superstructure Part of a bridge that spans an obstacle and bears the external loads directly (main steel, concrete, timber or FRP-structure).
Support Theoretical part of a structural element where forces are transferred to other
structural elements or the solid ground. Support system * (of a bridge) The arrangement of bearings and other structural devices and elements which
support the structure and provide for movements. Time Forcing Function A mathematical function (in MIDAS), describing the magnitude of a (dynamic
nodal) load in time. Unit Influence Line (own definition) Graph that illustrates the magnitude of a parameter due to a unit axle load as a
function of the longitudinal position of that load on the bridge deck.
Symbols
Some graphic symbols that have been used in this report need some explanation.
1 Applied load
2 Friction force in plane surfaces
3 Reaction force
4 Fixed bearing
5 Guided bearing
6 Free bearing
Abbreviations
DNL Dynamic Nodal Load
DOF Degree-Of-Freedom
FLM Fatigue Load Model
FPSI Friction Pendulum System Isolator
IGP Influence Generating Point
LIL Lorry Influence Line
PTFE PolyTetraFluoroEthylene
TFF Time Forcing Function
UHMWPE Ultra-High-Molecular-Weight PolyEthylene
UIL Unit Influence Line
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Table of contents
1 Introduction ......................................................................................................................... X
2 Theoretical background: function and historical development of structural bearings, kinematical behaviour and wear of sliding elements and internal seals, evaluation of current test procedures for serviceability and scope of this thesis .........................................1
2.1 The function of structural bearings.................................................................................1
2.1.1 The need for (free) supports in structural mechanics...............................................1
2.1.2 Bridge support systems and the function of structural bearings ...............................4
2.2 Historical development of structural bearings ................................................................6
2.3 Bearing types according to EN 1337 .............................................................................7
2.4 Geometry and kinematical behaviour of pot bearings and spherical bearings.................8
2.4.1 Geometry and global functional principles of pot bearings and spherical bearings.................................................................................................................8
2.4.2 Kinematical behaviour of sliding elements, a relation between displacements and relative displacements .....................................................................................9
2.4.3 Rotational behaviour of pot bearings.....................................................................13
2.5 General context of this thesis: wear of bearing components ........................................14
2.6 Design regulations for sliding elements, guides and internal seals ...............................15
2.6.1 Functional requirements and design issues for sliding elements and guides..........15
2.6.2 Current test for determining the coefficient of friction in sliding planes and guides ..................................................................................................................16
2.6.3 Functional requirements and design issues for internal seals ................................17
2.6.4 Current durability test and criteria for internal seals ...............................................17
2.7 Evaluation of current test procedures and request for research....................................18
2.8 Scope of this thesis .....................................................................................................18
2.9 Plan of action ..............................................................................................................19
3 Geometry, support system and traffic induced mechanical behaviour of the A27 bridge across the river Lek near Hagestein ...................................................................................21
3.1 Global geometry of the bridge, carriageway and nomenclature....................................22
3.2 Support system...........................................................................................................24
3.3 Analysis of the traffic induced mechanical behaviour and corresponding support displacements and reaction forces ..............................................................................25
3.3.1 Support displacements and reactions in the longitudinal/vertical plane as a result of vertical bending of the bridge deck ..........................................................26
3.3.2 Support displacements and reactions in the lateral/vertical plane due to lateral bending and torsion of the bridge deck and local deformations of stability frames .....................................................................................................28
3.3.3 Prediction of direction reaction forces and support displacements .........................31
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4 Explanation of the applied traffic/fatigue load model............................................................35
4.1 General introduction into traffic load models ................................................................35
4.2 Fatigue load models in EN 1991-2 and the applied measurement-based model...........35
4.3 Wheel prints and tyre configuration per axle ................................................................36
4.4 Overview of the applied fatigue load model, FLM M [75/25] .........................................37
5 Explanation about the finite element model.........................................................................39
5.1 Modelling the bridge as a finite element model ............................................................39
5.1.1 Element types and their characteristics.................................................................39
5.1.2 Physical properties of the finite elements ..............................................................40
5.1.3 Consideration of selection and combination of finite elements ...............................41
5.1.4 Explanation about the model’s composition ..........................................................42
5.1.5 The supports’ geometry and the applied boundary conditions in different analyses ..............................................................................................................43
5.2 Modelling self weight and superimposed dead load .....................................................45
5.3 Modelling lorry loads ...................................................................................................45
5.4 Methods for generating time-related displacements and reaction forces ......................45
5.4.1 A static analysis using influence line theory ..........................................................45
5.4.2 Time-history analysis, an examination of the dynamic response............................48
6 Examination of dead load results and validation of the model..............................................51
6.1 Examination of deflection and vertical reactions due to dead load................................51
6.2 Traffic induced deflection and validation of the model’s global stiffness........................52
7 Linear static analysis of support displacements and reaction forces ....................................53
7.1 Examination of UILs, and DOF selection for calculating accumulated displacements.............................................................................................................53
7.1.1 Presentation of UILs and DOF selection ...............................................................53
7.1.2 Discussion about the UILs and a summary of selected DOFs ...............................57
7.2 Examination of LILs and discussion about accumulated displacements .......................57
7.2.1 Explanation about the determination and representation of LILs............................57
7.2.2 Presentation and discussion of LILs and graphs for accumulated displacements ......................................................................................................59
7.2.3 Summary and interpretation of the calculated values for the accumulated displacements ......................................................................................................65
7.3 Influence of interacting lorries on the accumulated displacements ...............................68
7.4 Evaluation and general conclusions from the linear static influence line analysis .........72
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8 Linear dynamic analysis of support displacements and reaction forces ...............................75
8.1 A brief introduction into the need for a dynamic analysis..............................................75
8.2 Examination of the lowest natural frequencies of the bridge.........................................76
8.3 Expectations about dynamic amplifications..................................................................77
8.4 Presentation and evaluation of displacements obtained from a (frictionless) linear time-history analysis....................................................................................................79
9 Discussion about the applicability, effectiveness and accuracy of the methods of working ..............................................................................................................................83
10 Conclusions and recommendations ....................................................................................85
References................................................................................................................................87
ANNEX A: Research into the longitudinal translations of supports 13/14
ANNEX B: Calculation of equivalent main girders’ bottom flanges
ANNEX C: Modal superposition method in case of traffic loads
ANNEX D: Modelling friction with a Friction Pendulum System Isolator
ANNEX E: Weight specification of the bridge’s steel structure
ANNEX F: Influence lines for unit loads and single lorries (UILs and LILs)
ANNEX G: Results of the multiple lorry analysis
ANNEX H: Vibration mode shapes of the 6 lowest natural frequencies
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1 Introduction
Bearings are structural devices which, except for integral bridges, are installed between a
bridge’s superstructure and its substructure. They can be considered as an example of the
practical application of theoretical supports in structural mechanics. Principally they transmit
vertical forces and allow the superstructure to rotate locally about 1 or several axes. Additionally
they either transfer horizontal forces or allow local horizontal translations of the superstructure
with little resistance. The combination of such movements and contact stresses causes wear of
some bearing components as PTFE-sheets, internal seals and guides. Wear affects the
serviceability and material characteristics of bearing components which is the main cause that
bearings by far do not reach the design life of the bridge itself. They have to be renovated or
replaced several times in the bridge’s life which is rather expensive and therefore logically not
desired by governments, for example.
Internal seals are tested for serviceability; sliding elements and guides in order to examine the
development of the coefficient of friction in time. For both tests, very little variation in the test
parameters is prescribed which is exactly what wear is related to. It is, therefore, questioned
whether or not the tests results provide useful information about wear and the expected service
live of bearing components in reality. A research into wear by random loads and movements
perhaps result in an increase of the bearing’s service life.
In that context, it is asked to examine the movements and bearing loads of a steel plate-girder
bridge in the Netherlands which is regarded as representative for many others of the same type.
This thesis reports the results of this research.
As an extended introduction, chapter 2 creates an image of this thesis’ theoretical background.
The first part explains the need for structural bearings from elementary mechanics. The historical
development of bearings and the basic principles of load transfer and movements are described
briefly for several bearing types. This first part is meant for people that are not familiar with
structures and their components rather than senior structural engineers. The second part
discusses the context and scope of this thesis, including wear related parameters and recent test
procedures.
Chapter 3 describes the geometry and support system of the investigated bridge. Moreover, it
discusses the mechanical behaviour and corresponding reactions and support displacements.
Chapter 4 explains the applied traffic load model.
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Chapter 5 is about the finite element model and the methods of working. The first part deals with
the selection of finite elements, their material and stiffness properties, load application, and
boundary conditions. Subsequently, 3 methods are described for the examination of support
displacements and reaction forces.
Chapter 6 is a validation of the finite element model.
Chapter 7 provides the results of the static influence line analysis. Firstly, the results of the static
influence lines due to a unit load are given. Subsequently, the influence lines due to lorries are
presented and discussed. Graphs for the accumulated displacements are shown as well. Then, a
summary is given of the calculated values for the accumulated displacements. Thereafter, 2
parameter studies are discussed about the influence of lorry distances on the accumulated
displacements. Eventually, some conclusions are drawn about the results and applicability of this
method.
Chapter 8 presents the results of the linear dynamic analysis. This part starts with a small
introduction into dynamic amplifications. Subsequently, the lowest natural frequencies are
discussed, followed by some expectations about the dynamic amplifications of certain
parameters. Finally, the results of the dynamic run are presented followed by an evaluation and
conclusion.
Chapter 9 is a small evaluation of the different methods of working.
Chapter 10 gives the general conclusions and recommendations about this research.
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2 Theoretical background: function and historical development of
structural bearings, kinematical behaviour and wear of sliding elements
and internal seals, evaluation of current test procedures for
serviceability and scope of this thesis
2.1 The function of structural bearings
Most people know what a bridge is and some reasons why they are built. They probably can
indicate the most general parts of a bridge as well; most likely they will not mention the bearings
and support system. However, most bridges do have them. What is their function?
2.1.1 The need for (free) supports in structural mechanics
From mechanics it is known that the resultant force on a stationary object is exactly 0 (ΣF=0). We
call this a static or kinematic equilibrium. Structures deform under action of loads, thus local
displacements of structural elements do occur. However, the structure as a whole should
obviously not displace. Reaction forces, generated by the solid ground, on the foundation prevent
the structure from moving globally. The theoretical locations where such forces act, are called
supports. As an example, the simplest and most well-known equilibrium equation to be fulfilled for
structures, is that for vertical forces (ΣFvertical=0). The structure’s weight plus variable vertical loads
(e.g. snow, traffic, furniture), have to be resisted by the solid ground. Sometimes the first earth
layer is able to resist these loads; sometimes a pile foundation has to be built to transfer the loads
to a deeper layer that can withstand the load. The same reasoning applies to horizontal and
rotational equilibrium. A case in which they both have to be achieved, is a building subjected to
horizontal loads (e.g. wind) above ground level. Horizontal reactions ensure horizontal equilibrium
while vertical reactions prevent the structure from rotating. The vertical reactions are essential as
the structure tends to rotate due to the horizontal forces that do not have the same line of action.
A structure must satisfy all equilibrium equations for displacements, 3 for translations and 3 for
rotations. Theoretically, this can be achieved by only a single support that is able to restrain all
imaginable displacements. However, practically this will result in unrealistically heavy and
uneconomic structures and supports. Structures, therefore, have an adequately conceived
support system in which the loads, especially vertical loads, are distributed over several supports.
Common building structures, for example, are frequently supported by an array of piles that are
monolithically connected to the main structure. All these ‘fixed’ supports are able to transmit loads
in vertical and both horizontal directions. Additionally, they restrain rotations to a certain extent.
It seems wise to restrain displacements at many support locations in order to distribute the loads
over a lot of supports. This might be the case for vertical translations; for other displacements,
however, this can be a big mistake.
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In bridge structures, for example, many supports allow rotations and horizontal translations of the
superstructure in order to avoid restraining forces. These forces are generated on a structure
when its deformation is prevented, for instance, by supports. In some cases, restraining forces
might be large, hardly predictable or perhaps unknown. Nevertheless, structures and supports
need to be designed for these forces. Therefore it is better to avoid these restraining forces by
allowing displacements at several support locations.
Deformations are caused by loads and changing temperature. Temperature induced
deformations occur since a body’s volume is temperature dependent (material characteristic).
Structural elements elongate or contract proportional to the elements dimensions and the change
in temperature. Reaction forces are generated on a structure when it is not able to deform due to
the support restraints. This phenomenon is known as thermal action. Large restraining forces can
be generated in bridge structures because they have an elongated shape and are generally
subjected to notable temperature changes.
Consider a generic steel (S235) bar with a cross-sectional area of 2120mm2 and 2 fixed ends. The bar is subjected
to a temperature decrease of 20 degrees Celsius so that the bar tends to contract. Because the restraints prevent the bar ends from moving, a stress in the steel is introduced. These stresses and the axial force in the bar can now be calculated. The axial force is this case is equal to the restraining and reaction forces.
kN107N1010721204.50AF
mmN4.50000,210102.120ET3
barrestrain
25
=⋅≈×=×σ∆=∆
=×⋅×=×α×∆=σ∆ −
The yield strength of steel quality S235 is 235N/mm
2. The calculated tensile stress is 20% of the elastic capacity of
the material which is a significant part and can therefore not be disregarded; especially when the bar in case of compression is sensitive to buckling and the buckling stress is even lower than 235N/mm
2.
Furthermore, bridge superstructures are often supported on piers. Those piers are generally not able to restrain the deformations of the deck because of a relatively low bending stiffness. The deformation of the deck, therefore, causes a bending moment in the piers. This phenomenon is more important for larger span bridges as the displacement is proportional to the bridge span. For a 300 metre span bridge, for instance, the elongation might be more than 70 millimetres.
mm72000,300102.120LTL 5 =×⋅×=×α×∆=∆ −
It must be checked whether or not this deformation is detrimental to the pier structure or supports. Probably the support fails due to the high restraining forces.
Figure 2-1: effects of changing temperature on steel structures with fixed supports.
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Secondly, supports are not located on the neutral axis which means that fixed supports prevent
translations due to rotation and deformation of the cross section.
A bridge deck can be considered as a large beam subjected to bending. From structural mechanics it is known that the neutral axis does not elongate or shorten in case of pure bending. However, bridge supports are not located at the neutral axis but usually below the bottom fibre (e.g. a steel main girder’s bottom flange) which does elongate or shorten. When supports restrain displacements, they restrain the displacements of this bottom part. Therefore additional horizontal reaction forces are generated to neutralize this part’s displacements by means of contra-bending of the deck and an axial force. These forces can be estimated by means of a simple consideration based on basic mechanic’s formulae. The next (upper) formula gives the (resultant) rotation of the beam’s end due to a point load in the beam’s centre and a restraining moment at the beam’s ends. The lower formula gives the normal force generated to neutralize the resultant translation due to rotation of the beam’s ends. Note that it is assumed that the cross sections remain plane which will obviously not be the case in reality.
L
e2EA
L
uEAF
EI
LeF
2
1
EI
LP
16
1
bx
bx2
z
×θ××=×=
×××−
××=θ
Combining these formulae results in a formula for the restraining force.
+×
=
+
××
=
b
b
2
b
b
z
x
ee
i8
L
eeA
I8
L
P
F
The table below indicates for some 10 metre span IPE-sections the ratio between restraining force and applied force (in centre of span). Apparently the restraining force can be a multiple of the applied force. The span over height ratio illustrates that long slender structure experience high restraining forces.
SECTION i [mm] eb [mm] L/h Fx/Pz
IPE100 40.7 50 100 15.0
IPE200 82.6 100 50 7.4
IPE300 125 150 33 4.9
IPE400 165 200 25 3.7
IPE500 204 250 20 3.0
IPE600 243 300 17 2.5
Figure 2-2: effects of restrained deformations on horizontal reaction forces.
Both examples show that the elongated shape of a bridge deck is the main reason that not all
DOFs (Degrees-Of-Freedom) are restrained in bridge structures. Supports that allow structures to
deform freely are needed. However, avoiding restraining forces completely is too ambitious since
the complex 3-dimensional behaviour of structures causes deformations that can not always be
predicted.
The question arises how all loads can be transferred to the solid ground while the bridge as a
whole is not allowed to displace but restraining forces are avoided as much as possible. This can
be achieved by an adequate selection of restraints per support.
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2.1.2 Bridge support systems and the function of structural bearings
The necessity for an adequately conceived support system for bridges has been described in the
previous paragraph. The challenge is to choose per support the right DOFs and restraints to
satisfy all equilibrium equations for static equilibrium of the structure as a whole and additionally
avoid restraining forces as much as possible. This will be elucidated by means of a very simple 2-
dimensional example.
The grey area represents a bridge’s superstructure with 4 supports. The eccentricity of the supports to the neutral axis has not been taken into account. All supports have 3 rotational DOFs (not drawn) and less than 3 translational DOFs. Translational DOFs are indicated by black arrows, reaction forces by red arrows. The reaction forces are drawn in positive direction but may act in opposite direction. To be clear, no translations are possible at support 1 (fixed support), support 2 allows translations in x-direction only (guided support), supports 3 and 4 allow translations in x-direction as well as in y-direction (free supports). This example shows that both translational and rotational equilibrium can be achieved by restraining only 7 from 24 DOFs. In each direction at least 1 support should be able to generate a reaction force to fulfil translational equilibrium in that direction. Theoretically, supports 2, 3 and 4 are unnecessary to achieve translational equilibrium since support 1 is able to generate a reaction force in each direction. The reason for extra vertical supports is logic and already clarified but they have an additional function. They are able to generate a moment and thus rotation about the x-axis and y-axis is restrained as well and hence only rotation about the z-axis still has to be achieved. The reaction forces in y-direction at support 1 and 2 are able to restrain rotation about the z-axis. Conclusion: rotational restraints are unnecessary; a few translational restraints are able to fulfil all equilibrium equations.
Figure 2-3: working of a bridge support system.
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The example shows that not all DOFs have to be restrained. It may be questioned why support 2
is restrained in y-direction in stead of support 3 in x-direction. Firstly, the lever arm is larger and
consequently the reaction forces smaller (in case of an equal Mz). Sometimes support 3 is
restrained in longitudinal direction to distribute the total longitudinal load over 2 supports.
Although that seems to be a reasonable solution, a 3-dimensional analysis shows that additional
restraining forces act on the structure in case of eccentric loading.
The 2 girders with different loads represent a plate-girder bridge with an eccentric load (the girders are connected by means of an orthotropic deck and cross frames and thus experience the same rigid body translation). The right girder bears the highest load and accordingly deflects the most. Supports are connected to the bottom flanges on both sides of the girders in a system as shown. The left support of the right girder has been restrained for longitudinal translations. The cross section of this girder’s end therefore rotates about this point instead of the neutral axis. The neutral axis accordingly shifts longitudinally which results in a movement of the left girder as well because the girders are rigidly connected and the left girder is not restrained against translations. This translation can be seen clearer in the left upper corner of the figure. When the left support of the upper girder was restrained as well, it should generate a reaction force in order to restrain the translation of the girder. This situation does not occur in case of symmetric loading.
Figure 2-4: restraining forces due to eccentric loading.
Furthermore, it should be noted that free supports – i.e. displacements can occur – always react
to the bridge’s displacements to some small extent since infinitely frictionless materials do
(currently) not exist and hence reaction forces in that ‘free’ direction act on the bridge. High bridge
piers (often built in mountainous regions) are susceptible to this phenomenon as friction forces on
top may cause a significant bending moment near the foundation. Therefore designers and
manufacturers should strive for the least resistance against displacements as possible.
Yet, it is clear what a support should be able to do and how a bridge structure can be supported
adequately, but how do such supports look like? A controlled load transfer (i.e. loads must act on
locations where they are designed for) and displacements with little resistance are desirable.
Some supports should restrain displacements while others must allow them. Special structural
devices should ensure this, they are called structural bearings.
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2.2 Historical development of structural bearings
Bridges did not have sophisticated bearings in the past. In the beginning they had no bearings at
all (integral bridges), for instance, ancient masonry arch bridges. Later the need for deformation
capacity was increasing (especially for slender steel bridges) and supporting devices were
designed so that they could follow deformations of the bridge’s superstructure. Still, only
longitudinal translations and mainly rotations about the lateral axis were considered.
In the beginning, completely steel rocker bearings and roller bearings were applied. Point rocker
bearings (point tangency) and line rocker bearings (line tangency) enabled the structure to rotate
about several or one axis respectively but restrained translations. Translations were enabled by
using roller bearings that were able to roll longitudinally only. This was not a major problem as
bridges were narrow and cross frames rather stiff. Therefore lateral displacements were assumed
to be insignificant.
Roller bearings generally have little resistance against displacements. Vertical reaction forces are
transferred through contact between steel plates and the steel roll of the bearing. The small
contact area, due to the curved surface of these types, are subjected to high contact stresses
between the structural elements (Hertzian stresses). Horizontal reaction forces were transferred,
for example, through steel guides.
Bridges increased in width and span length which resulted in more significant lateral
displacements and rotations about the longitudinal axis. Due to the line tangency, roller bearings
and linear rocker bearings restrained these rotations causing an unequal stress field in the
contact surface which resulted in out-of-plane wear. Development of the roller bearing resulted in
higher steel qualities for the bearing. However, these qualities turned out to be sensitive to
corrosion in combination with fatigue. The fatigue problem was caused by stress fluctuations due
to rolling. The steel rollers cracked and the superstructure sometimes dropped several
centimetres down.
