UIC CODE 2nd edition, June 2009 Translation
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Design requirements for rail-bridges based on interaction phenomena between train, track and bridge Exigences dans la conception des ponts-rails liées aux phénomènes dynamiques d’interaction véhiculevoie-pont Anforderungen für die Planung der Eisenbahnbrücken in Bezug auf die dynamischen Wechselwirkungen Fahrzeug - Gleis - Brücke
Leaflet to be classified in Volumes: VII - Way and Works
Application: With effect from 1st June 2009 All members of the International Union of Railways
Record of updates 1st edition, July 1976
First issue, titled: "Bridges for high and very high speeds"
2nd edition, June 2009
Overhaul of leaflet to adapt to European norms
The person responsible for this leaflet is named in the UIC Code
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Contents Summary ..............................................................................................................................1 1-
Introduction ................................................................................................................. 2 1.1 - Role of rail-bridges................................................................................................ 2 1.2 - Purpose of this leaflet ........................................................................................... 2 1.3 - Train-track-bridge interaction................................................................................ 2 1.4 - European Regulations .......................................................................................... 2
2-
Definitions.................................................................................................................... 3 2.1 - List of symbols ...................................................................................................... 3 2.2 - Bridge deformations and displacements............................................................... 5
3-
Requirements for train traffic safety ......................................................................... 7 3.1 - Phenomena .......................................................................................................... 7 3.2 - Criteria .................................................................................................................. 7
4-
Requirements for structural strength ....................................................................... 9 4.1 - Phenomena .......................................................................................................... 9 4.2 - Criteria .................................................................................................................. 9
5-
Requirements for passenger comfort ..................................................................... 11 5.1 - Physical phenomena .......................................................................................... 11 5.2 - Criteria to verify................................................................................................... 12
6-
Regulatory provisions: summary ............................................................................ 14 6.1 - Static verifications............................................................................................... 14 6.2 - Additional dynamic verifications.......................................................................... 15
Appendix A - Verification procedures for dynamic calculation .................................... 16 A.1 - General ............................................................................................................... 16 A.2 - Conditions dictating dynamic calculations .......................................................... 17 A.3 - Fundamental hypotheses for dynamic calculation relating to the bridge ............ 19
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A.4 - Fundamental hypotheses relating to vehicles (excitation) .................................. 24 A.5 - Fundamental hypotheses relating to the track.................................................... 31 A.6 - Calculations ........................................................................................................ 31 Appendix B - Criteria to be satisfied in the case where a dynamic analysis is not required............................................................................................. 38 List of abbreviations ..........................................................................................................42 Bibliography .......................................................................................................................43
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Summary The procedures for verifying the strength of railway bridges are covered by detailed and comprehensive rules of calculation already in existence. In contrast, serviceability limit states, notably deformation ELS, are described only in network calculation rules or in UIC leaflets. In essence, bridges are deformable structures. These deformations must be controlled all the more accurately as trains travel at high, and very high speeds. The purpose of this leaflet is to specify the design requirements for rail-bridges as regards train/track/bridge interaction phenomena and in particular speed, thereby taking into account bridge resonance phenomena. It outlines the corresponding draft criteria and provides information on the phenomena to be controlled as well as the appropriate procedures for verifying the structures. This leaflet should be used in conjunction with UIC Leaflet 776-1.
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1 - Introduction 1.1 -
Role of rail-bridges
Rail bridges are designed to guarantee continuity of the rail platform so as to ensure the movement of traffic in the same conditions of safety and comfort as on normal tracks and at any traffic speed up to the crossing speed limit defined for this bridge and for all types of traffic scheduled to cross the structure.
1.2 -
Purpose of this leaflet
The procedures for verifying the strength of railway bridges are covered by detailed and comprehensive rules of calculation already in existence. In contrast, serviceability limit states, notably deformation ELS, are described only in network calculation rules or in UIC leaflets. In essence, bridges are deformable structures. These deformations must be controlled all the more accurately as trains travel at high, and very high speeds. The purpose of this leaflet is to specify the design requirements for rail-bridges as regards train/track/bridge interaction phenomena and in particular speed, thereby taking into account bridge resonance phenomena. It outlines the corresponding draft criteria and provides information on the phenomena to be controlled as well as the appropriate procedures for verifying the structures. This leaflet should be used in conjunction with UIC Leaflet 776-1 (see Bibliography - page 43).