Due to the increasing spans and loads, bearings needed to be bigger and bigger. In order to
reduce their size and avoid the problems with rocker bearings and roller bearings, new types of
bearings with a distributed load transfer were needed. The introduction and development of
synthetic, low-friction polymers as PTFE (Teflon) resulted in bearings with low-resistance sliding
planes that have a distributed load transfer. These new bearings, for example pot bearings, were
able to transfer higher loads.
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Initially, pot bearings consisted of a sand-filled steel pot that was able to follow the
superstructure’s rotation. The sand acted as a liquid due to the high pressure. Later, elastomers
were introduced in bearing design because of their deformation capacity and their economic
advantages. Therefore the sand was replaced by elastomeric material.
The use of entirely steel roller and rocker bearings in the past gradually changed into the frequent
use nowadays of partly-steel pot and spherical bearings and elastomeric bearings whose load
transfer is more distributed compared to the concentrated load transfer of their steel counterparts.
Figure 2-5: example of an old-fashioned roller bearing (left) and a modern spherical bearing (right).
2.3 Bearing types according to EN 1337
Different kinds of bearings exist, all with different characteristics and behaviour. There may be
differences, for example, in restraints, load bearing capacity, materials and deformation capacity.
They are usually distinguished in the way they allow displacements, i.e. by deformation of the
bearing (elastomeric bearings), rolling (roller bearings), rocking (rocker bearings) or sliding
(bearings with sliding elements).
EN 1337 shall be used for bearing design, manufacturing, installation, and etcetera. Part 1
subdivides structural bearings mainly in ‘Elastomeric bearings’; ‘Roller bearings’; ‘Pot bearings’;
‘Rocker bearings’; ‘Spherical and cylindrical PTFE bearings’ and ‘Guided bearings and restrained
bearings’. All of these types have a separate part for design rules. Furthermore, there are parts
especially for sliding elements, protection, inspection and maintenance, and transport, storage
and installation.
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2.4 Geometry and kinematical behaviour of pot bearings and spherical bearings
The geometry and kinematical behaviour of pot bearings and spherical bearings will be discussed
as they are frequently applied and this research predominantly concerns these types.
2.4.1 Geometry and global functional principles of pot bearings and spherical bearings
Pot bearings and spherical bearings are by nature rotational elements. Other features, e.g. sliding
elements, have to be added to enable translations. The rotational element is theoretically able to
rotate about every imaginable axis, but principally an axis in the horizontal plane. The main
difference between pot bearings and spherical bearings is the way they allow rotations. Pot
bearings have a steel pot with an enclosed, embedded, elastomeric pad which can deform to
follow the bridge’s rotations. A steel piston covers the pad and transfers the load to the pad and
pot. The elastomer acts as liquid due to high pressure it is subjected to. An internal seal must
avoid that the elastomer is extruded from the pot. Spherical bearings do not have an elastomeric
pad but have a spherical sliding surface that enables rotation of the bridge. This sliding element
may contain a dimpled PTFE-sheet. One part of these bearings is fixed to the bridge’s
substructure; the other part to the superstructure.
Figure 2-6: impression of a free pot bearing (left) and a free spherical bearing (right), impression without
nomenclature from manufacturer 'Maurer Söhne'.
An added sliding element controls the support’s translations (fixed, guided or free). The guided
and free types have a similar translational sliding element. German versions of fixed spherical
bearings contain a translational sliding element as well to avoid unequal stress distributions in the
curved surface. This will be discussed more detailed in the next paragraph.
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The sliding element is generally a combination of a polished austenitic steel plate, which is
connected to the backing plate (by means of full surface bonding, continuous fillet welds,
(counterpunched) screwing or riveting), and a PTFE-disc with lubricant-filled dimples which is
located in a recess in the piston of a pot bearing or in the spherical cap of a spherical bearing.
Due to the movements of the sliding plate and the pressure in the PTFE, the lubricant is smeared
out between the PTFE and the mating surface in order to lower the coefficient of friction.
Horizontal reaction forces are transferred through contact in the steel guides or cover ring.
Guides can be implemented in different ways. For example, a unidirectional guide can be
obtained by an internal guide (see left picture below) or an external guide (see middle picture
below).
Figure 2-7: principle of guided or fixed bearings according to European Standard EN 1337-1.
2.4.2 Kinematical behaviour of sliding elements, a relation between displacements and relative
displacements
NOTE: Some information given in this paragraph is based on a TU Delft master thesis [Driessen,
2003]. It is presented to give a better understanding of the behaviour of bearings with sliding
elements.
In mechanical modelling it is generally assumed that supports are infinitely small points where
reaction forces act or displacements occur. For global behaviour of a structure it is a safe
assumption. However, in this research the behaviour of bearings due to bridge use is investigated
and consequently the behaviour of the bearing itself might be considered as well. It is still
assumed that the mechanical behaviour of the superstructure is not influenced by the detailed
geometry of the bearings (only restraints and DOFs of the superstructure-bearing connection are
of interest), though, a distinction should be made between bridge displacements and the slide
path of the bearing’s sliding elements. This paragraph should make this distinction more clear.
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The slide path is the distance that the mating surface slides over the PTFE. This distance is the
result of relative displacement. Obviously, if the upper part of the bearings moves with the same
distance in the same direction as the lower part, no sliding occurs. This relative displacement is a
contribution to the accumulated sliding distance that can be estimated by considering the bridge’s
displacements only, assuming that those displacements are equal to the slide path. In that case
the elastic shear deformation of the PTFE and additional translations due the rotations will be
disregarded.
Elastic shear deformations of the PTFE, caused by friction, reduces the accumulated sliding
distance since no relative displacement between mating surface and PTFE occur. The question
is, however, whether or not these deformations should be considered significant.
The upper left picture shows schematically a PTFE sheet between a part of the superstructure and a part of the substructure without horizontal displacements. The picture in the middle shows the elastic deflection without sliding in the interface (displacement without sliding path). When the friction is not sufficient to restrain the displacement, the plate will slide over the PTFE (right picture). When the translation stops, the plate stays in its final position (left picture below). As the displacement reverses, the plate stays eccentric and follows the elastic deformation of the PTFE (middle picture) until the friction is again not sufficient anymore and sliding starts again (right picture below). From an existing spherical bearing it is known that the sliding polymer is an 8×Ø260mm PTFE disc with a UHMWPE ring. It is designed for a vertical load of 3300kN. Assume that the ring does not resist the elastic shear deformation of the PTFE disc and that the disc is located in a 6 millimetres deep recess in the steel spherical cap. A simple calculation shows an indication of the horizontal shear deflection caused by friction forces (3% friction). Values for PTFE according to EN 1337-2.
( )
( ) ( )mm027.02
260400
44.0103.01033008h
dE
1F8x
hxA
FG
12
EG
2
3
22=×
×π×
+××⋅×=×
×π×
ν+×µ××=∆
×γ=∆µ×=τγ×=τν+×
=
This example is a little bit conservative but it shows that elastic deformations are small.
Figure 2-8: elastic shear deformation of a PTFE sliding element.
The figure illustrates the elastic shear deformation of a PTFE-sheet. These deformations should
be subtracted from the displacement of the bridge. Here, the elastic shear deformation has been
described because the polymer’s modulus of elasticity is rather.
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Yet, the influence principle of elastic shear deformation on the accumulated sliding distance is
clearer. This distance in case of German spherical bearings is influenced by another
phenomenon as well. That is an additional translation due to the bearing’s geometry.
German fixed spherical bearings contain a translational sliding plane in addition to the spherical
surface, avoiding transmission of horizontal forces through the spherical surface. Horizontal loads
are transferred via guides or cover rings. Bearings without this plane (applied in Italy and the
United Kingdom, for example) transmit horizontal loads through the spherical surface which has a
much smaller radius than the German counterpart.
Bridges supported by the German bearings have 2 centres of rotation per support, one of the
bridge’s superstructure and one of the bearing’s spherical surface. When these 2 centres do not
coincide (which might be the case) and the distance between the sliding interface and both
centres is not equal, different translations occur for the upper part and bottom part since the
rotation of both parts should be equal (same interface).
Although the superstructure can not translate near fixed supports, the sliding plane of fixed
bearings experience translations!
A steel bridge’s superstructure rotates about an axis perpendicular to the paper (the right figure shows the deformed situation). The superstructure rotates about c1, between the 2 guides. The distance from c1 to the sliding plane is r1. For the spherical cap applies c2 and r2. Since the sliding plane is both, part of the superstructure and part of the spherical cap, both should experience the same rotation. The different distances of both centres of rotation to the sliding plane cause a difference in translation. This difference is a contribution to the slide path (s).
( )12
1221
rrs
uusru
−×ϕ=
−=×ϕ=ϕ=ϕ
Figure 2-9: rotational behaviour of a German guided or fixed spherical bearing.
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Free spherical bearings behave differently. Since they are not restrained, they follow the
deformation of the superstructure. Their only task is to support the superstructure vertically.
Therefore the horizontal displacement is related to the rotation of the cross section, the distance
from the neutral axis and the translation of the neutral axis which is caused by the cross sectional
rotation near the fixed support.
The fixed bearing story applies to free bearings as well, except for the superstructure’s centre-of-rotation. For free bearings it is located in the neutral axis of the superstructure. The right picture is drawn in deformed situation.
Figure 2-10: rotational behaviour of German free spherical bearings.
Importantly, these examples show the additional contribution to the accumulated sliding distance
caused by different centres of rotation. The difference in distance between them and the sliding
plane appears to be a predominant factor. Especially for free bearings, the translation and
rotation of the neutral axis itself is relevant as well.
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2.4.3 Rotational behaviour of pot bearings
As described earlier, pot bearings allow rotation by deformation of the elastomeric pad. This pad
acts as liquid due to the pressure and the fact that it is tightly enclosed by the steel pot and
piston. Therefore, depending on the velocity of movement, which is addressed in tests in order to
account for restraining moment and stresses for design purposes, the elastomer is ‘redistributed’
in the pot. The internal seal between the piston and pot prevents leakage by extrusion.
Rotation of the superstructure results in deformation of the elastomeric pad. As a result of the rotation and deformation of the pad, the internal seal moves vertically against the pot wall and will follow the rotation as well. Consequently, the seal is compressed to the pot wall unequally (due to minor horizontal displacement) and hence stress concentrations at the interface between seal and pot wall are introduced.
Figure 2-11: rotational behaviour of pot bearings.
The figure illustrates the rotational behaviour of pot bearings. The theory of additional
displacements due to rotations holds for pot bearings as well. The piston’s centre-of-rotation must
be located in the centre of the interface between pad and piston. Free pot bearings therefore
suffer from extra displacements, fixed pot bearings do not since their centre-of-rotation coincides
with that of the superstructure.
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2.5 General context of this thesis: wear of bearing components
Since they are constantly subjected to relative movements under high pressure, the internal
seals, guides and lubricated PTFE-sheets in sliding elements show wear. For example: loss of
lubricant, snow flakes and loss of material resulting in decreasing volumes and increasing
coefficients of friction. A decreasing volume of the polymers is obviously not desirable since it
may lead to metal-to-metal contact, causing cold welding damage. It has already been explained
what may happen when the coefficient of friction becomes too large and significant restraining
forces are introduced.
Despite the fact that manufacturers continue with bearing design improvements and research for
better material properties (more wear resistance), the bearings’ life still (by far) does not reach the
life of the bridge structure itself (EN 1990 proposes 100 years for bridges and 25 years for
bearings). Therefore they have to be replaced several times in a bridge’s life. As this is a costly
operation (due to jacking operation and new bearings), bridge owners desire an increase of the
bearing’s life.
Wear is highly related to the magnitude of relative displacements and velocities on one hand and
to the magnitude of contact pressure on the other hand. It may be compared with sandpapering a
piece of wood. Pushing hard and fast is more effective than soft and slow. The same reasoning
holds for wear of sliding elements. Movements under high contact pressure are more destructive
than movements under low contact pressure and a fast movement is more detrimental than slow
movements. Furthermore, for bearings also the relative shift (e / D) plays a role with respect to
loss of lubricant.
Wear is a hardly-predictable phenomenon, nevertheless, with a long-term test a lower threshold
can be obtained of the life expectancy of sliding elements and internal seals. For instance, the
sliding element should be able to fulfil its task at least until an accumulated sliding distance of a
certain length has been reached. Realistic input parameters for these tests are essential in order
to obtain realistic results, expectancies and possibly design rules.
EN 1337-2 prescribes some test procedures to determine the coefficient of friction for sliding
elements in long-term tests and short-term tests. EN 1337-5 prescribes tests to investigate the
serviceability of internal seals, whether or not they are able to avoid the elastomer to be extruded,
after a certain slide path or accumulated rotation. Whether or not the test procedures are
representative to investigate wear, should be examined. A project has been started to do this.
This master thesis is a part of that project.
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2.6 Design regulations for sliding elements, guides and internal seals
NOTE: some information, given in this paragraph, is taken from EN 1337-2 (sliding elements) and
EN 1337-5 (pot bearings).
2.6.1 Functional requirements and design issues for sliding elements and guides
The next functional requirement for sliding elements and guides is given in EN 1337-2.
‘Sliding elements and guides permit movements in plane or curved sliding surfaces with a
minimum of friction. Specific verification of friction resistance is required, as verification of
mechanical and physical properties alone is not sufficient to ensure that these components will
have the required characteristics. The performance of the sliding elements and guides is deemed
to be satisfactory if standardized specimens shown in annex D of specified material combinations
meet the requirements of this clause when tested as specified in specific friction tests described
in annex D.’
An important point of interest is the ambiguous expression ‘movements’. A better description is
‘relative movements’ since a sliding element permit one part of the bearing to move
independently of the other part. This distinction is particularly important for wear related problems.
However, for simplicity, the expression ´movements´ is used.
EN 1337-2 prescribes, among other things, properties for the sliding elements to minimize friction
and wear; for example, the mating surface roughness; lubricant properties; dimple pattern and
dimple shape. Furthermore, it prescribes maximum allowable values for the coefficient of friction
of the PTFE from initial type testing and third party controlled FPC (Factory Production Control)
as well.
There has been made a distinction between long-term requirements and short-term requirements
for the coefficient of friction. The long-term coefficients should not be exceeded after a certain
accumulated slide path, e.g. 10000 metres. Differently, the PTFE should have at most a
coefficient of friction as stated in the standard after a certain accumulated sliding distance (test
methods will be discussed in the next paragraph). An additional distinction has been made for
materials of the mating surfaces.
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Separation of sliding surfaces is not allowed under characteristic conditions since it, for instance,
may lead to wear due to contamination and loss of lubricant, and extrusion of PTFE. Therefore it
is required that the sliding surface is always under compression. In addition to that, the mating
surface must always cover the PTFE in order to avoid contamination and direct damage as a
result of planing of the PTFE by the steel mating surface.
Deformation of the backing plates should be considered. The total allowable deformation is
related to the magnitude of protrusion of the PTFE from its recess and the sheet diameter.
Obviously, deformed sliding planes result in stress reductions and increases, e.g. stress
increases on the perimeter, therefore deformation of the plates should be kept as less as
possible. Stress increases may lead to excessive wear of the PTFE and ‘lipping’.
2.6.2 Current test for determining the coefficient of friction in sliding planes and guides
Annex D of EN 1337-2 prescribes how the coefficient of friction for sliding elements has to be
determined. In short, a standardized specimen of PTFE or a composite is moved subsequently
forwards and backwards under lateral pressure of a fixed mating surface until a certain sliding
distance (accumulated) has been reached. The test conditions or parameters have been
prescribed for a short-term test and a long-term test. For this thesis, the long-term test is more
relevant since wear is a long-term phenomenon.
The test is divided into alternating phases, marked A or B, with different test procedures and
parameters. The odd phases, marked as type A in the standard, have to be tested for 1100
cycles per phase using a constant sliding speed of 0.4mm/s, a constant sliding distance per cycle
of 10mm, a constant contact pressure of 0.33 times the characteristic compressive strength of the
material, and different temperatures. The even phase, marked as type B in the standard, have to
be tested for 62500 cycles per phase using a variable (approximately sinusoidal) sliding speed
with an average of 2mm/s, a constant sliding distance per cycle of 8mm, a constant contact
pressure of 0.33 times the characteristic compressive strength of the material, and a constant
temperature of 21 degrees Celsius.
For curved surfaces, these phases should be performed together alternately 5 times (3 times A
and 2 times B) so that the total sliding distance is 2066 metres; plane surfaces have to be tested
together alternately 21 times (11 times A and 10 times B) corresponding to an accumulated
distance of 10242 metres.
After testing, the specimen shall meet the requirements about maximum values for the coefficient
of friction, given in EN 1337-2 as well.
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2.6.3 Functional requirements and design issues for internal seals
The functional requirements stated in EN 1337-5 apply to pot bearings as a whole. The next
requirement for internal seals is given.
‘The internal seal system shall prevent extrusion of the elastomer from the pot’
EN 1337-5 prescribes a kind of unity check regarding the accumulated sliding distance due to
traffic. It must be lower than a threshold value stated in the European Standard or obtained by
testing. The expected value due to traffic loads can be approximated by using the next formula.
T2vd,A sc2
DnS ×≤×α∆×=
In this formula is SA,d the expected accumulated sliding distance due to traffic, nv is the expected
amount of lorries passing the bridge in the bearing’s life (according to FLM3 in EN 1991-2), ∆α2 is
the additional rotation of the pot bearing due to traffic loads, D is the pad diameter, c is a factor
that accounts for the difference between a constant amplitude slide path in the test and a variable
amplitude in reality (in EN 1337-5 taken as 5), sT is the threshold value.
2.6.4 Current durability test and criteria for internal seals
The next procedure for a durability test is described in annex E of EN 1337-5. The piston of a pot
bearing must be loaded increasingly until a 35N/mm2 compressive stress in the elastomeric pad
has been reached. Subsequently, the piston has to be rotated alternately about an axis parallel to
the pad-piston contact surface with a frequency between 0.25Hz and 2.5Hz and a magnitude
between -0.0025rad and 0.0025rad. This has to be done at room temperature until the specified
total slide path of the seal (depending on seal type) against the pot wall has been reached. When
this is done, it has to be checked whether or not the elastomer has been squeezed out of the pot.
When no defects are detected, an additional compression test has to be performed. The piston
must be arranged such that the maximum clearance is achieved in the most adverse position.
Then an increasing vertical force has to be applied to the piston until a 60N/mm2 compressive
stress in the elastomeric pad has been introduced. After several hours, again it has to be checked
whether of not the elastomer has been squeezed out of the pot.
EN 1337-5 prescribes 2 acceptance criteria for this durability test. Wear of the seals is acceptable
as long as the bearing fulfils the next 2 criteria:
1. There shall be no extrusion of cohesive elastomeric material.
2. The compression deformation under the test load shall have not increased for at least 24
hours.
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2.7 Evaluation of current test procedures and request for research
Several topics have been discussed: the function and behaviour of structural bearings with sliding
elements and/or internal seals; the influence of contact pressure, velocity and movements on
wear and the bearing’s life, and design issues and test procedures for these components.
Bridge structures are often heavy structures in which dead load is a major part of the total load.
Nevertheless, variable loads might have a significant addition or subtraction to the total load on a
bearing, depending on the bridge geometry. Furthermore, variable loads are random loads that
change continuously. Therefore bridge deformations are not constant or equally repeating at all.
Consequently, it is very doubtful whether or not constant contact pressure, sliding distance and
velocity are representative for a real-life situation and thus investigating wear.
A project has been started to examine the current test procedures. As a start, a better
understanding is needed of the reaction forces on bearings and the (accumulated) displacements
that bearings are subjected to during their working life.
Since bearings are applied in all kind of bridges that all behave differently, it is not accurate to
study just a single arbitrary bridge. Actually, it is not accurate to study 1 bridge per type as well
since all bridges have a different geometry, connections, and etcetera. Though, as a start one of
each type in steel and concrete is examined.
2.8 Scope of this thesis
This thesis is about the mechanical behaviour of steel bridges. Representative bridge types are
assumed to be a plate-girder bridge, an arch bridge and a cable-stayed bridge. Only the plate-
girder bridge is analysed in this thesis. The behaviour of this bridge is examined using a finite
element model. The geometry and specific behaviour of the bearings are not taken into account
since they do not influence the externally applied forces and movements and their influence is
expected to be negligible. Therefore, elastic shear deflection and additional translations in case of
German variants are disregarded.
Clearly, bridge deformations are mainly caused by wind, thermal actions and traffic. Thermal
action will be disregarded since it does not contribute to ‘fast’ movements and the caused wear
path is relatively small; it only changes the location of load transfer. As traffic is a predominant
load, this thesis will only consider the influence of traffic on the bridge behaviour. Basis for the
traffic load model is a thesis about modified Eurocode traffic load models written by a TU Delft
master student [Otte, 2009].
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2.9 Plan of action
Using a finite element model of the bridge across the river Lek near Hagestein (the Netherlands),
the mechanical behaviour, including support displacements and reactions, of a representative
plate-girder bridge is studied.
As a start, a static analysis is performed in order to gain insight in the mechanical behaviour of
the bridge and to obtain information of support displacements due to traffic loads. Initially, only
single lorries are examined, later the influence of interacting lorries is investigated in a parameter
study. Furthermore, the difference between a static and dynamic approach is compared and the
influence of friction discussed.
As wear is a service-life phenomenon, it is reasonable to perform a geometrically and physically
linear analysis under characteristic conditions. However, friction in the sliding planes is a
nonlinear phenomenon and must therefore be studied carefully.
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3 Geometry, support system and traffic induced mechanical behaviour of
the A27 bridge across the river Lek near Hagestein
The motorway A27 bridge near Hagestein in the Netherlands is notorious for its detrimentally
flexible behaviour to its structural bearings. They have been replaced several times since the
bridge was constructed around 1978 as they were completely worn out. For this reason, this
bridge is considered firstly in the project. This chapter gives a description of its geometry and
mechanical behaviour.
Figure 3-1: bridge across the river Lek near Hagestein (Netherlands), view from north to south.
Figure 3-2: location of the bridge.
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3.1 Global geometry of the bridge, carriageway and nomenclature
For orientation purposes, 3 axis are defined. The direction of traffic flow (north to south) is called
the longitudinal or x-direction, the other horizontal direction (perpendicular to the x-direction) is
called the lateral or y-direction and the last one is logically called the vertical or z-direction.
The bridge crossing the river Lek near Hagestein (motorway A27 south of Utrecht) was
constructed around 1978. It consists of 2 identical, independent superstructures, 1 for each traffic
direction. The superstructures consist of composite approach bridges and a steel main bridge.
The behaviour of one of the main bridges is investigated. The western one is considered since
most historical measurement results are available about this part, which simplifies validation of
the finite element model. From now on, the considered part will be called ‘bridge’.
The bridge is a statically indeterminate, steel, plate-girder bridge. It has 2 main girders at a
distance of 12.6 metres from each other. The 162 metre main span is situated between 2 side
spans of 93.5 metre each (total bridge length is approximately 350 metres). Both main girders are
supported on 4 spherical bearings, thus 8 bearings support the whole bridge. The western main
girder is called the outer girder; the eastern one is called the inner girder. The inner girder is
positioned 315 millimetres higher than the outer girder in order to create a 1:40 lateral slope
which prevents the carriageway from standing water during rainfall. The bridge has been
composed of 18 metre welded sections before it was assembled with bolted connections.
The structural height of the bridge is approximately 3.5 metres over a big part of its length, but
close to the intermediate supports it starts increasing to almost 6 metres above the piers. Stability
in y-direction is provided by 4.5 metre spaced cross frames, composed of vertical stiffeners and
cross beams. These frames are constructed stiffer above the piers.
The main girders are composed of longitudinally stiffened plates and a built-up bottom flange.
The base bottom flange is 900 millimetres wide, additional wider plates have been added where
the bending moments increase. Near the intermediate supports, the bottom flange consists of 4
plates bolted and welded together to an accumulated thickness of 100 millimetres. Furthermore, it
is important to note that the outer girder is constructed somewhat heavier than the inner girder to
accommodate an additional traffic lane in the ‘future’ (this lane has not yet been built in practice).
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The 14.4 metres wide bridge deck is an orthotropic steel deck with closed trapezoidal stiffeners.
Initially, the road surface was an asphalt layer on top of the orthotropic steel deck, later it was
replaced by a high strength reinforced concrete layer of approximately the same weight to
minimize fatigue cracks in the deck plate.
Figure 3-3: initial carriageway and generic cross section of the bridge.
The carriageway is 12.25 metres wide, divided into an emergency lane, a slow lane and a fast
lane. The slow lane is located somewhat eccentric from the bridge deck’s centre line. Nowadays
the carriageway is divided differently to accommodate an extra traffic lane during rush hour.
Nevertheless, the old situation, as shown above, will be considered in the analyses.
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3.2 Support system
The original support system has been adapted in the past after having serious wear problems
due to thermal deformations of the bridge. The actual support system will be analysed.
Nowadays, all bearings are spherical bearings with 3 rotational DOFs. Translational restraints in
y-direction have been applied to all supports of the inner girder; bearing 06 on pier 7 is restrained
in x-direction as well to provide longitudinal fixation. All bearings below the outer are free to move.
The next figure shows all restraints of the bridge.
DEGREE-OF-FREEDOM
SUPPORT ux uy uz φx φy φz
01 O O X O O O
02 O X X O O O
05 O O X O O O
06 X X X O O O
09 O O X O O O
10 O X X O O O
13 O O X O O O
14 O X X O O O
X = restraint O = degree-of-freedom
Figure 3-4: support conditions of the bridge.
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3.3 Analysis of the traffic induced mechanical behaviour and corresponding support
displacements and reaction forces
This paragraph is related to the verification of the output generated by the finite element model. It
is rather difficult to examine the support displacements by hand since the bridge deforms 3-
dimensionally. Support rotations are equal to the rotation of the cross section plus in-plane
deformation; translations, depend on the distance to the centre-of-rotation as well.
xzu
yzu
zxy
zyx
×ϕ+×ϕ−=
×ϕ−×ϕ=
It is quite complicated to predict all displacements beforehand correctly, therefore, deformations
due to major deformation phenomena (bending, torsion and rigid body translation) will be
analyzed. In order to obtain reliable predictions, 7 situations will be considered: 4 support loads
and 3 span loads. For convenience, these terms are used. Support loads are loads that act on
the bridge deck above the supports; span loads act on the bridge deck as well, though in the
bridge span.
As a reminder and useful summary, the table below gives all nomenclature for the parameters.
DESCRIPTION SYMBOL NOMENCLATURE
Translation in longitudinal direction ux x-translation
Translation in lateral direction uy y-translation
Translation in vertical direction uz z-translation
Rotation about longitudinal axis φx x-rotation
Rotation about lateral axis φy y-rotation
Rotation about vertical axis φz z-rotation
Reaction force in longitudinal direction Fx x-reaction
Reaction force in lateral direction Fy y-reaction
Reaction force in vertical direction Fz z-reaction
Figure 3-5: nomenclature of DOFs and reaction forces.
This nomenclature is used further since it is compact and clear.
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3.3.1 Support displacements and reactions in the longitudinal/vertical plane as a result of
vertical bending of the bridge deck
The next figure shows very simple models of the bridge with predicted deformations in the xz-
plane for several load situations. The bridge might be considered as a 2D portal frame in which
the upper beam represents the neutral axis of the main girder and the columns the rigid
connection between neutral axis and supports. Although in reality the neutral axis is not straight
at all, for simplicity, vertical shifts are ignored.
Figure 3-6: predicted deformations for several load situations. Note that friction is drawn for increasing
deflection as it depends on the direction of support motion.
In situations with support loads (1,3,5,7), no displacements in this plane occur since the load is
expected to be transferred to the supports below. Only vertical reactions below the load are
expected.
In the other situations (2,4,6), however, the main girder bends and thus cross sections rotate and
consequently supports translate, due to the supports’ eccentricities. Nevertheless, it can not be
said directly whether or not the supports will move as the movements might be restrained by
friction in the sliding planes. In the initial stage of loading, a free support can be considered as
fixed as long as the ‘restraining force’ on the support, due to the vertical load, is smaller than the
static friction force. This situation is comparable to that as explained in figure 2-2. A similar
formula as given in that figure can be derived for the bridge. The derived ‘restraining force’ can be
compared to the friction force and it can be examined whether or not the support will move.
However, the bridge is statically indeterminate and far from prismatic; therefore an analytical
formula would be too ambitious. Still, it is assumed that the restraining force is mostly bigger than
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the applied vertical load and friction force, and hence the support will move. Once the support
starts sliding, the final displacement will not differ largely from an unrestrained movement.
Importantly, friction acts opposite to the direction of motion. Therefore, its direction changes when
the vertical motion due to the traffic load changes direction (when the load passes somewhere
mid-span). The drawn arrows in figure 3-6 therefore represent the direction of friction for
increasing vertical span deflection. The friction magnitude is proportional to the vertical reaction
acting on the support. This vertical reaction is always a combination of live load and dead load
and is therefore much larger at the intermediate supports than at the end supports. Therefore the
x-reaction at support 06 probably acts in opposite direction compared to the friction force at
supports 09 and 10.
In case of bridge deformations (span loads), fixed support 06 restrains the bridges bottom flange
in stead of the neutral axis. Therefore, a rotation of that cross section inevitably induces a
longitudinal shift of the neutral axis which is caused by an x-reaction on the fixed support. This
translation must be added to the other supports´ x-translation due to local cross sectional
rotations.
na;xii;zii;yi;x uyzu +×ϕ−×ϕ=
Displacements of supports below the outer girder are probably larger than those of the inner
girder as the load acts closer to the outer girder. Therefore it is loaded heavier and consequently
experiences larger displacements.
Considering this xz-plane, support loads cause z-reactions, span loads cause x-translations, y-
rotations and z-reactions as well. Furthermore, x-translations suffer from frictional resistance
which will cause x-reactions at support 06.
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3.3.2 Support displacements and reactions in the lateral/vertical plane due to lateral bending
and torsion of the bridge deck and local deformations of stability frames
Displacements in the cross sectional plane are mainly caused by 3 major deformation
phenomena: bending of cross frames, twisting of the bridge (torsion) and lateral bending. The
cross section is asymmetric and the load acts eccentric. These facts are important to evaluate the
mechanical behaviour in lateral direction.
The torsional rigidity of the main girders is low; the bending stiffness of stability frames relatively
large. Therefore support loads mainly deform the local frames and will not affect other frames,
certainly no other pier frames. It is therefore assumed that the displacements at the non-loaded
supports are 0 in case of support loads. That is not the case for y-reactions. Figure 3-7 gives a
very simplified view on the bridge and makes these situations more clear. It shows the behaviour
of the bridge under support loads.
Figure 3-7: predicted deformation due to support loads. The upper figures show the deformation when the
local support would not have been restrained. The bold lines in those figures represent the inner girder’s
bottom flange which is shown from above in the bottom part. The red arrows represent the reaction forces
that are introduced by a fictitious translation of the actually restrained supports. Note that the bottom flange
is not straight in the final deformation.
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For example, a load above supports 13 and 14 deforms the end portal (figure 3-7, upper
situation). The other frames are not deformed. As the figure shows, supports 13 and 14 tend to
translate laterally, though, support 14 is restrained in this direction. This fictitious translation must
be neutralized by means of a fictitious support translation which is visualized in the bottom part of
the figure. Comparable to support settlements, y-reactions are introduced in this kind of
situations.
Until now, friction was disregarded. Yet it will be considered. Similar to ‘longitudinal’ friction,
‘lateral’ friction may influence the behaviour as well. The same principle holds for restraining
forces and friction forces as for the x-direction, however, the magnitude of the restraining force is
totally different. Yet it is not a beam structure but a portal frame, which must be analyzed
differently. The next formula is derived for the restraining force when the longitudinal and lateral
behaviour are uncoupled. It is based on simple mechanical formulae (assume that the bending
stiffness of column and beam is equal).
×+××
=
z3
2yz8
y
P
F 2
z
y
In this formula is z the height of the frame and y the distance between main girders. For the
intermediate piers, assume z is 5.5 metres and y is 12.6 metres. Then the y-reaction is
approximately 4.5 times smaller than the applied load. In that case it is likely that, for lightly
loaded lorries, the restraining force is smaller than the friction force and hence no y-translations
occur. However, friction is a hardly understandable phenomenon. It is very questionable whether
or not this theoretical restraint will occur since these supports probably already slide, due to the
longitudinal motion, and thus are unable to restrain the y-translations anymore. Furthermore it is
very hard to implement friction in a model. Therefore, it is assumed that the y-translations are not
restrained by friction in models were friction is disregarded. Although the translations are not
restrained by friction, the friction force does affect the y-reactions. How this is incorporated will be
discussed later.
Support loads cause y-translations of the local portal (right part of figure 3-7), x-rotations, z-
reactions and y-reactions due to restrained deformation of the bottom flange and lateral friction.
Theoretically support loads on end supports cause z-rotations as well. Consider the bottom flange
as a large cantilever beam with an applied displacement at the free end, at this location a rotation
may be expected.
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Figure 3-8: deformation due to torsion. The blue lines visualize the in-plane rotation about the shear centre.
The hatched regions do not displace which clarifies the torsional rotation of the cross section. Additionally,
a blue in-plane centre line is connected through a red longitudinal line which shows the deviation compared
to the undeformed situation.
Twisting of the bridge is caused by an eccentric load on the bridge span. The slow lane is located
somewhat closer to the outer girder than the inner girder, causing a torque acting on the bridge.
The cross section rotates about its shear centre, that is located somewhere above the bridge
deck, resulting in vertical and lateral movements of the bottom flanges and x-translations and z-
rotations near the supports.
Lateral bending may occur since the outer girder is constructed heavier than the inner girder
which means an asymmetric cross section. Therefore lateral bending due to vertical loads occurs
in addition to vertical bending. Displacements due to this lateral bending may be added to the
displacements due to torsion as they act in the same direction.
The figure below shows the lateral deflection of the bridge’s bottom flange due to torsion and
lateral bending. It also elucidates the rotations about the vertical axis.
Figure 3-9: lateral displacement of the bridge’s centre line due to a vertical span load (indicated by a cross)
and effect on rotations about vertical axis.
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3.3.3 Prediction of direction reaction forces and support displacements
3.3.3.1 Rotations about the lateral axis: φy
The most straightforward sign prediction of displacements is the examination of y-rotations. The
direction can be obtained from figure 3-6 directly. These rotations are caused by vertical bending.
3.3.3.2 Rotations about the longitudinal axis: φx
Bending of stability frames and twisting of the bridge, due to span loads, cause x-rotations.
Support loads result in positive rotations for the supports below the inner girder and negative
rotations for supports below the outer girder (figure 3-7). Span loads probably do not cause these
rotations since the cross section above the supports does not rotate due to torsion. Therefore
they are expected to be 0. Perhaps the portals deform (distorsion) due to these span loads which
cause an x-rotation, this will be disregarded.
3.3.3.3 Rotations about the vertical axis: φz
Support loads on end portals cause a rotation comparable to cantilever beams (outwards). In that
case supports 14 and 01 rotate in a negative sense; supports 13 and 02 in a positive sense. It is
expected that the intermediate supports do not rotate in case of support loads. Rotations due to
span loads can be obtained simply from figure 3-9, they are caused by torsion and lateral
bending.
3.3.3.4 Longitudinal translations: ux
The source of x-translations has been described in paragraph 3.3.1 and 3.3.2 (bending and
torsion). They are composed of 2 rotations and a rigid body motion of the whole bridge (which is
also a result of rotations).
0606;yii;zii;yna;xii;zii;yi;x eyzuyzu ×ϕ+×ϕ−×ϕ=+×ϕ−×ϕ=
This translation is likely dominated by longitudinal bending or torsion. Therefore the z-rotation will
be neglected for this prediction. The eccentricity of support 06 in the equation above is defined
positive since the neutral axis is located on the positive z-direction compared to the centre-of-
rotation of the support itself. The eccentricities of the other supports are negative since those
supports are located on the negative z-side related to the centre-of-rotation (neutral axis). The
equation therefore can be transformed in the next equation in which the eccentricities are both
positive constants.
0606;yii;yi;x eeu ×ϕ+×ϕ−=
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It might be that these components have the same direction (supports 09,10,01 and 02), this is not
the case for supports 13 and 14. Whether the total translation is positive or negative depends on
the ratio between the rotation (rφ) of supports 13,14 and 05,06 on the one hand, and on the ratio
between their eccentricities (re) to the neutral axis on the other hand.
0606;y14/1314/13;y14/13;x eeu ×ϕ+×ϕ−=
This eccentricity to the neutral axis is approximately 2300 millimetres at the end supports and
2750 millimetres at the intermediate supports (annex A) which means a ratio of 1.2. As long as
the rotation of supports 13 and 14 is larger than 1.2 times the rotation of supports 05 and 06, the
x-translation of supports 13 and 14 is dominated by rotation of the local cross section. Otherwise
this rotation is dominated by the translation of the neutral axis.
For example, a load in the first side span causes a positive rotation at supports 13,14,05 and 06.
The rotation ratio is larger than the eccentricity ratio, consequently the x-translation is negative.
3.3.3.5 Lateral translations: uy
The direction of y-translations is obvious in case of support loads. The supports below the inner
girder are restrained and consequently do not translate. Supports below the outer girder move in
negative direction, though, only when the load acts above the support itself, otherwise it is
assumed to be 0 (a support load only deforms the local stability frame).
Span loads cause a rotation of the cross section about the shear centre. A clear prediction for the
direction of y-translations can not be given as torsion is resisted by vertical reactions only.
3.3.3.6 Vertical reaction forces: Fz
It is assumed that support loads will be transferred directly to the supports on that pier. Span
loads cause positive and negative reactions, mainly as a result of torsion and bending.
Importantly, the bearings are not able to generate negative reactions. Nevertheless, they are
indicated by a negative sign as they lower the total vertical reaction (including dead load) which is
still positive. Explanation of the direction of vertical reactions is expected to be obvious and
therefore omitted.
3.3.3.7 Lateral reaction forces: Fy
The y-reactions that act on the structure in case of support loads have been explained in figure 3-
7. In case of span loads it is too complicated and perhaps impossible to predict their direction
since torsion and lateral bending not necessarily induce lateral reactions. They may be a result of
restrained deformations of the bottom flange; however, it is rather difficult to predict the
deformations due to these 2 phenomena.
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3.3.3.8 Longitudinal reaction forces: Fx
The bridge contains only 1 support that is able to transfer x-reactions. Therefore static equilibrium
is only possible between applied loads and the single longitudinally restrained support. Neglecting
friction in the other supports during sliding and considering a static analysis, no x-reactions will
act on the bridge as no loads are applied in that direction.
0F0F 06;xx =→=∑
However, the structure does not behave statically but dynamically. As mentioned before, the
neutral axis shifts longitudinally due to the vertical load acting on the bridge. This longitudinal
motion induces x-reactions at support 06 (in a dynamic analysis). These forces, however, are
hardly predictable.
x06;xxx amFamF ×=→×=∑
Additionally, neglecting friction in the other supports is not realistic as well. Friction between
sliding elements may be considered similarly as restraining forces since friction restrains
movements to some extent. Those friction forces affect the x-reaction of the fixed support and
must be included in the equilibrium equation. Friction is proportional to the normal force (vertical
reaction force) by a coefficient of friction and is always opposite to the direction of movement.
frictionx06;xxx FamFamF +×=→×=∑
It is quite difficult to predict the direction of the longitudinal reaction force. In figure 3-6 it can be
seen that in some load situations all supports move in the same direction and accordingly all
friction forces act in opposite direction. It is therefore most likely that the x-reaction acts in the
same direction as the longitudinal motion. As mentioned earlier, especially the direction of friction
in supports 09 and 10 is relevant since the vertical reaction force due to dead weight is large on
those supports. Therefore it will be assumed that the longitudinal reaction force acts in the same
direction as the longitudinal translation at supports 09 and 10.
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3.3.3.9 Summary of predictions
LOAD SITUATION PARAMETER SUPPORT
1 2 3 4 5 6 7
14 0 - 0 - 0 + 0
13 0 - 0 - 0 + 0
10 0 + 0 - 0 + 0
09 0 + 0 - 0 + 0
05 0 - 0 + 0 - 0
02 0 + 0 - 0 + 0
ux
01 0 + 0 - 0 + 0
13 - + 0 0 0 0 0
09 0 0 - 0 0 0 0
05 0 0 0 0 - 0 0 uy
01 0 0 0 0 0 + -
14 + 0 0 0
13 - 0 0 0
10 0 + 0 0
09 0 - 0 0
06 0 0 + 0
05 0 0 - 0
02 0 0 0 +
φx
01 0 0 0 -
14 0 + 0 - 0 + 0
13 0 + 0 - 0 + 0
10 0 - 0 + 0 - 0
09 0 - 0 + 0 - 0
06 0 + 0 - 0 + 0
05 0 + 0 - 0 + 0
02 0 - 0 + 0 - 0
φy
01 0 - 0 + 0 - 0
14 - + 0 - 0 + 0
13 + + 0 - 0 + 0
10 0 - 0 + 0 - 0
09 0 - 0 + 0 - 0
06 0 + 0 - 0 + 0
05 0 + 0 - 0 + 0
02 0 - 0 + 0 - +
φz
01 0 - 0 + 0 - -
Fx 06 0 + → - 0 - → + 0 + → - 0
14 - → + + → - - → + + → -
10 + → - - → + + → - - → +
06 - → + + → - - → + + → - Fy
02 + → - - → + + → - - → +
14 + + 0 - 0 + 0
13 + + 0 - 0 + 0
10 0 + + + 0 - 0
09 0 + + + 0 - 0
06 0 - 0 + + + 0
05 0 - 0 + + + 0
02 0 + 0 - 0 + +
Fz
01 0 + 0 - 0 + +
Figure 3-10: predictions about the direction of parameters. Positive directions have been marked by +,
negative by -, no result by 0 and empty spaces indicate an unpredictable direction.
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4 Explanation of the applied traffic/fatigue load model
4.1 General introduction into traffic load models
EN 1991 (Eurocode 1) prescribes all kinds of loads that have to be considered in the design of
structures. Part 2 of this code only covers traffic loads (pedestrians, road and railway traffic) on
bridges. These loads have to be applied according to several prescribed traffic load models that
have been developed for, for example, strength verifications and fatigue assessment. Different
models apply to both purposes. Models for strength verification generally prescribe maximum
loads that are expected to occur, based on a certain probability. Fatigue load models, however,
prescribe representative loads for daily traffic in order to obtain insight in the (remaining) fatigue
life of a structure.
This research is part of an investigation into wear and fatigue of structural bearings. Therefore a
fatigue load model is used. Rijkswaterstaat requested to use a fatigue load model which has been
developed by a TU Delft master student in 2009 [Otte, 2009]. This model and the background for
use are explained in this chapter.
4.2 Fatigue load models in EN 1991-2 and the applied measurement-based model
EN 1991-2 describes 5 fatigue load models. All models are different and are intended to be used
for different purposes. However, they have in common that they outline loads that have to be
applied for obtaining stress ranges in fatigue assessment. Models 1 and 2 are intended to be
used to determine whether or not the fatigue life may be considered as infinite; models 3 to 5 for
fatigue life assessment, i.e. finite fatigue life. Furthermore, models 1 to 3 are intended to be used
to determine a maximum and minimum stress, resulting from different load arrangements on the
bridge; models 4 and 5 to determine stress range spectra.