1.3 -
Train-track-bridge interaction
In order to properly assess these phenomena, it is best to examine the effects of both primary and secondary suspensions of the vehicles as well as the associated masses, the behavioural effects of the track and the deformability of the bridge deck and its supports. This leaflet also contains more simple alternative methods giving acceptable results using common calculations. Aside from its vertical component which is the most critical and which constitutes the greater part of this leaflet, train-track-bridge interaction also has a lateral component that has a bearing on lateral vehicle behaviour through the effects of the suspension, while also exercising an influence, albeit to a lesser degree, on the track and on the bridge.
1.4 -
European Regulations
These phenomena have been studied in far greater detail as part of the preparatory work into "Eurocodes" European regulations which now make bridge dimensioning possible by looking at the effects of train-track-bridge interaction, irrespective of the proposed speed of traffic up to 350 km/h and irrespective of the type of trains to be operated.
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2 - Definitions 2.1 -
List of symbols
E
=
Young's modulus of the material [kN/mm2]
Ec
=
Static modulus [kN/mm2]
Ecq
=
Dynamic modulus [kN/mm2]
Ecm
=
Secant modulus of elasticity
G
=
Shear modulus [kN/mm2]
I
=
Moment of inertia of the deck cross section
Ic
=
Intermediate cracking or 'partially cracked' state of inertia
IG
=
Gross moment of inertia of the uncracked transformed section
ICR
=
Moment of inertia of the fully cracked transformed section
Lc
=
Characteristic distance (e.g. span length or vehicle length)
L
=
Length of the deck
Mcr
=
Serviceability limit state cracking moment
MA
=
Maximum moment due to service loads at serviceability limit state
P
=
Maximum axle load of the load train (articulated train)
P’
=
Maximum axle load of the load train (conventional train)
Vcrit
=
Critical speed in relation to the resonance phenomenon
Vlim
=
Speed limit giving the upper limit where no dynamic calculations are necessary
V
=
Actual train speed (in general) [km/h]
Vpro
=
Speed of project
Vligne
=
Maximum line speed
Φ2
=
Dynamic increment coefficient for rail bridges (tracks with superior maintenance)
Φ3
=
Dynamic increment coefficient for rail bridges (tracks with standard maintenance)
Φ
=
Dynamic increment coefficient
a
=
Acceleration of the deck
amax
=
Maximum acceleration of the deck
b
=
Length of longitudinal distribution of a load across a sleeper and ballast
bv
=
Vertical acceleration in the vehicle
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d
=
Axle spacing of the bogies of the load train
D
=
Bogie spacing of the load train
fck
=
Characteristic compressive strength of the concrete [kN/mm2]
m
=
Mass of deck per unit length
nj
=
Natural bending frequency of row j of the unloaded deck [Hz]
n0
=
First natural bending frequency of the unloaded deck [Hz]
nT
=
Natural torsion frequency
t
=
Distortion of the deck
δ0
=
Deflection calculated at mid-span of the deck due to permanent loads (own weight + superstructure) applied in the direction of the deflection
α
=
Classification coefficient
δdyn
=
Deflection at mid-span under dynamic operating loads
δstat
=
Deflection at mid-span under static operating loads
δH
=
End displacement of the supports under operating loads
ϕ
=
Dynamic increment component for real trains
ϕ’
=
Dynamic increment for the real train and for a track without irregularities
ϕ"
=
Dynamic increment for the real train taking into account track irregularities
θstat
=
Rotation of the end of the deck under the influence of static operating loads
θdyn
=
Rotation of the end of the deck under the influence of dynamic operating loads
ζ
=
Damping coefficient or % of critical damping
ν
=
Poisson's coefficient
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2.2 -
Bridge deformations and displacements
Bridge deformations and displacements occur under the effect of external action applied through the spanned rail tracks, the deck supports or even directly onto the deck. These deformations and displacements are described below.