In this project it is not relevant to obtain maximum or minimum displacements only, especially the
accumulated sliding distance is desired. Therefore model 4 or 5 should be used rather than
model 1, 2 or 3. Model 4 specifies a set of 5 standard lorries with prescribed axle distances and
axle loads. Model 5 is based on recorded data.
Otte proved that fatigue load model 4, as described in EN 1991-2, is very conservative compared
to actual traffic on Dutch motorway bridges. He compared the model with traffic measurements of
the Moerdijk Bridge in the Netherlands, this bridge is located in the most intensively used Dutch
motorway and regarded as a reference location for the Netherlands. He used the measurements
for deriving several fatigue load models which can be regarded as adapted versions of FLM4, the
adapted versions act as FLM5. The traffic load model used in this thesis is one of Otte’s models.
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Otte proposed several models for different purposes: FLM M [100]; FLM M [75/25]; FLM M
[75/20/5]; and a reduced version of the current fatigue load model 4. The latter one specifies the
same lorry types as FLM4 in EN 1991-2. The characteristic axle loads, however, are adapted to
the measured values on the Moerdijk Bridge. The other models (FLM M) prescribe slightly
different lorries and loads compared to this reduced version of FLM4. These models have the
lorry types (axle configuration) in common, however, each model describes different (sets of) axle
loads. FLM M [100] describes lorries without subdivision per type while FLM M [75/25] and FLM
M [75/20/5] subdivide each type in 2 or 3 load sets respectively. The subdivision has been done
since not all lorries of the same type are loaded equally, which is of crucial importance in fatigue
life assessment.
The numbers in this nomenclature indicate the relative amount of lorries that must be taken into
account per load set. In FLM M [75/25], for example, 75% of the lorries should be considered with
load set 1, representing lower values for axle loads, while 25% of the lorries must be considered
with load set 2, representing higher values.
Any load set needs to be calculated separately. Therefore, one should question whether or not
any additional load set will contribute substantially to the results. For fatigue assessment of
directly loaded fatigue sensitive details as in orthotropic decks, for instance, especially the most
heavily loaded lorries are relevant to consider since they affect the fatigue life disproportionately
(logarithmic scale in SN-curves). For these cases, considering several load sets is more accurate
and beneficial and thus model FLM M [75/20/5] is recommended as the 5% represents very
heavily loaded lorries.
This project is about general movements of structural bearings. Therefore it seems unnecessary
to consider this rather extended model for only the 5% heaviest loads and thus is it decided to
apply model FLM M [75/25].
4.3 Wheel prints and tyre configuration per axle
EN 1991-2 and Otte prescribe different wheel prints and tyre configurations. However, as relevant
these parameters are for fatigue assessment of orthotropic steel decks, as irrelevant is it for the
global behaviour of the bridge. Therefore an estimated value for the tyre distances is accurate
enough. A 2 metre tyre distance per axle is chosen. Wheel prints are completely disregarded.
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4.4 Overview of the applied fatigue load model, FLM M [75/25]
LOAD [kN]
LORRY OCCURANCE (%) NUMBER OF AXLES
AXLE SPACING [m] 75% LOW (L) 25% HIGH (H)
5.2 50 80 V11 10 2
- 60 115
TOTAL 5.2 110 195
3.8 75 100
1.3 75 140 V12 10 3
55 115
TOTAL 5.1 205 355
3.8 70 80
6.6 55 95
1.3 40 70 T11O2 15 4
- 40 70
TOTAL 11.7 205 315
3.8 70 90
5.6 80 140
1.3 50 100
1.3 50 100
T11O3 50 5
- 50 100
TOTAL 12.0 300 530
4.2 85 105
1.3 100 145
4.2 70 110
3.8 70 110
1.3 60 95
V12A12 5 6
- 60 95
TOTAL 14.8 445 660
2.8 70 90
1.3 60 105
5.6 85 135
1.3 70 120
1.3 70 120
4.2 70 120
1.3 60 100
T12O3A2 10 8
- 60 100
TOTAL 17.8 545 890
Figure 4-1: definition of lorries according to FLM M [72/25]. Total number of lorries per year is 3 million.
The figure shows, for instance, that 375,000 lorries (0.5×0.25×3,000,000) of the type 530kN
T11O3 may be expected in a year.
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5 Explanation about the finite element model
The bridge has been modelled in ‘MIDAS Civil 2010’, a finite element program especially for
bridge design. This chapter gives a brief description of some program features and the way it has
been used for this project. The geometry of the model, selection of finite elements and their
properties, boundary conditions, loads and analysis types are discussed.
5.1 Modelling the bridge as a finite element model
5.1.1 Element types and their characteristics
The program provides line elements (e.g. truss, cable and beam), plane elements (e.g. plate and
plane-stress) and solid elements. The studied bridge is mainly composed of steel plates which
are subjected to all kinds of mechanical deformations as bending and torsion, for example.
Therefore, plate and beam elements are the most appropriate elements for modelling the bridge.
Figure 5-1: two finite elements in ‘MIDAS Civil 2010’, a general beam element (left) and a plate element (right).
A general beam element is defined by 2 nodes and has bending stiffness, shear stiffness, axial
stiffness and torsional stiffness. Furthermore, it has 6 DOFs per node (3 translations and 3
rotations). Output is given in axial forces, shear forces, torques, bending moments and stresses.
A plate element is defined by 4 nodes and has 6 DOFs per node as well. They have axial
stiffness, in-plane and out-of-plane shear stiffness and out-of-plane bending stiffness.
Furthermore, plate elements have thickness properties which can be classified in stiffened and
unstiffened properties. Obviously, unstiffened plates represent ordinary plain steel plates without
stiffeners. For these elements only a plate thickness has to be assigned.
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Using stiffened plate elements prevents the user from a lot of effort in either calculating cross
sectional properties for the plates as e.g. bending stiffness, or in modelling the stiffeners in the
computer program. It is possible in MIDAS to define a plate thickness stiffened through a specific
type of stiffener, for instance, a trapezoidal stiffener (see figure below). Care needs to be taken
that the stiffened direction of the element coincides with the actual direction of the stiffeners
(element’s local axis).
Figure 5-2: input screen for stiffened plate elements (for trapezoidal stiffeners).
Orthotropic steel decks, for instance, can easily be modelled in this way. The finite element
behaviour is in accordance with the orthotropic properties of the real structure. Output for plate
elements is given in plate forces or stresses and bending moments.
5.1.2 Physical properties of the finite elements
Since the purpose of this project is to investigate the elastic deformations of the bridge, only
linear-elastic stiffness properties of the material are relevant. Because all steel grades, applied in
the bridge, have the same physical stiffness properties (Young’s modulus of E=210MPa) and
wear is considered in the Serviceability Limit State (SLS), accordingly all material behaves
linearly elastic, the same steel grade has been used for all elements.
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5.1.3 Consideration of selection and combination of finite elements
In finite element modelling, an efficient model of the structure is desirable. Although a very
detailed and complex model might be more accurate, it has very important disadvantages. The
computation time and required memory is disproportionately larger. An engineer must, therefore,
strive for a solution which provides reliable results with a relatively simple model. A smart
combination of elements is there part of. The number of nodes should be kept as less as
possible, since the required memory and computation time directly depend on the number of
DOFs.
For example, an H-shaped section can be modelled using truss elements, beam elements, plate
elements, plane-stress elements or a combination of them. Beams in bending are generally
represented by beam elements, though, sometimes plane elements are more appropriate. A huge
disadvantage of using plane elements for beam structures is the increasing number of nodes and
consequently the number of DOFs. Using plane-stress elements or plate elements for girder
webs and beam elements for flanges enables the use of plane elements with a relatively small
number of DOFs.
1 BEAM 1 PLATE + 2 BEAM 5 PLATE
NUMBER OF NODES* 1 2 6
NUMBER OF DOFs* 6 12 36
Figure 5-3: modelling an H-shaped beam using different elements (* number of nodes per cross section).
In this project the global behaviour of the bridge has been investigated which implies that in most
cases only the assigned stiffness per node is relevant. Therefore, it is decided to model the
structure using plate and beam elements wherever possible. However, the application and
consequence of using beam elements rather than plate elements, in order to reduce the number
of nodes, must be assessed carefully.
As explained earlier, using stiffened plate elements is very efficient for orthotropic decks and thus
will be applied for modelling the deck plate. The main girders are modelled by stiffened plate
elements too since they are longitudinally stiffened as well.
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5.1.4 Explanation about the model’s composition
This paragraph outlines the model’s construction. The model has been built-up with a coarse
mesh and a refinement near the supports. The cross frame distance has been used as general
element length in spans. In lateral direction, the division of elements is based on the axle tyre
distance of approximately 2 metres which is beneficial for modelling the lorry loads. Additionally,
this division appears to be sufficient for simulating lateral bending correctly. The main girders are
divided in 5 elements vertically, based on the local geometry at the intermediate supports. An
acceptable bending simulation of the vertical stiffeners has been achieved as well.
Figure 5-4: complete view on the model.
INDICATOR DESCRIPTION STRUCTURAL ELEMENT TYPE OF FINITE ELEMENT
1 Orthotropic deck plate Stiffened plate element
2 Stiffened web main girder Stiffened plate element
3 Cross beam (kamplaten) Stiffened plate element
4 Cross beam web Plate element
5 Vertical web stiffener in cross frame Plate element
6 Vertical web stiffener near support (not in this figure) Plate element
7 Bottom flange main girder Beam element
8 Flanges cross beam and vertical web stiffener of cross frame Beam element
Figure 5-5: clarification of used finite elements of a generic cross section.
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A very important detail is the connection between vertical web stiffener of the main girder and the
cross beam. In reality there is a kind of rigid connection by means of a small welded corner plate
(figure 3-1). Initially, the vertical stiffeners, web and flange, were represented by a single T-
shaped beam element. Since beam elements are defined by 1 node per cross section, stiffness
was assigned to the node connecting the cross beam with main girder only. It has been found
that a significant difference exist between this approach and the use of plane elements for the
stiffener’s webs in which the second node of the cross beam, in lateral direction, is connected as
well. It was, therefore, decided to use plane elements for the vertical web stiffeners.
The geometry near the supports has been modelled more accurately. Besides mesh refinement,
web stiffeners for load introduction (6) and plates between deck plate and end beam have been
modelled.
Figure 5-6: model of the stability frames (end frame left, intermediate frame right) above the supports.
5.1.5 The supports’ geometry and the applied boundary conditions in different analyses
5.1.5.1 Support geometry for all analyses
There is a difficulty concerning the modelling of supports for this project. For examination of the
global behaviour of a bridge, a detailed model of the bearings is needless. However, for
investigating the bearings’ behaviour precisely, a detailed model of the bearings is essential. This
project is more or less a combination of both. As in this case the behaviour of the bridge is
central, the geometry of the bearings has been disregarded.
Looking forward to a time-history analysis, reaction forces need to be obtained. For some
inexplicable reason, MIDAS is not able to provide these results in such analyses. However,
element forces can be obtained. Therefore, extra (circular) beam elements have been added
below the main girders’ bottom flanges representing a kind of backing plate. These elements
must prevent the user from summing up all element forces from elements connected to the
support node. Yet, the axial force in the beam element represents the vertical reaction force and
both shear forces in the beam element represent the 2 horizontal reactions.
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Furthermore, in reality the bearings provide a stiffened planar surface below the girders’ bottom
flanges. It is not clear what influence this ‘stiff’ plane may have for the support displacements
(probably none). To be sure, it is decided to apply linear constraints (master-slave nodes) to the
adjacent horizontal nodes to achieve a stiff plane.
Figure 5-7: application of linear constraints (left) and additional beam elements below bottom flanges(right).
5.1.5.2 Boundary conditions in the linear static analysis
Friction is a nonlinear phenomenon. Therefore, it has been disregarded in the linear static
analysis. Boundary conditions in these kind of analyses are just nodal constraints. The boundary
conditions have been applied to the lower nodes of the circular beam elements.
5.1.5.3 Boundary conditions in the linear dynamic analysis
Since friction is a nonlinear phenomenon, it has been disregarded in the linear dynamic analysis
as well. Therefore, the same boundary conditions hold as for the linear static analysis.
5.1.5.4 Boundary conditions in the nonlinear dynamic analysis
In a nonlinear dynamic analysis, friction can be included. However, it is not very easy to model
friction. Modelling friction in MIDAS can only be done using the FPSI-feature (Friction Pendulum
System Isolator, a general link property). However, using this feature correctly requires a lot of
specific knowledge which is currently not sufficient to judge the results correctly. A simple beam
test model has been considered in order to validate the applicability of this feature for modelling
frictional resistance in the supports. This research has been described in annex D. As discussed
in that annex, the knowledge is still insufficient and therefore no nonlinear analysis will be
performed. Additional research is recommended to model friction and examine its influence on
the global behaviour of the bridge, especially on the dynamic results.
FINAL VERSION
Page 45
5.2 Modelling self weight and superimposed dead load
Dead load is actually not necessary to incorporate in the static analysis since it has an irrelevant
contribution to the results. However, it is needed in a dynamic analysis (mass matrix).
Additionally, for validation of the model it may be helpful to incorporate it.
Self weight is calculated directly by the program when a load case is created with a unit gravity
vector in vertical direction. The superimposed dead load (which is estimated on 1.75kN/m2) has
been applied as a pressure load (distributed face load) on the deck in another load case.
In a dynamic load case these static loads need to be transformed into nodal masses.
5.3 Modelling lorry loads
The irrelevance of wheel prints has been discussed in chapter 4. Nodal (point) loads, therefore,
represent wheel loads. A difference exists between a static and dynamic analysis which is
discussed in the next paragraph.
5.4 Methods for generating time-related displacements and reaction forces
Time-related displacements can be obtained in different ways. In this thesis both, a static and
dynamic analysis is performed. For both cases the analysis procedure is explained.
5.4.1 A static analysis using influence line theory
Time-related parameters can not be obtained from a static analysis. However, the results are
related to the load position. In a spreadsheet program these results may be transformed in time-
dependent parameters. The next paragraph discusses the possibility of modelling a lot of static
load cases. Section 5.4.1.2 outlines the procedure that has been followed, using a more powerful
feature in MIDAS.
5.4.1.1 Ordinary static analysis using separate load cases
Ordinary static load cases, representing a lorry at 1 specific location on the bridge, can be made.
A disadvantage is that the axle distances mostly do not coincide with the nodal distances (point
loads can be applied to nodes only). A solution to overcome this problem is to divide the wheel
loads into 2 nodal loads which are related to each other as reaction forces of a simply supported
beam induced by an in-span point load. This method is quite labour-intensive since the nodal load
values must be calculated for all lorry types and locations on the bridge.
Another method, which is probably more efficient, is to apply unity loads and subsequently use
these values as influence values in a spreadsheet program in which all kind of loads and axle
configurations can be considered. This method is less labour-intensive as the previous one, has
the advantage of easy modification in a spreadsheet program, but uses results of nodal loads
only. Probably more results (loads between nodes) are desired.
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MIDAS has a built-in influence line feature which is a powerful tool in order to obtain a lot of
influence data that can be adapted in a spreadsheet program. This built-in feature is the ‘moving
load analysis’.
5.4.1.2 Moving load analysis, generating influence lines statically
Perhaps the name is quite contradictory, but the moving load analysis is a static analysis. The
user must specify traffic line lanes or traffic surface lanes that represent the actual traffic lanes on
the bridge. All kinds of moving load codes, including EN 1991-2, are implemented in order to
reduce the amount of work for the user. Based on the design rules, the program calculates all
maximum and minimum values of, for example, bending moments and shear forces, induced by
lorries that have been prescribed by the codes.
This part of the feature is unnecessary for this project. However, the moving load analysis
provides influence lines for reaction forces and nodal displacements as well. An advantage using
this tool is that the user may specify the number of IGPs (Influence Generating Points). It enables
the user to obtain more results per element than just 2 nodal influence values. It may be
questioned whether or not the extra IGPs are necessary. However, this feature has been used for
obtaining the influence values.
After completion of the standard model, the next steps have been undertaken in order to obtain
influence lines for different single lorry types:
1. Modelling traffic lanes on the bridge: only 1 traffic line lane has been applied on the
bridge. This lane represents the slow lane on the bridge. The number of IGPs per
element has been set on 3, so just 1 extra IGP is added (in addition to the 2 existing
nodes). The traffic lane has been applied such that both wheel loads act on 1 metre on
either side of the actual slow lane centre line.
2. Performing Moving Load Analysis and obtaining results: MIDAS generates the results in
millimetre, radian or kilonewton per unit wheel load (rather than per unit axle load). A
problem that had to be solved was the fact that the output file provided values with a
limited number of decimals caused by a unity load which is relatively small compared to
the loads that act in reality on the bridge. The limited number of decimals made some
adjacent values indistinguishably small or even zero. The solution was rather easy: the
Young’s modulus was reduced by a factor of 1 million which enlarged the deformations
with the same factor (this is only possible in a linear static analysis). Later the results
were scaled back to normal values, however, having more decimals yet.
3. Modification of MIDAS results into useful values: the results have been loaded in a
spreadsheet program and modified into average values per unit axle load. These values
are used as input for the (unit) influence line graphs.
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4. Validation of the model: validation by comparing the influence lines with existing
measurements (as much as possible) and checks for dead load deflection and reaction
forces (chapter 6).
5. Selection of representative DOFs for calculating accumulated displacements: an
evaluation or analysis of the influence lines must clarify which graphs are useful for wear
investigations. Clearly, examination of 8 supports is superfluous since some supports
behave comparably.
6. Linear interpolation of discrete IGP-values for intermediate axle locations: according to
the principle of influence lines, the influence of a lorry is equal to the sum of all
independent axle contributions. It is, therefore, needed that for each position of a lorry all
contributions of the independent axles can be obtained. However, the axle distances are
not equal to the distances between IGPs (figure 5-8) and thus not all axle contributions
can be determined from the IGPs themselves. A linear interpolation of the IGP-values for
intermediate locations secures a possible influence value for each axle location. A step
size of 0.1 metre is sufficient since the axle distances differ with the same amount.
7. Generation of lorry influence values: summation of all axle contributions (multiplication of
axle loads and influence values) results in lorry influence values per location of the lorry.
8. Generation of lorry influence lines: having the influence values per lorry, the lorry
influence lines can be drawn. The influence values are plotted as a function of time.
Therefore, a lorry speed is needed which is taken as 90km/h (25m/s).
9. Generating graphs for lorry sliding distances: based on the influence values, the sliding
distance per lorry can be estimated. It should be noted that this will be an upper bound
value since lorries may influence (and reduce) each others support movements.
10. Evaluation of the results: evaluation of the results is important for judging the correctness
of the results.
Figure 5-8: axle locations of a T11O3-lorry compared to the location of IGPs.
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5.4.2 Time-history analysis, an examination of the dynamic response
A time-history analysis is generally performed for examining the response of a structure in time,
including dynamic effects. The dynamic equilibrium equation is solved for every time step either
linearly or nonlinearly. Only a brief description of a time-history analysis is given, detailed
information can be found in other literature.
5.4.2.1 Analysis methods in MIDAS for performing a time-history analysis
MIDAS calculates the dynamic response either by means of direct integration or by applying the
modal superposition method. The latter method is based on the assumption that the particular
solution is a summation of synchronized motions [Spijkers, 2006]. This method can only be
applied properly when eigenvectors are used that correspond to the load induced deformation.
Annex C shows that in case of complex structures (in which local instabilities may occur) the
applicability of this method is highly questionable. Since in this project results must be obtained
that not always can be linked to corresponding mode shapes (local deformations), it is unwisely to
use this method. Using Ritz vectors may be a solution to overcome the problem with modal
superposition. This is not examined and, therefore, not applied.
The Newmark method has been applied. This method is generally more time-consuming but the
accuracy is independent on the number of mode shapes. Additionally, the constant acceleration
method is unconditionally stable and hence a relatively large time step can be used.
5.4.2.2 Determination of the time step
As mentioned before, the integration scheme is unconditionally stable. This does not imply that
the method is accurate for all time step sizes! The time step is chosen such that it does not affect
the shape of TFFs (discussed in section 5.4.2.4) and the accuracy of the results. A time step of
∆t=0.01s appears to be an appropriate choice.
5.4.2.3 Structural damping and local damping in the bearings
In dynamic analyses, the effect of structural damping must be taken into account. Welded steel
structures (which is the case of the bridge near Hagestein) are assumed to be damped by 1 to
2% structural damping. In this case it is chosen to apply ‘Mass & Stiffness Proportional damping’
(Rayleigh damping) of 2% for the frequencies 0.50Hz and 0.80Hz.
Additionally, friction in the bearings’ sliding elements causes local damping as well (possibly
much). In case of analyzing the vertical deflection of the deck, this damping is possibly irrelevant.
In case of the support movements, however, it might be of crucial importance. Therefore the
influence of friction must be examined as well. This has been discussed in section 5.1.5.4.
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5.4.2.4 Modelling lorry loads
Loads are time dependent and therefore can vary in time (what they generally do). In MIDAS they
can only be applied as nodal loads. These DNLs (Dynamic Nodal Loads) are linked to TFFs
(Time Forcing Functions) which describe the behaviour of the load in (local) time.
In this model it is decided to apply DNLs on every cross frame (79 pairs, mostly 4.5 metres
distance). The axle load is divided in 2 equal loads which behave according to a TFF that
represent the passage of a complete lorry.
Figure 5-9: TFF for a T11O3(L) lorry driving 90km/h (25m/s) for a node between 2 elements of 4.5 metres.
TFFs differ when the adjacent element sizes differ and in case of other axle configurations or axle
loads. The TFF for the first and last node are related to 1 element only causing an unrealistic,
infinitely small application time for the load. This may also affect the dynamic response of the
structure. It is therefore assumed that the load increases over a length of 0.5 metres,
representing an expansion joint.
5.4.2.5 Obtaining time-history results from MIDAS
MIDAS is able to provide graphs for time-history results. Therefore no manual post-processing is
needed. Nodal displacements can be obtained directly while reaction forces must be obtained
from element forces as they cannot be obtained directly.
Since a reaction force is equal to the sum of all element forces (in 1 direction) in 1 particular
node, it is needed to ask for all independent element forces which is a lot of work. Therefore, a
small beam element has been inserted between the bottom flanges and the supports. These
elements might be regarded as backing plates of the bearings. From these elements, the axial
forces and shear forces can be obtained which represent the reaction forces.
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Page 50
FINAL VERSION
Page 51
6 Examination of dead load results and validation of the model
It is always difficult to be convinced about the correctness of results, obtained from a finite
element model. The availability of measurements can help to validate a model. In this case,
however, not many useful measurements have been found. The validity is, therefore, supported
by some quick dead load checks and measurements which were available. The influence of mesh
refinement and element selection has been checked as well but are not discussed further.
6.1 Examination of deflection and vertical reactions due to dead load
It is known that this steel structure weights about 300-350 kilograms per square metre bridge
deck. This is supported by a simple calculation of which the results are summarized in annex E.
MIDAS calculates weight automatically. The sum of vertical reactions due to dead load can be
compared to the values given in the annex. The next calculation applies to the total weight that
has been calculated by MIDAS.
2
3
deck
zMIDAS;av
z
mkg3264.141.34981.9
1016060
Ag
Fm
kN16060F
≈××
⋅=
×=
≈
∑∑
This value corresponds approximately to the result given in annex E and thus can the calculation
in MIDAS be regarded as reliable.
The vertical deflection in mid-span is about 370 millimetres for self weight only and approximately
600 millimetres including superimposed dead load. The vertical deflection is hard to calculate
manually because of the change in geometry and stiffness and is therefore omitted. A deflection
of 370 millimetres in case of a 162 metre span is not unlikely. This does also apply to the load
combination including superimposed dead load.
Figure 6-1: vertical deflection due to self weight and superimposed dead load.
FINAL VERSION
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6.2 Traffic induced deflection and validation of the model’s global stiffness
In the 90’s, a measurement had been performed to measure strains in the bridge’s
superstructure. They additionally measured the vertical deflection of the inner girder due to a
145kN passing lorry (the lorry drove 50km/h and the bridge was closed for other traffic). The next
figure shows the measured deflection of the inner girder in mid-span (left) and the corresponding
influence line (right) that can be obtained from the finite element model.
-10
-5
0
5
10
15
20
0 50 100 150 200 250 300 350
Figure 6-2: measured and modelled deflection of the inner girder in mid-span due to a 145kN lorry.
The maximum deflection is about 15 millimetres. The influence line from the model, however,
shows a deflection of approximately 18.5 millimetres. Although the model already might be
regarded as quite accurate, one particular explanation supports the justification of the model’s
correctness. It is the fact that it is not known on what lateral position the lorry drove on the bridge.
Another measurement report described that the lorry in that test drove against the edge marking.
The model’s influence line, however, has been determined from the slow lane’s centre line (1
metre on either side). The lateral shift (0.50m-0.75m) to the outer girder would result in a
significant decrease of the inner girder’s vertical bending deflection since it will be loaded lighter.
Additionally the shift enlarges the induced torque causing extra upwards deflection of the inner
girder which must be subtracted from the bending deflection.
The measured deflection shows an asymmetric line which is not the case in the model. The
symmetry of the model’s line is explicable since the model itself is symmetric and obtained from a
static analysis. The asymmetric measured line is inexplicable as the actual bridge is also
‘symmetric’. Perhaps inertia plays a role (which might be relevant as well for the difference in
vertical deflection). Nevertheless, the simulation of the vertical deflection is acceptable.
Other checks are hardly possible. Detailed evaluation of the local deformations and
displacements show no significant deviations.
FINAL VERSION
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7 Linear static analysis of support displacements and reaction forces
In this chapter, all influence lines for unit loads are presented (UIL). Because some supports
behave comparably, simultaneously a selection of DOFs is made which supports are
representative for calculating accumulated displacements. The UILs are thereafter transformed
into LILs (Lorry Influence Lines) and analyzed. Although friction has not been taken into account
in the finite element model, graphs for horizontal reactions include friction. This friction is
calculated separately and manually.
Some reminders:
• The next coordinates apply to the supports;
o End supports 13 and 14 at 0 metres;
o Intermediate supports 09 and 10 at 93.5 metres;
o Intermediate supports 05 and 06 at 255.5 metres;
o End supports 01 and 02 at 349 metres;
• The free supports 13, 09, 05 and 01 are located below the heavier loaded outer girder;
the guided supports 14,10 and 02 as well as the fixed support 06 below the inner girder.
7.1 Examination of UILs, and DOF selection for calculating accumulated displacements
7.1.1 Presentation of UILs and DOF selection
Figure 7-1 must be examined carefully as it includes lorry induced reactions only and thus does
not provide any information about the actual contact stress. The eccentric traffic lane results in
larger influences for supports below the outer girder than those for the inner girder. Furthermore,
no support selection has to be made since no further calculations are done for this parameter.
UNIT INFLUENCE LINES FOR VERTICAL REACTION FORCES: Fz
-0.20
0.00
0.20
0.40
0.60
0 50 100 150 200 250 300 350
Location of the unit load from bridge entrance [m]
Infl
ue
nc
e [
kN
/kN
ax
le l
oa
d]
UNIT 01 UNIT 02 UNIT 05 UNIT 06 UNIT 09 UNIT 10 UNIT 13 UNIT 14
Figure 7-1: UILs for z-reactions.
FINAL VERSION
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UNIT INFLUENCE LINES FOR LATERAL REACTION FORCES: Fy
-0.06
-0.03
0.00
0.03
0.06
0 50 100 150 200 250 300 350
Location of the unit load from bridge entrance [m]
Infl
ue
nc
e [
kN
/kN
ax
le l
oa
d]
UNIT 01 UNIT 02 UNIT 05 UNIT 06 UNIT 09 UNIT 10 UNIT 13 UNIT 14
Figure 7-2: UILs for y-reactions.
Because the supports below the outer girder are free, only inner girder supports show results for
lateral reactions. As for the vertical reactions, no selection has to be made for this parameter.
UNIT INFLUENCE LINES FOR LONGITUDINAL TRANSLATIONS: ux
-0.020
-0.010
0.000
0.010
0.020
0 50 100 150 200 250 300 350
Location of the unit load from bridge entrance [m]
Infl
ue
nc
e [
mm
/kN
ax
le l
oa
d]
UNIT 01 UNIT 02 UNIT 05 UNIT 06 UNIT 09 UNIT 10 UNIT 13 UNIT 14
Figure 7-3: UILs for x-translations.
Except for 13 and 14, all supports behave similarly. Therefore, considering only 2 supports will be
sufficient for accumulated displacement calculations. Supports 09 and 13 represent this DOF best
and hence are selected.
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Page 55
UNIT INFLUENCE LINES FOR LATERAL TRANSLATIONS: uy
-0.024
-0.016
-0.008
0.000
0.008
0 50 100 150 200 250 300 350
Location of the unit load from bridge entrance [m]
Infl
ue
nc
e [
mm
/kN
ax
le l
oa
d]
UNIT 01 UNIT 02 UNIT 05 UNIT 06 UNIT 09 UNIT 10 UNIT 13 UNIT 14
Figure 7-4: UILs for y-translations.
In contrast to lateral reactions, yet the supports below the outer girder show results. Due to the
static analysis, the graph starts and ends diverging. This is obviously not realistic but will be
ignored for now. Selection of 1 end support and 1 intermediate support is sufficient, that will be
supports 13 and 09.
UNIT INFLUENCE LINES FOR ROTATIONS ABOUT THE LONGITUDINAL AXIS: φx
-5.00E-06
-2.50E-06
0.00E+00
2.50E-06
5.00E-06
0 50 100 150 200 250 300 350
Location of the unit load from bridge entrance [m]
Infl
ue
nc
e [
rad
/kN
ax
le l
oa
d]
UNIT 01 UNIT 02 UNIT 05 UNIT 06 UNIT 09 UNIT 10 UNIT 13 UNIT 14
Figure 7-5: UILs for x-rotations.
All intermediate supports behave similarly, the same holds for all end supports. Therefore only 1
end support and 1 intermediate support are selected, supports 13 and 09 are most relevant.
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Page 56
UNIT INFLUENCE LINES FOR ROTATIONS ABOUT THE LATERAL AXIS: φy
-3.00E-06
-1.50E-06
0.00E+00
1.50E-06
3.00E-06
0 50 100 150 200 250 300 350
Location of the unit load from bridge entrance [m]
Infl
ue
nc
e [
rad
/kN
ax
le l
oa
d]
UNIT 01 UNIT 02 UNIT 05 UNIT 06 UNIT 09 UNIT 10 UNIT 13 UNIT 14
Figure 7-6: UILs for y-rotations.
Looking carefully, both intermediate and end supports behave comparably. Therefore considering
1 support only is sufficient. Support 13 is selected since it experiences the largest rotations.
UNIT INFLUENCE LINES FOR ROTATIONS ABOUT THE VERTICAL AXIS: φz
-2.00E-06
-1.00E-06
0.00E+00
1.00E-06
2.00E-06
0 50 100 150 200 250 300 350
Location of the unit load from bridge entrance [m]
Infl
ue
nc
e [
rad
/kN
ax
le l
oa
d]
UNIT 01 UNIT 02 UNIT 05 UNIT 06 UNIT 09 UNIT 10 UNIT 13 UNIT 14
Figure 7-7: UILs for z-rotations.
For this DOF it can not be said that all end or intermediate supports behave equally. A selection
of 4 supports is needed to consider all kinds of behaviour. Supports 13, 14, 09 and 10 are
chosen.
FINAL VERSION
Page 57
7.1.2 Discussion about the UILs and a summary of selected DOFs
The graphs meet the expectations given in chapter 3. The magnitude of the parameters in the xz-
plane (ux, φy) develop very globally while cross plane parameters (uy, φx, Fy) develop locally. This
is very relevant in examining the behaviour due to (single and multiple) vehicles with a certain
length. The similar behaviour of some supports enables an analysis of a selection of supports for
the accumulated displacement calculations (figure 7-8). The number of DOFs which should have
been considered has been reduced from 35 to 11.
DISPLACEMENT SUPPORT DOF END SUPPORT INTERMEDIATE SUPPORT
13 ux;13 X x-translation
09 ux;09 X
13 uy;13 X y-translation
09 uy;09 X
13 φx;13 X x-rotation
09 φx;09 X
y-rotation 13 φy;13 X
13 φz;13 X
14 φz;14 X
09 φz;09 X z-rotation
10 φz;10 X
Figure 7-8: summary of selected DOFs for the calculation of accumulated displacements.
7.2 Examination of LILs and discussion about accumulated displacements
7.2.1 Explanation about the determination and representation of LILs
Expecting that the reader is familiar with influence line theory, only a brief description is given
about the strategy that has been followed to transform UILs into LILs.
A UIL is a graph showing the magnitude of a parameter due to a unit load as a function of the
location of that load on the structure. A LIL, however, shows the magnitude of a parameter, due
to a specific lorry, as a function of the lorry location (on a bridge deck). In a linear static
consideration, the lorry influence value is just a summation of all independent axle loads times the
influence value for unit loads at the location of the axle. This summation has been done in a
spreadsheet program. All LILs are plotted as a function of time that passed since the first axle
entered the bridge.
The LILs for DOFs show the lorry induced displacements only since dead load displacements do
not cause wear.
FINAL VERSION
Page 58
The reaction forces need some more explanation. Firstly, z-reactions are calculated and plotted
including vertical dead load reactions in order to illustrate the (limited) contribution of lorry loads
to the total vertical reaction. The vertical axis has been divided into 10 percent divisions which
directly illustrates the deviation from the unloaded situation.
Before explaining the graphs for horizontal reactions, some words on friction are given. The
‘restraining’ forces that act on the supports before sliding are assumed to be so large that friction
does not restrain movements at all. Additionally, in case of longitudinal sliding, friction does also
not restrain lateral movements anymore. This assumption enables estimation of the magnitude of
friction in both horizontal directions and consequently the effect on the fixed and guided supports
by means of a simplified consideration.
Guided supports only move longitudinally; free supports, however, are able to move in 2
horizontal directions. Therefore, friction in these supports can act in every direction in the
horizontal plane. Its direction is opposite to the direction of motion. Knowing this direction (from
the influence lines) and the vertical reaction, the 2 main parameters of the friction force are known
(direction and magnitude) and the friction forces in both horizontal directions can be calculated.
The magnitude of friction is equal to the vertical reaction times the coefficient of friction, which
has to be estimated (in this case µdyn=0.015 is assumed, based on the limit values for sliding
elements in EN 1337-2). Note that another choice for µdyn directly affects the results.
Assumptions have been made regarding the lateral reactions of guided and fixed supports. It is
assumed that lateral friction of free supports is taken by the adjacent guided (or fixed) support
only. This assumption facilitates the calculation of lateral reactions that are highly dominated by
friction. Graphs for y-reactions, therefore, show the distribution due to lorry loads including
frictional resistance of the adjacent support.
Lateral dead load reactions, obtained from the finite element model, are not included as they are
fictitious. Dead load deformation occurs before installation of the bearings and thus do lateral
reactions not occur.
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Page 59
7.2.2 Presentation and discussion of LILs and graphs for accumulated displacements
It would be needless to show all generated graphs. Therefore, only several typical graphs of the
most common lorry (T11O3(L)) are discussed. This lorry’s weight can also be regarded as
average for all lorries. All generated UILs and LILs can be found in annex F (digitally).
LORRY INFLUENCE LINES FOR LONGITUDINAL TRANSLATIONS (ux), vlorry = 90km/h
-6.0
-3.0
0.0
3.0
6.0
0 2 4 6 8 10 12 14 16
Time [s]
Tra
ns
lati
on
[m
m]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
Figure 7-9: LILs for x-translations due to a 90km/h driving T11O3(L) lorry.
The graphs for longitudinal translations do not contain (clearly visible) unexpected or inexplicable
results. The graphs are almost scaled versions of the UILs by a factor equal to the total lorry load.
Since the lines develop rather gradually, the maximum values can be calculated by considering
the lorry as a single point load. This is confirmed by a simple calculation. For example, support 09
has a unit influence value of 0.009 millimetres per kilonewton axle load at 50 metres from the
bridge entrance (figure 7-3). The 300kN lorry then causes a 2.7 millimetre translation which is
almost the same as shown above. In conclusion, the maximum lorry influence values for these
kinds of graphs can be calculated from the unit influence lines and a single point load that is
equal to the total lorry load.
If one regard this lorry as an average vehicle and this can be compared with the long-term friction
tests, we can observe 2 things. From maximum to minimum displacement about 7.5 millimetre
translation occurs. This is almost the sliding distance that is prescribed in the test procedure (type
B). The movement velocity in that test is 2 millimetres per second. Here it is 7.5 millimetres in
about 5 seconds which is a 1.5 millimetres per seconds movement velocity.
The current test parameters meet the actual situation in this particular case (for x-translations).
FINAL VERSION
Page 60
The simplistic calculation of the maximum displacements does not apply to lateral translations
and other DOFs that contain relatively small influence zones. Neglecting the vehicle length in
those cases is unacceptable because the influence lines have a very large gradient in the
normative zones (figure 7-10). This may result in unacceptable overestimated values. Consider
the same calculation as for the longitudinal translations. The unit influence value for support 09 at
100 metres from the bridge entrance (figure 7-4) is about 0.012 millimetres per kilonewton axle
load. This would result in a 3.6 millimetre translation for the 300kN lorry instead of 2.4 millimetres
that is given in the LIL below. The lorry influence value for the end supports can not be calculated
this way either (6.5 millimetres instead of 2.1). Therefore, the maximum lorry influence values for
these kinds of DOFs must be determined by means of an extended calculation.
LORRY INFLUENCE LINES FOR LATERAL TRANSLATIONS (uy), vlorry = 90km/h
-4.5
-3.0
-1.5
0.0
1.5
0 2 4 6 8 10 12 14 16
Time [s]
Tra
ns
lati
on
[m
m]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
Figure 7-10: LILs for y-translations due to a 90km/h driving T11O3(L) lorry.
Another problem that arises is that some (static) LILs show hysterical behaviour at the bridge
entrance and end. These singularities in the graphs are caused by a non-zero start (or end) value
of the corresponding UIL and the lack of inertia in a static analysis. Due to the inertia of a stability
frame, a support remains displaced as 2 axles enter the frame in rapid succession. In this static
analysis, displacements are calculated disregarding inertia and consequently the displacements
may differ suddenly (singularities). This will not be the case in a dynamic analysis in which inertia
is considered. This phenomenon affects the calculation results of the accumulated displacements
as well since this behaviour results in extra displacements. This will be discussed later.
FINAL VERSION
Page 61
The accumulated displacements have been calculated by summing up all contributions per
coordinate. As a check: the accumulated x-translation of support 09 is approximately 19.9
millimetres in the next graph. This must be equal to twice the sum of the maximum displacements
in figure 7-9.
( ) mm8.194.28.47.22s 09;x =++×=
SLIDING DISTANCE PER SINGLE LORRY FOR LONGITUDINAL TRANSLATIONS (ux), vlorry = 90km/h
0.0
5.3
10.5
15.8
21.0
0 2 4 6 8 10 12 14 16
Time [s]
Sli
din
g d
ista
nc
e [
mm
]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
SLIDING DISTANCE PER SINGLE LORRY FOR LATERAL TRANSLATIONS (uy), vlorry = 90km/h
0.0
4.5
9.0
13.5
18.0
0 2 4 6 8 10 12 14 16
Time [s]
Sli
din
g d
ista
nc
e [
mm
]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
Figure 7-11: development of the accumulated sliding distance for longitudinal and lateral translations due to
a 90km/h driving T11O3(L) lorry.
Both figures above show the development of the accumulated slide path during passage of the
lorry. Apparently, the development is completely different for the x-direction than for the y-
direction (in x-direction is more gradual than in y-direction).
FINAL VERSION
Page 62
The calculated sliding distance of the y-translations for end supports is too large because of the
singularities in the LILs. In order to obtain more reliable results, a dynamic analysis is suggested.
A rough estimate for these values can be obtained when the behaviour is compared to that of the
intermediate supports. The unit influence lines show that the influence zone is much smaller than
that for the intermediate supports but the influence value is larger. That means that the
displacement largely depends on the lorry length and axle configuration. The axle distance
reduces the maximum displacement but increases the sliding distance since it causes repeated
displacements. Therefore, as long as no better results are obtained, the total lateral sliding
distance is assumed to be comparable to the intermediate supports.
EN 1337-5 prescribes test parameters for internal seals. The amplitude of rotation in that test is
0.0025rad; the frequency between 0.25Hz and 2.5 Hz. Figure 7-12 shows clearly that the
rotations of an average lorry are considerably smaller than the test parameters. This figure
illustrates that a half period corresponds to about 4 seconds and thus a frequency of 0.125Hz.
The values in the standard are larger than the actual values for this average lorry.
LORRY INFLUENCE LINES FOR ROTATIONS ABOUT THE LATERAL AXIS (φy), vlorry = 90km/h
-9.00E-04
-4.50E-04
0.00E+00
4.50E-04
9.00E-04
0 2 4 6 8 10 12 14 16
Time [s]
Ro
tati
on
[ra
d]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
Figure 7-12: LILs for y-rotations due to a 90km/h driving T11O3(L) lorry.
The graphs for the other rotations show very large gradients, especially for end supports. This is
important regarding selection of bearings types. Note that z-rotations cannot be neglected which
is done in current tests.
Words on the sliding distances of internal seals can hardly be given. The sliding distance
depends largely on the bearing geometry (radius) and thus differs for every single bearing. The
graphs for the other rotations and lateral reactions look similar and thus are omitted in this
section.
FINAL VERSION
Page 63
Figure 7-13 illustrates the development of vertical reactions due to a passing lorry. Dead load
reactions have been included in these LILs as it clarifies the contribution of the lorry load to the
total vertical reaction and thus the fluctuation in contact stresses in the sliding planes. Because
the dead load reactions of intermediate supports and end supports differ much, 2 graphs have
been made so that the contribution of lorry loads is clearly visible.
LORRY INFLUENCE LINES FOR VERTICAL REACTION FORCES OF END SUPPORTS (Fz), vlorry = 90km/h
0
375
750
1125
1500
0 2 4 6 8 10 12 14 16
Time [s]
Re
ac
tio
n f
orc
e [
kN
]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 13 T11O3(L) 14
LORRY INFLUENCE LINES FOR VERTICAL REACTION FORCES OF INTERMEDIATE SUPPORTS (Fz), vlorry = 90km/h
0
2750
5500
8250
11000
0 2 4 6 8 10 12 14 16
Time [s]
Re
ac
tio
n f
orc
e [
kN
]
T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10
Figure 7-13: LILs for vertical reactions due to a 90km/h driving T11O3(L) lorry.
The graphs suggest that the vertical reactions do not change significantly. The vertical reactions
on end supports increase about 15 to 20% and less than 5% in case of intermediate supports.
For the heaviest loaded lorries this is about 10% for intermediate supports and 55% for end
supports.
FINAL VERSION
Page 64
Finally a small note about the longitudinal and lateral reactions is given. The analysis procedure
has been described in paragraph 7.2.1. The magnitude of friction has been adapted to the
direction of motion which can be seen in the next figures. These graphs are schematic
visualizations of the friction forces on the non-fixed supports and the resultant force on the fixed
or guided supports. It gives a global indication of the maximum reaction forces on the guides.
LORRY INFLUENCE LINES FOR LONGITUDINAL (FRICTION AND REACTION) FORCES (Fx), vlorry = 90km/h
-250.0
-125.0
0.0
125.0
250.0
0 2 4 6 8 10 12 14 16
Time [s]
Fo
rce
[k
N]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
LORRY INFLUENCE LINES FOR LATERAL (FRICTION AND REACTION) FORCES (Fy), vlorry = 90km/h
-125.0
-62.5
0.0
62.5
125.0
0 2 4 6 8 10 12 14 16
Time [s]
Fo
rce
[k
N]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
Figure 7-14: longitudinal and lateral reactions due to friction and a 90km/h driving T11O3(L) lorry.
These graphs should be considered carefully. For example, the lateral reaction on support 10 is
largest after 4 seconds and after 10.5 seconds. However, it was assumed that the friction was
taken by the adjacent support only and therefore only occurs once. This is due to the fact that the
direction of friction is related to the direction of motion. Since at 10.5 seconds support 09 moves
somewhat laterally and not longitudinally, it gives a lateral friction force at support 09 and hence a
lateral reaction force at support 10 which does probably not occur in reality.
FINAL VERSION
Page 65
7.2.3 Summary and interpretation of the calculated values for the accumulated displacements
The previous paragraph describes how the graphs should be interpreted. As a summary, the
calculated values for the accumulated displacements are listed below.
TRANSLATION [mm] ROTATION [·10-4
rad] LORRY
ux;13 ux;09 uy;13 uy;09 φx;13 φx;09 φy;13 φz;13 φz;14 φz;09 φz;10
(L) 4.4 7.3 5.5 3.1 10.0 2.7 11.2 5.6 4.6 2.4 2.6 V11
(H) 7.9 13.0 9.8 5.5 17.7 4.8 19.9 10.1 8.3 4.3 4.6
(L) 8.3 13.7 10.2 6.0 18.5 5.2 20.8 11.3 9.3 4.6 4.9 V12
(H) 14.3 23.7 17.8 10.5 32.0 9.2 36.1 20.0 16.7 8.0 8.6
(L) 8.2 13.6 9.8 4.7 17.8 4.2 20.6 9.3 8.3 4.0 4.0 T11O2
(H) 12.5 20.9 15.3 7.2 27.6 6.4 31.8 14.8 13.3 6.1 6.2
(L) 11.9 19.9 14.7 6.9 26.6 6.2 30.2 13.9 12.8 5.8 5.9 T11O3
(H) 21.1 35.1 26.0 12.7 47.1 11.3 53.4 25.3 23.3 10.5 10.7
(L) 17.6 29.4 21.5 10.0 39.1 8.9 44.6 21.1 17.9 8.4 8.4 V12A12
(H) 26.2 43.6 31.9 14.8 58.1 13.3 66.2 31.6 27.0 12.6 12.7
(L) 21.4 35.9 26.2 11.3 47.5 10.2 54.3 24.7 22.7 9.7 9.6 T12O3A2
(H) 34.9 58.6 42.8 18.9 77.8 17.0 88.7 40.4 37.3 16.1 16.0
ACC. TRANSLATIONS [km] ACC. ROTATIONS [·103rad]
ux;13 ux;09 uy;13 uy;09 φx;13 φx;09 φy;13 φz;13 φz;14 φz;09 φz;10
YEAR 40.4 67.3 49.6 23.8 9.0 2.1 10.2 4.8 4.3 2.0 2.0
25 YEARS 1010 1683 1240 595 225 523 255 120 108 50 50
Figure 7-15: calculated values for the accumulated displacements. The total values per year are calculated
for 3 million lorries (per year).
Remarkably, the accumulated sliding distance in y-direction is larger than in x-direction for the
end supports. This might be a result of the singularities which has been discussed in paragraph
7.2.2. This value (and φx;13, φz;13, φz;14 as well) must therefore be regarded as unreliable.
In the final stage of this thesis, it was found that the actual sliding distance is (obviously) a
combination of x-translations and y-translations and thus can not be considered separately. Until
now the translations (and rotations as well) were uncoupled. It still holds that the accumulated
displacements are equal to the values given in figure 7-15. However, the sliding distance in case
of free supports is not equal to the accumulated values in both horizontal directions. Consider a
Pythagoras triangle: draw an internal curve between the 2 ends of the hypotenuse. This curve (s)
represents the actual path of the bridge’s superstructure due to a combination of x-translations
(ux) and y-translations (uy). Then, the next equation applies to the sliding distance.
yx2y
2x uusuu +≤≤+
FINAL VERSION
Page 66
In case of support 09, however, almost all lateral displacement occurs between 3 and 5 seconds
(figure 7-11). In this period approximately 25% of the total sliding distance in x-direction occurs.
That means that the total sliding distance (in both horizontal directions together) can be estimated
by the next equation. Importantly, this equation only applies to horizontal translations of
intermediate supports! Other equations may be derived for the other DOFs and other supports.
( ) 2y
2xx
2y
2
xxxy ss0625.0s75.0ss25.0s75.0s +×+×=+×+×=
For example, if one is interested in the actual sliding distance of support 09, one can calculate:
km6.798.233.670625.03.6775.0s 2209;xy =+×+×=
Yet it is possible to compare the calculated sliding distances with the test specifications given in
EN 1337-2. As described in paragraph 2.6.2, this standard prescribes an accumulated sliding
distance of circa 10000 metres for plane surfaces which is considerably smaller than the
calculated values. Some steps will be checked for support 09.
1. The simulated global mechanical behaviour (bending in combination with torsion) of the
bridge was found to be quite accurate compared to measurement results (chapter 6). The
model’s influence line was based on a scaled version of a UIL for vertical deflections. As
a result, it might be assumed that the UILs for x-translations are correct as well since they
are largely related to the vertical bending deformation.
2. Since the UILs for x-translations are assumed to be correct, and it has been found that
the lorry influence values for this DOF can be obtained by scaling the unit influence
values by the total lorry load (paragraph 7.2.2), the weighted sliding distance for an
average lorry can be calculated.
3. The (weighted) average lorry load can be calculated. For every type of lorry i:
kN340n
nG...nG
n
nG
Gtotal
)H(2A3O12T)H(2A3O12T)L(11V)L(11V
total
12
1iii
average ≈×++×
=
×
=∑=
4. The weighted average lorry load is about 340kN. The maximum unit influence values for
x-translations of support 09 are subsequently 0.009, -0.016 and 0.008. With these values,
the sliding distance per average lorry can be calculated:
( ) mm5.22008.0016.0009.02340s average;09;x ≈++××=
5. Since this value is average, the accumulated sliding distance per year can be calculated
by multiplying this value by 3 million (total amount of considered lorries per year). This
results in a sliding distance of 67.5km/year which has been found in the table as well.
FINAL VERSION
Page 67
As shown above, the calculated values seem to be calculated correctly. That does not imply that
the results are correct. Actually, the calculated accumulated values represent an upper bound for
the sliding distances. These results are based on single lorries that pass the bridge independently
which means that lorries do not reduce each others sliding distance, which is the case for lorries
driving in adjacent spans. However, considering 3 million lorries separately is unrealistic. This
amount of lorries per year means an average lorry interval of about 10 seconds. Driving 25m/s
means an average lorry distance of 250 metres which is smaller than the bridge’s length.
Therefore, several lorries on the bridge inevitably do occur. This influence must be examined by
means of a parameter study or a traffic simulation. As a start, a parameter study will be
performed and discussed in the next paragraph.
The values in figure 7-15 are very large compared to EN 1337. Still a reduction can be expected
in addition to the interference of multiple lorries. Firstly, 3 million lorries per year is representative
to the most intensively used bridge in the Netherlands. In Hagestein this is about 2 and 2.5
million. The composition is comparable and thus a reduction of perhaps 20-30% is realistic.
Regarding rotations, these values do not give any information about the sliding distances of
internal seals of pot bearings or spherical surfaces. Therefore a pot diameter or radius of the
spherical surface, for example, is needed. Consider a diameter of 0.5 metre, then for y-rotations
hold a sliding distance of 5000 metres per year. A similar calculation for the x- and z-rotations can
only be done for intermediate supports since the end supports contain unrealistic singularities in
the graphs. For all these DOFs hold a sliding distance of 1000 metres per year in case of a 0.5
metre radius.
FINAL VERSION
Page 68
7.3 Influence of interacting lorries on the accumulated displacements
The previous paragraph described that for some parameters the lorry influence values can not be
calculated accurately by multiplying the unit influence value by the total lorry load. Reason for this
is the relatively large vehicle length compared to the influence zone of those parameters. This
principle applies to the other DOFs with complete lorries as well since these loads act on
relatively large distances. Accordingly, a reduction of the lorry influence values might be expected
for those DOFs as well when several lorries drive on the bridge simultaneously.
Therefore, a parameter study about the influence of vehicle distances on the accumulated
displacements has been performed. This has been done for a 2 lorry situation, type T11O3(L) has
been examined only, and a 3 lorry situation. In the analysis of 2 lorries, the vehicle distance was a
multiple of 50 metres until it exceeded the sum of the bridge length and vehicle length (400
metres). The 400 metre analysis represents a situation in which the second vehicle enters the
bridge when the first one left it. Since the x-translation of support 09 has been discussed in the
previous paragraph, this DOF will be discussed here as well.
DISPLACEMENT DUE TO 2 T11O3(L) LORRIES FOR SEVERAL LORRY DISTANCES: ux;09
-10
-5
0
5
10
0 5 10 15 20 25 30 35
Time [s]
Tra
ns
lati
on
[m
m]
50 100 150 200 250 300 350 400
Figure 7-16: development of the x-translation of support 09 due to 2 T11O3(L) lorries for several distances.
This figure is shown to give an impression about the development of the displacement due to
multiple lorries. The 400 metre line covers the other lines over the first several metres in which
the second lorry did not enter the bridge yet. It is probably not very clear, however, looking
carefully it can be found that 2 lorries on the bridge change the development significantly,
especially for vehicle distances smaller than 300 metres. The 400 metres series repeats itself
twice since the 2 lorries do not drive simultaneously on the bridge. The 50 metres series is almost
equal to a scaled single lorry analysis.
FINAL VERSION
Page 69
A better illustration is the development of the accumulated sliding distance. In the figure below,
the sliding distance for several vehicle distances are compared. Yet, a significant difference is
observable between independently and simultaneously driving lorries. The 400 metres line,
representing independent lorries, ends at about 40 millimetres which is (correctly) twice the
sliding distance of a single lorry calculation. The 100 metres line, however, ends at a sliding
distance of 22.5 millimetres which is only 56% of this value for the 400 metre lorry distance.
ACCUMULATED DISPLACEMENT DUE TO 2 T11O3(L) LORRIES FOR SEVERAL LORRY DISTANCES: ux;09
0
10
20
30
40
0 5 10 15 20 25 30 35
Time [s]
Ac
cu
mu
late
d t
ran
sla
tio
n [
mm
]
50 100 150 200 250 300 350 400
Figure 7-17: accumulated sliding distance for x-translations of support 09 due to 2 T11O3(L) lorries.
The average lorry interval was previously taken as 10 seconds (250 metres). For that distance,
the reduction is negligible. However, the 10 seconds interval is not realistic as most lorries pass
during daytime and perhaps in groups as well. Therefore the average lorry distance may become
in the range between 50 and 150 metres which will cause a significant reduction of the calculated
values in paragraph 7.2.3. Before drawing some conclusions, a parameter study with 3 lorries will
be discussed.
In the parameter study for 3 lorries, a 125 metre vehicle distance is examined, i.e. the first and
third lorry drive 250 metres from each other while the second vehicle drives on a variable
distance in between. The variable distance is fixed in each analysis, though, it varies between
several analyses. The distances that mark the series are related to the distance between lorry 1
and 2. The distance between lorry 1 and 3 is always 250 metres.
FINAL VERSION
Page 70
ACCUMULATED DISPLACEMENT DUE TO 3 T11O3(L) LORRIES FOR SEVERAL LORRY DISTANCES: ux;09
0
15
30
45
60
0 5 10 15 20 25 30 35
Time [s]
Ac
cu
mu
late
d t
ran
sla
tio
n [
mm
]
50 75 100 125 150 175 200 225
Figure 7-18: accumulated sliding distance for x-translations of support 09 due to 3 T11O3(L) lorries, driving
90km/h. The distances given in the legend represent the distance between the first and second lorry. The
distance between lorry 1 and 3 is always 250 metres.
This figure illustrates that the sliding distance for 3 lorries at a 125 metre interval is just 30
millimetres for 3 lorries which is 10 millimetres per lorry instead of 20 (50%).
These studies support the need for a traffic simulation in order to gain insight into the actual
sliding distances. The values in figure 7-15 should possibly be reduced by 30% to achieve a first
impression of the sliding distances (30% is an estimation since the percentages above are based
on the most adverse situation).
The previously described parameter study applies to x-translations and rotations about the lateral
axis (φy). It does not apply to the other DOFs since their relevant influence zones are
considerably smaller. Therefore, a multiple lorry consideration does not result in a reduction of the
accumulated sliding distance.
FINAL VERSION
Page 71
Figure 7-19 confirms the negligible influence of multiple lorries for lateral translations. The
calculated values in paragraph 7.2.3 can therefore be considered as useful.
DISPLACEMENT DUE TO 2 T11O3(L) LORRIES FOR SEVERAL LORRY DISTANCES: uy;09
-3
-2
0
2
3
0 5 10 15 20 25 30 35
Time [s]
Tra
ns
lati
on
[m
m]
50 100 150 200 250 300 350 400
ACCUMULATED DISPLACEMENT DUE TO 2 T11O3(L) LORRIES FOR SEVERAL LORRY DISTANCES: uy;09
0
5
10
15
20
0 5 10 15 20 25 30 35
Time [s]
Ac
cu
mu
late
d t
ran
sla
tio
n [
mm
]
50 100 150 200 250 300 350 400
Figure 7-19: development of the y-translation and accumulated sliding distance of support 09 due to 2
T11O3(L) lorries, driving 90km/h.
The influence on the accumulated values are also negligible for φx and to a lesser extent φz
(annex G).
FINAL VERSION
Page 72
7.4 Evaluation and general conclusions from the linear static influence line analysis
The next items can be said about the static analyses:
• The UILs meet the predictions in chapter 3. Especially the DOF graphs give a useful
impression of the development of support displacements due to a certain point load that
passes the bridge in longitudinal direction. The graph for z-reactions must be examined
carefully as important dead load reactions are not included. The graph for y-reactions
must be considered carefully as well because of the neglect of lateral friction.
• The LILs give the actual displacements of supports due to several prescribed lorries.
Although MIDAS calculated the influences frictionless, the final displacements are not
expected to be influenced largely by the lack of friction (possibly 5-10% for intermediate
supports, based on a non-reported quick-scan study).
• The vertical reactions due to lorry loads are small compared to the dead load reactions
for the intermediate piers, the maximum increase is 10% (annex F-170). Lorry loads can
not be neglected for the end supports, the increase can be more than 50% (annex F-
169). However, a maximum of 20% applies to the majority of lorries (70%).
• The lateral reactions are calculated manually and therefore give only a schematic
indication of the maximum lateral reactions due to lorry loads and (lateral) friction.
Because MIDAS did not consider friction, the cooperation of supports due to frictional
resistance has been disregarded as well. The lateral reactions therefore do possibly not
represent reality. It is sometimes questioned which part of the time a guide is loaded.
Because the cooperation between supports is not examined and the development of
lateral reactions is accordingly not representative, this question can not be answered.
Lateral reactions are mainly a result of friction.
• For x-reactions, the same story applies as for lateral reactions.
• For x-translations and y-rotations, the maximum influences for lorries can be calculated
by multiplying the unit influence values by the total lorry load. The lorry influence values
for the other parameters cannot be calculated likewise as they are overestimated.
• The x-translations and movement velocity, induced by an average lorry, are comparable
to the test parameters in EN 1337-2 type B.
• The results for y-rotations are significantly smaller than the prescribed test parameters,
(magnitude of rotation and the frequency) in EN 1337-5.
• Rotations about the vertical axis cannot be neglected.
FINAL VERSION
Page 73
• The calculated accumulated displacements are much larger than the prescribed values in
EN 1337. However, the values must be considered as unrealistic upper-bound values as
it is shown that several lorries on the bridge mostly reduce these values that are based
on single lorries. A multiple lorry consideration illustrates that these values for x-
translations and y-rotations can be reduced by maximally 50% but 30% is probably more
realistic. Therefore an extended parameter study or traffic simulation is recommended to
find the actual values. Still, the 30% reduction gives much larger values than EN 1337.
As a result of the small influence zones of the other parameters (compared to the vehicle
length and lorry distance), their accumulated displacements are hardly influenced by
interacting lorries. Their calculated values are therefore already quite realistic. These
values for end supports can not be considered as correct due to the hysterical behaviour
and must be examined in a time-history analysis.
FINAL VERSION
Page 74
FINAL VERSION
Page 75
8 Linear dynamic analysis of support displacements and reaction forces
8.1 A brief introduction into the need for a dynamic analysis
From structural dynamics it is known that the dynamic response of a structure due to dynamic
loads depends on the ratio between load frequency and a certain natural frequency of the
structure (frequency-response function). Results obtained from dynamic analyses generally differ
from those in similar static analyses. In general, relatively low frequency loads cause an almost
static response while very high frequency loads may cause a negligible response (due to inertia
of the structure). Loads with a frequency in the range of specific natural frequencies may cause a
response that is (much) larger than the static response of a comparable load. Importantly,
dynamic loads only amplify deformations when the load causes similar deformations as the
vibration mode shape that corresponds to the considered natural frequency (e.g. horizontal loads
obviously do not amplify vertical motions even though the frequency might be the same).
Although dynamic behaviour is more an issue in case of periodic loads (e.g. wind), transient loads
(e.g. traffic) may cause dynamic amplifications as well. An examination of the response due to the
dynamic traffic load is therefore needed.
The dynamic behaviour of a bridge must be assessed globally as well as locally. Globally, the
dynamic response of the whole bridge is regarded. A well-known example of detrimental global
dynamic amplification is the collapse of the Tacoma Narrows Bridge due to wind excitation.
Whether or not the considered plate-girder bridge is sensitive to global dynamic amplification due
to traffic, is discussed in the next paragraphs.
An example in local sense is the amplification of stresses in cross beams as a result of
successively passing axle loads on cross beams. Yet, only local frequencies are relevant. This
local behaviour may apply to lateral displacements of stability frames as well since those
displacements are related to local deformations only (chapter 7).
As mentioned before, amplification depends on the ratio between load frequency and natural
frequencies. Therefore it is useful to study the natural frequencies of the bridge before starting the
examination of support displacement. As a reminder, natural frequencies are proportional to a
structure’s stiffness and inversely proportional to its mass (damping disregarded).
m
kn =ω
FINAL VERSION
Page 76
8.2 Examination of the lowest natural frequencies of the bridge
Figures 8-1 and 8-2 show the first 2 vibration mode shapes of the bridge near Hagestein.
Apparently both are (almost similar) twisting mode shapes, which is a result of the very low
torsional rigidity of these kind of bridges (and similar decks of other bridge types). Main difference
between both frequencies is which main girder deforms mostly. For the first frequency it is the
inner girder while it is the outer girder for the second frequency. This may be explained by the fact
that the outer girder is somewhat stiffer and accordingly has a bit higher frequency.
Figure 8-1: first vibration mode shape (ωn=0.52Hz)
Figure 8-2: second vibration mode shape (ωn=0.54Hz)
FINAL VERSION
Page 77
The next figure shows the first 20 eigenfrequencies (natural frequency) of the structure. The
vibration mode shapes of the first 6 corresponding frequencies are given in annex H.
NUMBER FREQUENCY [Hz] PERIOD [s] NUMBER FREQUENCY [Hz] PERIOD [s]
1 0.52 1.92 11 2.23 0.45
2 0.54 1.86 12 2.53 0.40
3 0.97 1.03 13 2.86 0.35
4 1.02 0.98 14 3.20 0.31
5 1.06 0.94 15 3.64 0.27
6 1.41 0.71 16 3.77 0.27
7 1.44 0.69 17 4.04 0.25
8 1.48 0.68 18 4.17 0.24
9 1.79 0.56 19 4.44 0.23
10 2.05 0.49 20 4.98 0.20
Figure 8-3: first 20 natural frequencies of the bridge.
8.3 Expectations about dynamic amplifications
Previous chapters describe that the behaviour of supports is caused by local and/or global
mechanical behaviour. DOFs that are mostly related to global deformation of the bridge are x-
translations, y-rotations and to a lesser extent z-rotations. Those parameters constantly change
during passages of vehicles over the bridge. The other DOFs (y-translations and x-rotations) only
displace when the load acts close to the support itself. The examination of dynamic amplification
can therefore be split into a global analysis and a local consideration.
Starting globally, it has to be found whether or not the period of a passing lorry (almost) matches
to a natural period that corresponds to the lorry induced global deformation. Consider figure 8.2
and imagine a lorry driving from left to right over the bridge. The bridge will deform almost
identically to what the figure illustrates. It is, therefore, assumed that the second mode shape is
most sensitive to the lorry induced deformation and possibly cause dynamic amplification.
As long as the lorry is in the first span, it amplifies the downwards motion. Once it enters the
second span, it acts in the same direction as the reversed motion. The period of the lorry load is
therefore the time needed to pass 2 spans (full cycle). Since the side span and intermediate span
differ in length, it is easier to consider a half period for 1 span. Thus, half a period of the lorry load
is about 3.75 seconds for a side span. A full period of 7.5 seconds then applies to the lorry load
which corresponds to a load frequency of 0.13Hz. Although there is a lack of knowledge about
the frequency-response function of this structure, this frequency is significantly smaller than the
second natural frequency and thus little amplification of the x-translations and y-rotations is
expected. The same holds for rotations about the vertical axis, however, this DOF is also
influenced by local deformations which may result in local amplifications.
FINAL VERSION
Page 78
For the examination of this local behaviour, static influence lines can be useful to determine load
frequencies. Consider, for example, the lateral translation of support 09 in figure 8-4. The
displacement is negative between 3 and 5 seconds. A passing lorry can be interpreted as a
periodic load that act between these 2 moments in negative y-direction on the support. The 2
seconds may then be interpreted as a half period (similar as for the global behaviour). The total
period is then 4 seconds and the frequency 0.25Hz. This is very small as well, especially
compared to higher frequencies that correspond to lateral movements. Therefore little dynamic
amplification is expected for this DOF.
Both in global and local sense, little dynamic amplification is expected. However, although the
frequencies do not match well with natural frequencies, displacements in the dynamic analysis
may still be somewhat larger than those calculated in static analyses. A dynamic analysis in
MIDAS must clarify whether or not this is the case.
LORRY INFLUENCE LINES FOR LATERAL TRANSLATIONS (uy), vlorry = 90km/h
-4.5
-3.0
-1.5
0.0
1.5
0 2 4 6 8 10 12 14 16
Time [s]
Tra
ns
lati
on
[m
m]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
Figure 8-4: influence lines for lateral translations due to a 90km/h driving T11O3(L) lorry.
FINAL VERSION
Page 79
8.4 Presentation and evaluation of displacements obtained from a (frictionless) linear
time-history analysis
The support translations obtained from a time-history analysis in MIDAS are given in figure 8-5.
The graphs look very similar to the static influence lines. However, inconceivable wave patterns in
the influence lines are clearly visible. The behaviour in x-direction could still be possible, though,
in lateral direction it seems more unrealistic. Presumably this is a result of neglecting friction. We
have to keep in mind that the translations represent the relative displacements in the sliding
surfaces in which friction influences the local behaviour in the bearing largely. The time-history
analysis does not include local effects and may therefore be rather optimistic.
DYNAMIC INFLUENCE LINES FOR LONGITUDINAL TRANSLATIONS (ux), vlorry = 90km/h
-6.0
-3.0
0.0
3.0
6.0
0 2 4 6 8 10 12 14 16
Time [s]
Tra
ns
lati
on
[m
m]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
DYNAMIC INFLUENCE LINES FOR LATERAL TRANSLATIONS (uy), vlorry = 90km/h
-4.5
-3.0
-1.5
0.0
1.5
0 2 4 6 8 10 12 14 16
Time [s]
Tra
ns
lati
on
[m
m]
T11O3(L) 01 T11O3(L) 02 T11O3(L) 05 T11O3(L) 06 T11O3(L) 09 T11O3(L) 10 T11O3(L) 13 T11O3(L) 14
Figure 8-5: dynamic influence lines for x- and y-translations due to a 90km/h driving T11O3(L) lorry.
FINAL VERSION
Page 80
The actual mid-span deflection of the intermediate span is probably less influenced by friction in
sliding planes. Therefore the time-history results are validated by examining movements of the
inner girder’s bottom flange at this location. This may gain more insight into the correctness of
this analysis. Figure 8-6 shows the movements in vertical as well as lateral direction.
Figure 8-6: movements in the yz-plane of the inner girder's bottom flange due to a 90km/h driving T11O3(L)
lorry.
The graph shows a conceivable development of both translations. Remarkably, both lines
illustrate that the behaviour in mid-span is more gradual compared to the support translations.
One would expect that the movements in the sliding plane are more gradual since friction
prevents the sliding elements from hysterical movements. The support movements are therefore
assumed to be unreliable.
Still, the global (average) values in this dynamic run match quite well with the statically
determined values. The global average behaviour is therefore expected to be correct. The static
influence lines then might be used as a global indication of the maximum or minimum
displacements, general support movements and possibly for sliding path calculations as well.
According to figure 8-5, small dynamic amplifications are observed: in x-direction about 5%; in y-
direction nearly 10%. Besides this amplification, performed quick scan studies, not reported in
this thesis, about the influence of friction on the displacements show that the reduction of
displacements is possibly 5 or 10% as well. Although these latter values are very rough
estimates, the resulting dynamic displacements then possibly equal the static values.
FINAL VERSION
Page 81
Furthermore, the lateral behaviour of the end supports is better than that from a static analysis. It
even shows a small reduction of the lateral translations compared to the static analysis.
Generally concluded is a dynamic analysis an efficient approach, providing results without the
need for modification in spreadsheet programs. Additionally, a time-history analysis is able to
include inertia and probably friction. Therefore the behaviour becomes much more realistic and
the results more reliable. However, the calculation time is much larger per analysis and much
more finite element analyses have to be performed compared to a static moving load analysis in
which the unit influence values are calculated only once (the computation time of this single 15
second analysis was about 30 minutes). When friction is included, a nonlinear analysis has to be
performed which need even more calculation time. The time-history analysis is therefore
recommended for single lorry studies only and perhaps a parameter study with 2 or 3 lorries. An
extended traffic simulation is probably infeasible.
Still, it is recommended to study the applicability of the FPSI-feature in MIDAS for modelling
friction in the sliding surfaces. It will possibly provide more realistic results of the support
displacements and lateral reactions due to cooperation between supports.
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Page 82
FINAL VERSION
Page 83
9 Discussion about the applicability, effectiveness and accuracy of the
methods of working
The static ‘moving load analysis’ in MIDAS provides a lot of useful influence data which can be
transformed into influence lines for support movements and reaction forces. The benefit of using
this MIDAS feature instead of making a lot of static load cases, is that easily more data points can
be obtained in addition to nodal values. Using a spreadsheet program, the data can be adapted
into all kinds of parameters and series as influence lines for single lorries and even for multiple
lorries. Calculating the accumulated displacements is not possible in MIDAS and must therefore
be done in a spreadsheet program anyway. This method, using a spreadsheet, is very clear and
easy. Additionally, graphs can be formatted similarly and clearly. Using Visual Basic (macro’s in
the spreadsheet) is even more useful as it works more systematically, this is recommended for
future research as well. Following a standard procedure results in a very efficient and effective
method in which a lot of data can be generated in similar representations. The accuracy of this
method is good as it uses data from MIDAS itself. The global behaviour and magnitude of x-
translations and y-rotations is very representative. The behaviour of the other DOFs of the
intermediate supports is still acceptable. The calculation of these y-translation, x-rotation and z-
rotation of the end supports, however, is not correct. The neglect of inertia causes singularities in
the influence lines which affect the calculation of the accumulated displacements. This is 1 of the
main disadvantages of this method. Furthermore, disregarding friction and inertia will cause a
neglect of cooperating supports. Supports are yet considered simultaneously, though, separately
as well. Another disadvantage is that although the finite element analysis is only performed once,
exporting the influence data from MIDAS and importing into the spreadsheet is unfortunately
rather time-consuming. Totally it is a lot of work but it results in a lot of very useful data. A
possible traffic simulation will not result in excessive computation time.
The linear time-history analysis, in which friction is still disregarded but inertia is included, may
provide results that do not need further modification into useful values. The input is already the
data you want to be studied. Using a spreadsheet for the input parameters (TFFs and DNLs)
enables the performance of multiple analyses in rapid succession. MIDAS subsequently provides
the results directly in graphs. However, MIDAS is not able to show reaction forces in time in a
time-history analysis. Beam elements have to be added to the supports in order to save a lot of
effort (chapter 5) and provide the desired results. The general behaviour of the bridge appears to
be satisfactory (mid-span movements), however, the local behaviour of the supports seems to be
rather unrealistic, especially in lateral direction. Furthermore, the calculation time for short-term
linear analyses is already quite long (30 minutes for a 15 second analyses), a possible traffic
simulation will therefore even result in a very long analysis.
FINAL VERSION
Page 84
Including friction is probably the key to success in the time-history analysis. The applicability of
the FPSI has been proved for a simple beam structure (annex D). However, the input parameters
were still based on trial and error. The applicability in a bridge structure is therefore not examined.
As a short conclusion. The static approach is very reliable and provides (mostly) useful results.
However, it is rather time-consuming. Nevertheless, the time-history analyses last long as well.
Additionally, the results of the dynamic run are questionable. Including friction (using the FPSI-
feature) might be the key to success. It is therefore recommended to study the influence of friction
in the sliding planes on the support displacements, especially in a dynamic analysis. The results
can subsequently be used for validation of the results, obtained from the static analysis. If it
proves impossible to use the FPSI-feature, perhaps another finite element program including
interface elements is an option.
FINAL VERSION
Page 85
10 Conclusions and recommendations
This chapter provides conclusions and recommendations, based on the reported results in the
previous chapters. Note that the conclusions apply to this plate-girder bridge only.
• A linear static analysis of vehicle induced deformations is sufficiently accurate in order to
gain a clear understanding of the bridge’s mechanical behaviour, support displacements
and (vertical) reactions. Although for this bridge, a 5 to 10% dynamic amplification has
been found in a frictionless time-history analysis, the resultant effect of dynamic
amplification and frictional resistance is expected to be negligible for intermediate
supports but it should be noted that it depends on the actual coefficient of friction. For
end supports this must be studied further.
• Considering single lorries, the most relevant translation in longitudinal direction and
corresponding velocity match the input parameters for long-term friction test type B (EN
1337-2). The test parameters for internal seals (EN 1337-5) appear to be too large
compared to this bridge as they correspond to the results of the heaviest lorry only.
• The development of translations in longitudinal direction and rotations about the lateral
axis depends largely on the position of several lorries on the bridge. Because of the static
indeterminacy of the bridge, these parameters are reduced largely when lorries drive at
realistic distances between 100 to 150 metres (corresponding to a span length). A traffic
simulation is therefore needed in order to obtain the actual behaviour. This does not
apply to the other degrees-of-freedom since they are only influenced by local lorries.
• The accumulated displacements that have been calculated for 1 year are much larger
than the distances that are used in long-term friction and durability tests in EN 1337. It
has been proved that the accumulated translation in longitudinal direction and rotation
about the lateral axis due to actual traffic, are considerably smaller compared to
independent vehicles. Still, the current test distances are much smaller than the
calculated values. The reduction due to actual traffic is negligible for the other degrees-
of-freedom.
• Rotations about the lateral axis are most relevant regarding the amplitude but they
develop very gradually. Both other rotations, however, are smaller in magnitude but
develop with (much) higher velocities. This can be of crucial importance for some bearing
types.
• Vertical reactions of intermediate supports increase maximally 10% due to single lorries.
For end supports this can be up to 55%, however, an increase less than 20% applies to
70% of the lorries. Regarding single lorries, the constant pressure in friction tests for
sliding elements is thus realistic for the intermediate supports but conservative for end
supports.
FINAL VERSION
Page 86
• Horizontal reactions are, due to the dominating dead load reactions, mainly caused by
frictional resistance of the movable (intermediate) supports. Since friction is not included
in the finite element calculation, the cooperation of supports is neglected. This results
probably in other guide loads. It is, therefore, not possible to indicate what part of the time
a certain guide is loaded, which is sometimes questioned. A nonlinear finite element
calculation including friction is recommended to study the influence of friction on the
horizontal reactions.
What can the results be used for? Firstly they provide insight in the global mechanical behaviour
of a plate-girder bridge and the corresponding support displacements. Additionally, they can be
used for bearing type selection as some types are limited applicable regarding, for example,
angular velocities (pot bearings). Furthermore, the influence lines for horizontal reactions can be
used in order to obtain a fatigue load spectrum on guides as the magnitude per lorry and the
number of lorries is known. Guides can then be designed for a certain fatigue life.
Eventually some personal recommendations for further research are given. Firstly the nonlinear
behaviour of the bridge, including friction in the bearings, should be studied in order to examine
the cooperation between supports and the actual horizontal reactions (and displacements). In this
case, either the applicability of the Friction Pendulum System Isolator (general link) in MIDAS
Civil has to be studied further, or another finite element program (using interface elements, for
example) should be used. The nonlinear results might then be compared to the static influence
lines in order to assess the validity of the static approach. The actual (lateral) behaviour of end
supports must be studied further as well as the results are not satisfactory. Furthermore, I would
suggest measuring the support movements (translations and rotations) of a free end support and
free intermediate support due to a single lorry. These results can be compared to the results from
both a static and nonlinear dynamic analysis in order to check both approaches. Finally, when the
support displacements are expected to be reliable, I recommend modeling several bearings
which are subjected to a displacement history, obtained from the bridge behaviour. This model
can then be used in order to examine the actual movements and contact stresses of bearing
components. For the translations in longitudinal direction and rotations about the lateral axis, a
traffic simulation is needed in order to examine the actual values for the accumulated
displacements.
FINAL VERSION
Page 87
References
Design standards
EN 1337-2, ‘Structural bearings – Part 2: Sliding elements’, March 2004
EN 1337-5, ‘Structural bearings – Part 5: Pot bearings’, March 2005
EN 1990, ‘Eurocode – Basis of structural design’, July 2001
EN 1991-2, ‘Eurocode 1: Actions on structures – Part 2: Traffic loads on bridges’, September 2003
Delft University of Technology master of science theses
Driessen, Jeroen, ‘Slijtage van brugopleggingen’, 2003
Otte, Adriaan, ‘Proposal for modified Fatigue Load Model based on EN 1991-2’, 2009
Delft University of Technology lecture books
Spijkers, J.M.J., Vrouwenvelder, A.W.C.M., Klaver, E.C., ‘Dynamics of Structures, Part 1 Vibration of structures’,
December 2006
ANNEX A
Research into the longitudinal translation of support 13/14
A Delft University of Technology thesis by V. Bos
Master of Science ‘Civil Engineering’
January 31, 2011
FINAL VERSION
Page A-1
ANNEX A Research into the longitudinal translation of supports 13/14
This annex has been written to clarify when the longitudinal motion is dominated by the rigid body
motion and when by local bending of the neutral axis. The paragraphs contain basic mechanics
calculations. Therefore, not all details and steps have been described.
A.1 Relationship between rotations about the lateral axis of supports 13/14 and 05/06
The parameter ‘a’ in the next formulae represents the location of the load in the considered span.
A.1.1 Ratio between rotations supports 13/14 and 05/06 for load between 13/14 and 09/10
The bridge (neutral axis) is statically indeterminate. Suppose that the girder has a constant
bending stiffness along its length. The next formulae hold for the rotations of the supports (these
formulae are basic mechanical formulae and therefore not explained).
( )
( )
EI6
LM
EI3
LM
EI3
LM
EI6
LM
EI6
LM
EI3
LM
EI3
LM
LEI6
aLaP
EI6
LM
LEI6
L2aL3xaP
306/0502/01
306/05206/05210/09L06/05
L06/05
206/05210/09110/09
1
221L
10/09L
10/09
110/09
1
211
2
14/13
32
21
×
×−θ
×
×=
×
×−
×
×−θ=θ
×
×+
×
×=
×
×−
××
−××−θ=θ
×
×+
××
×+××−××θ
It is possible to rewrite (using mathematical software) the above equations into a ratio in which
only the length of the spans (side span and main span) and the location of the load is included.
( ) sms
2sm
3ss
2m
2msm
2s
06/05
14/13
m2
s31
LLaL2
LL14L6LL6aL3aLL10aL6r
LL
LLL
××+×
××−×−××−××+×××+××−=
θ
θ=
=
==
ϕ
FINAL VERSION
Page A-2
A.1.2 Ratio between rotation supports 13/14 and 05/06 for load between 09/10 and 05/06
The next formulae hold for the rotations of the supports.
( )
( )
( )mssm
ms2msm
06/05
14/13
306/0502/01
306/05206/05210/09
2
222L
06/05L
06/05
206/05210/09
2
222
2110/09L
10/09L
10/09
110/0914/13
LL2La2aL32
LL4L3La2aL3r
EI6
LM
EI3
LM
EI3
LM
EI6
LM
LEI6
aLaP
EI6
LM
EI3
LM
LEI6
L2aL3aaP
EI3
LM
EI6
LM
32
21
××+××+×××
××−×−××+××−=
θ
θ=
×
×−θ
×
×=
×
×−
×
×−
××
−××−θ=θ
×
×+
×
×+
××
×+××−××=
×
×−θ=θ
×
×θ
ϕ
A.1.3 Ratio between rotation supports 13/14 and 05/06 for load between 05/06 and 01/02
The next formulae hold for the rotations of the supports.
( )
( )
sm
s
06/05
14/13
306/05
3
223
02/01
306/05
3
233
2206/05210/09L
06/05L
06/05
206/05210/09110/09L10/09
L10/09
110/0914/13
L4L3
Lr
EI6
LM
LEI6
aLaP
EI3
LM
LEI6
L2aL3aaP
EI3
LM
EI6
LM
EI6
LM
EI3
LM
EI3
LM
EI6
LM
32
21
×+×−=
θ
θ=
×
×−
××
−××−θ
×
×+
××
×+××−××=
×
×−
×
×−θ=θ
×
×+
×
×=
×
×−θ=θ
×
×θ
ϕ
FINAL VERSION
Page A-3
A.2 Eccentricities of supports to the neutral axis
Eurocode 1993-1-5 has been used to determine the effective width of the orthotropic deck in
order to calculate the location of the neutral axis. In the next calculations, the lateral deck plate
slope has been neglected.
A.2.1 Eccentricity end supports to the neutral axis
Figure 3.1 in the Eurocode is applicable since the spans do not differ more than 50%.
m475.795.9385.0L85.0L sE =×=×=
The area of a standard single deck plate stiffener (2/325/6) is approximately 4560mm2; the area
of the stiffener between girder web and deck plate (2/275/10) approximately 6770mm2; the deck
plate thickness is 10mm.
STRUCTURAL PART b0 [mm] Asl [mm2] α0 κ β1 β0 beff [mm]
Internal flange 6300 48985 1.333 0.106 0.933 0.734 4625
Flange outstand 900 7945 1.372 0.016 1 1 900
MIDAS Civil 2010 finite element software is able to calculate the section properties of stiffened
plates. This tool has been used to calculate the section properties of the deck (deck plate
including trapezoidal stiffeners). The calculation results have been used to draw the next picture.
The relative thickness should be interpreted as the thickness of a plane plate (without stiffeners)
with the same cross-sectional area as the stiffened plate.
Figure A-1: cross section of the inner girder at the end supports
FINAL VERSION
Page A-4
With these data, the location of the neutral axis can be calculated.
mm229211483440e
mm1148409003115106.175525
3420409005.1842311510856.175525
A
S*y
end
*y
=−=
=×+×+×
××+××+××==
A.2.2 Eccentricity intermediate supports to neutral axis
The Eurocode figure 3.1 is still applicable.
( ) ( ) m875.631625.9325.0LL25.0L msE =+×=+×=
The stiffeners are the same as for the end supports; the deck plate thickness is 12mm.
STRUCTURAL PART b0 [mm] Asl [mm2] α0 κ β2 beff [mm]
Internal flange 6300 48985 1.284 0.127 0.566 3567
Flange outstand 900 7945 1.317 0.019 1 900
Figure A-2: cross section inner girder at intermediate supports
FINAL VERSION
Page A-5
The bottom flange has been composed of 4 steel plates of different dimensions. To be accurate,
all of them will be calculated separately. However, in order to reduce the size of the equations,
first a comparable cross section have been calculated which is incorporated in the next equation.
mm275330095762e
mm30091009605375166.194467
57131009605.2974537516786.194467
A
S*y
ermediateint
*y
=−=
=×+×+×
××+××+××==
A.2.3 Ratio between eccentricities
The eccentricities have been calculated for the inner girder. The ratio can now be calculated as
well.
20.12292
2753
e
er
end
ermediateinte ===
A.3 Examination of the ratios of rotation and eccentricity
As long as the ratio of rotations is larger than the ratio of eccentricities, the longitudinal translation
of supports 13 and 14 is dominated by local bending of the neutral axis. Otherwise it is dominated
by the longitudinal movement of the neutral axis. The next figure shows the ratios as a function of
the load location on the girder.
0
2
4
6
8
10
12
14
0 50 100 150 200 250 300 350
rφ1 rφ2 rφ3 re
Figure A-3: ratio of rotation and eccentricity as a function of the load location
ANNEX B
Calculation of equivalent main girders’ bottom flanges
A Delft University of Technology thesis by V. Bos
Master of Science ‘Civil Engineering’
January 31, 2011
FINAL VERSION
Page B-1
ANNEX B Calculation of equivalent main girders’ bottom flanges
The main girders’ bottom flanges are mainly composed of several steel plates with different sizes.
In the model, these plates are transformed in a beam element.
∑
∑
=
=
×
=n
1ii
n
1iii
rep
t
tb
b
NUMBER OF PLATES PLATE DIMENSIONS Σb×t Σt brep
1 900×40 36000 40 900
900×40 2
940×20 54800 60 913
900×40
940×20 3
980×20
74400 80 930
900×40
940×20
980×20
INN
ER
GIR
DE
R
4
1020×20
94800 100 948
1 1200×40 48000 40 1200
1200×40 2
1240×20 72800 60 1213
1200×40
1240×20 3
1280×20
98400 80 1230
1200×40
1240×20
1280×20
OU
TE
R G
IRD
ER
4
1320×20
124800 100 1248
ANNEX C
Modal superposition method in case of traffic loads
A Delft University of Technology thesis by V. Bos
Master of Science ‘Civil Engineering’
January 31, 2011
FINAL VERSION
Page C-1
ANNEX C Modal superposition method in case of traffic loads
The applicability of the modal superposition method in case of traffic loads is discussed in this
annex. It will be shown that this method must be applied carefully.
The next figure shows schematically a 2-span beam that have been modelled in MIDAS (beam
elements). Each nodal load is described by the TFF (Time Forcing Function) in the right part of
the figure. The TFF describes the development of the load in local time (the local time starts when
the global arrival time is reached). In the left part, the arrival times (global time) are shown. It
represents a moving load from left to right with a period of 2 seconds. Note that the loads as
shown below are 0 at the time they arrive at the node and maximum 0.1s later (see TFF). At
every moment the total load on the structure is equal to 1.
The maximum vertical static deflection of this load is 2.59 millimetres. The next figure shows the
vertical deflection of node 6 (mid-span left), obtained from MIDAS (maximum corresponds to
static value). Only 5 mode shapes have been considered in the modal superposition method.
What does this graph show? The deflection line corresponds to a static influence line that may be
expected in this case. However, no deflection occurs in the first 0.1 second. This is explicable
since the load on the support does not cause deformations. At t=0.1s, the load at node 2 starts
increasing and the deflection starts increasing as well. At t=1.1s, the load is above support 2 and
FINAL VERSION
Page C-2
thus the deflection is 0. So far, no problems arise and thus the method may be regarded as
applicable. Yet, the influence line for vertical reactions will be discussed.
For some reason, MIDAS does not provide (vertical) reactions in a time history analysis.
Therefore, the vertical shear force in the adjacent element is examined (which should be the
same as the vertical reaction of the end supports). This is where problems arise. The next figure
shows the ‘vertical reaction’ of support 1 in time.
At t=0.1s, the load is exactly above support 1 and hence the vertical reaction must be 1.
However, in this case the shear force in the adjacent element is obviously 0. The figure shows
that the shear force starts increasing when the load at node 2 starts increasing. The figure is
therefore correct, however, it does not show the vertical reaction in time correctly. A trick must be
done to overcome this problem.
Between the horizontal beam and each support, a small vertical beam element is added so that
the vertical loads above the supports cause an axial force in that element which can be provided
by MIDAS in a graph.
It should be noted that this situation is mechanically slightly different and therefore must be done
carefully.
FINAL VERSION
Page C-3
The figure below shows the development of the vertical reaction force of support 1 (axial force in
the vertical element). Still, the magnitude starts increasing from t=0.1s in stead of t=0.0s.
An even better example of this problem is the vertical reaction of support 2 which is shown below.
The vertical reaction approaches 1 when the load is near the support. When it enters the support,
however, it goes back to almost 0 which is completely nonsense.
This illustrates the main problem of using the modal superposition method in this kind of
situations. The reason that the result is incorrect, is the basic assumption of the method. The
point is that the response is calculated using the vibration mode shapes that correspond to the
deformation that is induced by the load. In case of a load above the element, a mode shape
related to axial deformation of the vertical element is needed to calculate the response and thus
the axial force. However, these mode shapes are generally related to frequencies that are much
higher than the ordinary bending frequencies. Since the number of considered frequencies must
FINAL VERSION
Page C-4
be specified in MIDAS, possibly the relevant frequency is out of the regarded frequency
spectrum. To overcome this problem, more frequencies are needed (so that the discussed
frequencies are regarded as well). The next figures show the same graphs for the analysis in
which all relevant frequencies are regarded (44 frequencies).
Yet, the maximum vertical reaction force is equal to the applied load which is correct. What can
be learned from this example? It is of crucial importance to consider all relevant frequencies in
order to obtain correct results. Probably only the first 5 or 10 frequencies are enough to obtain
reliable results for the vertical deflection in mid-span since that DOF is related to the lowest
frequencies. However, for other parameters it might be that much higher frequencies are needed.
The intermediate frequencies probably are completely irrelevant, though they do be calculated by
the program (those frequencies may represent local instabilities as plate buckling, for example).
Therefore it might be that, in case of large and complex structures, the relevant frequencies are
preceded by hundreds of needless frequencies. In that case it might be better to apply direct
integration instead of the modal superposition method.
ANNEX D
Modelling friction with a Friction Pendulum System Isolator
A Delft University of Technology thesis by V. Bos
Master of Science ‘Civil Engineering’
January 31, 2011
FINAL VERSION
Page D-1
ANNEX D Modelling friction with a Friction Pendulum System Isolator
This annex illustrates that the friction pendulum system isolator (a finite element feature) can
probably be applied to model friction in the supports. However, the examination of this feature’s
functioning is based on trial and error rather than engineering knowledge as the input parameters
are not straightforward and unique. Actually they can differ without affecting the result. In this
annex, the result of a simple beam loaded by 2 point loads is compared to an analytical solution.
Consider the next situation. A simply supported beam is loaded by 2 loads, 1 in mid-span and 1
above the free support (in order to increase the vertical reaction and consequently friction). The
bottom flange of the section will slide over the supporting surface due to the deformation caused
by the load in mid-span. In MIDAS, this support is modelled by a friction pendulum system
isolator in which 5% friction is taken into account.
The point load in mid-span develops sinusoidally in time as shown below. The load above the
support is constant in time and has a magnitude of 25000 Newton.
Time Forcing Function of the applied load in mid-span
0
5000
10000
15000
20000
25000
30000
0 5 10 15 20 25 30
Time [s]
Fo
rce [
N]
Fapplied
The restraining force, caused by the support’s eccentricity, is linearly related to this applied load
by the equation below (see chapter 2).
+×
×=
b
b
2
ee
i8
LPF
FINAL VERSION
Page D-2
The friction in the free support is equal to the vertical reaction force times the coefficient of
friction. The vertical reaction is equal to the load above the support plus half of the load in mid-
span.
With these parameters, the development of friction and consequently the axial force in the beam
and horizontal reaction at the fixed support can be examined. The next steps apply. Note that in
this graph compression is positive and tension negative.
1. The load in mid-span starts increasing. The restraining force is initially smaller than the
frictional resistance and thus no sliding occurs. The beam will bend though the support
remains at its position (stick). The support acts as a fixed support and the axial
(compression) force in the beam is equal to the restraining force. The shape of the graph
is therefore equal to the shape of the applied load (sinusoidally).
2. After a small period (in this case nearly 1 second), the restraining force becomes larger
than the frictional resistance and the free support starts sliding (slip). The axial force in
the beam in now equal to the friction force in the support (dynamic coefficient of friction).
The threshold value is about 5% of the load above the support (1250N) since that load is
dominating the vertical reaction at that moment. This will continue until the maximum
value of the applied load is reached (at 12.5 seconds).
3. Subsequently, the applied load starts decreasing (after 12.5 seconds) and consequently
the vertical deflection decreases. Although the restraining force and friction change
direction (friction outwards and restraining force inwards), there is still a compression
force in the beam and an inwards friction force (12.5-13.5s). At this moment, the support
does not move but the axial force in the beam gradually turns into tension (relaxation of
axial stresses in the beam). In this period the deformation of the beam is taken by ‘elastic
deformation of the support’. This phenomenon may be compared to the behaviour of an
elastomeric bearing which has been described in chapter 2. As long as this
FINAL VERSION
Page D-3
transformation occurs and the backwards restraining force is smaller than the friction
force, the support remains at its position.
4. After about 14 seconds, except that all forces act opposite, the same story applies as in
note 1. The restraining force for the inwards motion becomes larger than the friction force
in the sliding interface and the support starts sliding inwards. In this phase, the axial force
is again equal to the dynamic friction force.
This is a theoretical evaluation of the friction behaviour in the beam. This has been compared to a
model in MIDAS with a friction pendulum system isolator. The next graph shows both, the MIDAS
calculation result and the analytical solution. Since the parameters are ‘chosen’ by trial and error
and thus do not have any contribution to the validity, they are omitted.
Result of the MIDAS calculation with a friction pendulum system isolator
-2000
-1000
0
1000
2000
0 5 10 15 20 25 30
Time [s]
Ax
ial
forc
e [
N]
Faxial
γ=1
The graph shows a remarkable similarity with the analytical solution. Still, it does not match
perfectly (of course this must not necessarily be the case). Scaling the load by a factor 1000
should (linearly) analytically give a result which is 1000 times larger. This does not apply to the
generated solution by MIDAS. The next graph shows again the solutions (scaled to a load factor
1).
Result of the MIDAS calculation with a friction pendulum system isolator
-2000
-1000
0
1000
2000
0 5 10 15 20 25 30
Time [s]
Axia
l fo
rce
[N
]
Faxial
γ=1000
γ=1
FINAL VERSION
Page D-4
The blue line matches exactly with the analytical solution which is very remarkable since it should
not make any sense what loads are considered. Of course the friction pendulum is a nonlinear
element which may give different results, though this can not be clarified analytically.
It may be assumed that the friction pendulum system isolator is applicable to model friction
correctly. However, at this moment, a lack of knowledge about the input parameters prevents
wisely application of the feature in the bridge. More research into the application of this feature is
needed to perform a nonlinear analysis skilfully. At this moment, this has not be done and is
advised for further research.
ANNEX E
Weight specification of the bridge’s steel structure
A Delft University of Technology thesis by V. Bos
Master of Science ‘Civil Engineering’
January 31, 2011
FINAL VERSION
Page E-1
ANNEX E Weight specification of the bridge’s steel structure
In this annex, the weight of the steel structure is specified briefly (calculation has been omitted),
incorporating significantly contributing elements only. Sometimes rather conservative
assumptions have been made which may cause small deviations. The result, however, is a
sufficiently accurate lower bound estimation for verification of the dead load calculation by
MIDAS.
DESCRIPTION WEIGTH [kN]
Deck plate 10mm including trapezoidal stiffeners (relative thickness is 17.6mm) 6230
Deck plate 12mm including trapezoidal stiffeners (relative thickness is 19.6mm) 800
Web outer girder, varying height (3115mm-5375mm) and thickness (10mm-16mm) 1200
Web inner girder, varying height (3115mm-5375mm) and thickness (10mm-16mm) 1175
Bottom flange outer girder, varying width (900-1020mm) and thickness (40-100mm) 1985
Bottom flange inner girder, varying width (900-1320mm) and thickness (40-100mm) 1450
Cross beams, height 1225mm 2130
Vertical stiffeners in cross frames 250
TOTAL = 15220
The average mass per square metre bridge deck can be calculated (for all n structural elements).
23
deck
n
1ii
av mkg3034.141.34981.9
1015220
Ag
G
m ≈××
⋅=
×=∑=
As some structural elements (details) are omitted in the calculation, it is expected that this value
is somewhat larger.
ANNEX F
Influence lines for unit loads and single lorries (UILs and LILs)
A Delft University of Technology thesis by V. Bos
Master of Science ‘Civil Engineering’
January 31, 2011
FINAL VERSION
Page F-1
ANNEX F Influence lines for unit loads and single lorries (UILs and LILs)
This annex presents all UILs and LILs that have been generated. The graphs’ legend explains
what lorry is considered. Note that the vertical scale is different for each graph!
<< FOR ENVIRONMENTAL REASONS, ONLY GRAPHS FOR THE T11O3(L) LORRY ARE INSERTED >>
<< ALL 175 GRAPHS ARE SUPPLIED DIGITALLY >>
LO
RR
Y IN
FL
UE
NC
E L
INE
S F
OR
LO
NG
ITU
DIN
AL
TR
AN
SL
AT
ION
S (
ux),
vlo
rry =
90km
/h
-6.0
-3.0
0.0
3.0
6.0
02
46
81
01
21
41
6
Tim
e [
s]
Translation [mm]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 9
2
LO
RR
Y IN
FL
UE
NC
E L
INE
S F
OR
LA
TE
RA
L T
RA
NS
LA
TIO
NS
(u
y),
vlo
rry =
90km
/h
-4.5
-3.0
-1.5
0.0
1.5
02
46
81
01
21
41
6
Tim
e [
s]
Translation [mm]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 9
3
LO
RR
Y IN
FL
UE
NC
E L
INE
S F
OR
RO
TA
TIO
NS
AB
OU
T T
HE
LO
NG
ITU
DIN
AL
AX
IS (�
x),
vlo
rry =
90km
/h
-7.5
0E
-04
-3.7
5E
-04
0.0
0E
+0
0
3.7
5E
-04
7.5
0E
-04
02
46
81
01
21
41
6
Tim
e [
s]
Rotation [rad]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 9
4
LO
RR
Y IN
FL
UE
NC
E L
INE
S F
OR
RO
TA
TIO
NS
AB
OU
T T
HE
LA
TE
RA
L A
XIS
(�
y),
vlo
rry =
90km
/h
-9.0
0E
-04
-4.5
0E
-04
0.0
0E
+0
0
4.5
0E
-04
9.0
0E
-04
02
46
81
01
21
41
6
Tim
e [
s]
Rotation [rad]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 9
5
LO
RR
Y IN
FL
UE
NC
E L
INE
S F
OR
RO
TA
TIO
NS
AB
OU
T T
HE
VE
RT
ICA
L A
XIS
(�
z),
vlo
rry =
90km
/h
-3.0
0E
-04
-1.5
0E
-04
0.0
0E
+0
0
1.5
0E
-04
3.0
0E
-04
02
46
81
01
21
41
6
Tim
e [
s]
Rotation [rad]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 9
6
LO
RR
Y IN
FL
UE
NC
E L
INE
S F
OR
LO
NG
ITU
DIN
AL
(F
RIC
TIO
N A
ND
RE
AC
TIO
N)
FO
RC
ES
(F
x),
vlo
rry =
90km
/h
-25
0.0
-12
5.0
0.0
12
5.0
25
0.0
02
46
81
01
21
41
6
Tim
e [
s]
Force [kN]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 9
7
LO
RR
Y IN
FL
UE
NC
E L
INE
S F
OR
LA
TE
RA
L (
FR
ICT
ION
AN
D R
EA
CT
ION
) F
OR
CE
S (
Fy),
vlo
rry =
90km
/h
-12
0.0
-60
.0
0.0
60
.0
12
0.0
02
46
81
01
21
41
6
Tim
e [
s]
Force [kN]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 9
8
LO
RR
Y IN
FL
UE
NC
E L
INE
S F
OR
VE
RT
ICA
L R
EA
CT
ION
FO
RC
ES
OF
EN
D S
UP
PO
RT
S (
Fz),
vlo
rry =
90km
/h
0
37
5
75
0
11
25
15
00
02
46
81
01
21
41
6
Tim
e [
s]
Reaction force [kN]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 9
9
LO
RR
Y IN
FL
UE
NC
E L
INE
S F
OR
VE
RT
ICA
L R
EA
CT
ION
FO
RC
ES
OF
IN
TE
RM
ED
IAT
E S
UP
PO
RT
S (
Fz),
vlo
rry =
90km
/h
0
27
50
55
00
82
50
11
00
0
02
46
81
01
21
41
6
Tim
e [
s]
Reaction force [kN]
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
AN
NE
X F
- 1
00
SL
IDIN
G D
IST
AN
CE
PE
R S
ING
LE
LO
RR
Y F
OR
LO
NG
ITU
DIN
AL
TR
AN
SL
AT
ION
S (
ux),
vlo
rry =
90km
/h
0.0
5.3
10
.5
15
.8
21
.0
02
46
81
01
21
41
6
Tim
e [
s]
Sliding distance [mm]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 1
01
SL
IDIN
G D
IST
AN
CE
PE
R S
ING
LE
LO
RR
Y F
OR
LA
TE
RA
L T
RA
NS
LA
TIO
NS
(u
y),
vlo
rry =
90km
/h
0.0
4.5
9.0
13
.5
18
.0
02
46
81
01
21
41
6
Tim
e [
s]
Sliding distance [mm]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 1
02
AC
CU
MU
LA
TE
D R
OT
AT
ION
S P
ER
SIN
GL
E L
OR
RY
FO
R R
OT
AT
ION
S A
BO
UT
TH
E L
ON
GIT
UD
INA
L A
XIS
(�
x),
vlo
rry =
90km
/h
0.0
0E
+0
0
7.5
0E
-04
1.5
0E
-03
2.2
5E
-03
3.0
0E
-03
02
46
81
01
21
41
6
Tim
e [
s]
Accumulated rotation [rad]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 1
03
AC
CU
MU
LA
TE
D R
OT
AT
ION
S P
ER
SIN
GL
E L
OR
RY
FO
R R
OT
AT
ION
S A
BO
UT
TH
E L
AT
ER
AL
AX
IS (�
y),
vlo
rry =
90km
/h
0.0
0E
+0
0
9.0
0E
-04
1.8
0E
-03
2.7
0E
-03
3.6
0E
-03
02
46
81
01
21
41
6
Tim
e [
s]
Accumulated rotation [rad]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 1
04
AC
CU
MU
LA
TE
D R
OT
AT
ION
S P
ER
SIN
GL
E L
OR
RY
FO
R R
OT
AT
ION
S A
BO
UT
TH
E V
ER
TIC
AL
AX
IS (�
z),
vlo
rry =
90km
/h
0.0
0E
+0
0
4.5
0E
-04
9.0
0E
-04
1.3
5E
-03
1.8
0E
-03
02
46
81
01
21
41
6
Tim
e [
s]
Accumulated rotation [rad]
T1
1O
3(L
) 0
1T
11
O3
(L)
02
T1
1O
3(L
) 0
5T
11
O3
(L)
06
T1
1O
3(L
) 0
9T
11
O3
(L)
10
T1
1O
3(L
) 1
3T
11
O3
(L)
14
AN
NE
X F
- 1
05
ANNEX G
Results of the multiple lorry analysis
A Delft University of Technology thesis by V. Bos
Master of Science ‘Civil Engineering’
January 31, 2011
FINAL VERSION
Page G-1
ANNEX G Results of the multiple lorry analysis
This annex presents all LILs for multiple lorry considerations.
<< FOR ENVIRONMENTAL REASONS, ONLY GRAPHS FOR THE LONGITUDINAL TRANSLATION OF SUPPORT 09
ARE INSERTED >>
<< ALL 44 GRAPHS ARE SUPPLIED DIGITALLY >>
DIS
PL
AC
EM
EN
T D
UE
TO
2 T
11O
3(L
) L
OR
RIE
S F
OR
SE
VE
RA
L L
OR
RY
DIS
TA
NC
ES
: u
x;0
9
-10-505
10
05
10
15
20
25
30
35
Tim
e [
s]
Translation [mm]
DIS
TA
NC
E 5
0D
IST
AN
CE
10
0D
IST
AN
CE
15
0D
IST
AN
CE
20
0D
IST
AN
CE
25
0D
IST
AN
CE
30
0D
IST
AN
CE
35
0D
IST
AN
CE
40
0
AN
NE
X G
- 1
AC
CU
MU
LA
TE
D D
ISP
LA
CE
ME
NT
DU
E T
O 2
T11O
3(L
) L
OR
RIE
S F
OR
SE
VE
RA
L L
OR
RY
DIS
TA
NC
ES
: u
x;0
9
0.0
12
.5
25
.0
37
.5
50
.0
05
10
15
20
25
30
35
Tim
e [
s]
Sliding distance [mm]
DIS
TA
NC
E 5
0D
IST
AN
CE
10
0D
IST
AN
CE
15
0D
IST
AN
CE
20
0D
IST
AN
CE
25
0D
IST
AN
CE
30
0D
IST
AN
CE
35
0D
IST
AN
CE
40
0
AN
NE
X G
- 2
DIS
PL
AC
EM
EN
T D
UE
TO
3 T
11O
3(L
) L
OR
RIE
S F
OR
SE
VE
RA
L L
OR
RY
DIS
TA
NC
ES
: u
x;0
9
-10-505
10
05
10
15
20
25
30
35
Tim
e [
s]
Translation [mm]
DIS
TA
NC
E 5
0D
IST
AN
CE
75
DIS
TA
NC
E 1
00
DIS
TA
NC
E 1
25
DIS
TA
NC
E 1
50
AN
NE
X G
- 3
AC
CU
MU
LA
TE
D D
ISP
LA
CE
ME
NT
DU
E T
O 3
T11O
3(L
) L
OR
RIE
S F
OR
SE
VE
RA
L L
OR
RY
DIS
TA
NC
ES
: u
x;0
9
0
13
25
38
50
05
10
15
20
25
30
35
Tim
e [
s]
Sliding distance [mm]
DIS
TA
NC
E 5
0D
IST
AN
CE
75
DIS
TA
NC
E 1
00
DIS
TA
NC
E 1
25
DIS
TA
NC
E 1
50
AN
NE
X G
- 4
ANNEX H
Vibration mode shapes of the 6 lowest natural frequencies
A Delft University of Technology thesis by V. Bos
Master of Science ‘Civil Engineering’
January 31, 2011
FINAL VERSION
Page H-1
ANNEX H Vibration mode shapes of the 6 lowest natural frequencies
This annex presents the first 6 vibration mode shapes of the bridge.
Figure H-1: mode 1 (ωn=0.52Hz)
Figure H-2: mode 2 (ωn=0.54Hz)
FINAL VERSION
Page H-2
Figure H-3: mode 3 (ωn=0.97Hz)
Figure H-4: mode 4 (ωn=1.02Hz)
FINAL VERSION
Page H-3
Figure H-5: mode 5 (ωn=1.06Hz)
Figure H-6: mode 6 (ωn=1.41Hz)