2.2.1 -
Static deformations
The vertical operating loads applied to the bridge cause the deck to bend, resulting in a vertical displacement of every point on the surface of the deck. In general, maximum displacement occurs at the point in the middle of the deck, or at mid-span. This displacement is known as the deflection of the deck. When the loads are static, the deflection reading δstat is called the static deflection. The vertical deflection of the deck considered for each span (isostatic or continuous bridge or succession of decks) is important in determining the final vertical radii of the track. The deflection of the deck described above causes rotation of the ends of the deck. Fig. 1 shows the rotation of each deck along a transversal axis or the total relative rotation between the adjacent ends of the deck. Static operating loads are used to define a rotation of θstat.
θ2 θ1
θ3
Fig. 1 - Definition of angular rotation of the ends of the decks The deck demonstrates transversal horizontal static deflection in response to certain actions. This is important in determining the final horizontal radii of the track. Because of the horizontal deflection of the deck (or the succession of decks) it is possible to observe a horizontal rotation of the decks around a vertical axis at their ends. This has a bearing on the horizontal geometry of the track. Whenever the deck supports a non-centred track or several tracks, of which one is loaded, it undergoes torsion as a result of the operating loads. Distortion of the deck is measured along the axis of each track in proximity to a bridge or on the bridge. Distortion tstat under static operating loads is measured on a track 1 435 mm wide and over a distance of 3 m (cf. Fig. 2). s 3m
Fig. 2 - Definition of deck distortion
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As shown in Fig. 3 - page 6, deflection of the deck under operating loads causes the end of the deck behind the support structures to lift. There is also a longitudinal displacement of the ends of the upper surface of the deck as a result of rotation of the end of the deck. The deformability of the bridge support structures causes longitudinal horizontal displacements of the bridge. Displacement covers the entire bridge in case of a single deck but it is relative in case of a series of decks.
δ H2
δ H1 α neutral axis fixed support structures
δH1: End displacement of the fixed support structures δH2: End displacement of the mobile support structures
mobile support structures
Fig. 3 - Definition of end displacements of a deck
2.2.2 -
Dynamic deformations
All the deformations and displacements described earlier as taking place under static loads show different values under dynamic loads (in general, all the more higher if train crossing speeds are greater), whether they are vertical or horizontal deflections under operating loads, vertical and horizontal end rotations, or longitudinal end displacements as well as lifting of the ends of the deck. Under dynamic loads, these deformations are expressed as follows: δdyn, δHdyn, θdyn. Distortion also takes on a different value under the dynamic effect of operating loads. This is expressed as follows: dynamic distortion tdyn. In general, the value retained is the maximum value obtained for a given speed.
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3 - Requirements for train traffic safety 3.1 3.1.1 -
Phenomena Quality of the wheel-rail contact
Excessive deformation of the bridge can jeopardise train traffic safety by causing unacceptable changes in the vertical and horizontal geometry of the track, excessive rail stress and excessive vibrations in the bridge support structures. In the case of ballasted bridges, excessive vibrations could destabilise the ballast. Excessive deformation may also affect the loads imposed on the train/track/ bridge system, as well as create conditions that lead to passenger discomfort.
3.1.2 -
Track stability
Relative displacements of the track and of the bridge, caused by a possible combination of the effects of train braking/starting, deflection of the deck under operational loads, as well as thermal variations, lead to the track/bridge phenomenon that results in additional stresses to the bridge and the track. It is important to ensure track stability as this may be compromised by additional stresses in the rail during compression (risk of buckling of the track, especially at bridge ends) or traction (risk of rail breakage). It is also important to minimise the forces lifting the rail fastening systems (vertical displacement at deck ends), as well as horizontal displacements (under braking/starting) which could weaken the ballast and destabilise the track. It is also essential to limit angular discontinuty at expansion joints and at points and switches in order to reduce any risk of derailment.
3.2 3.2.1 -
Criteria Distortion
Distortion of the deck is calculated with the characteristic value of load model UIC 71 and with load diagrams SW/0 or SW/2 as necessary multiplied by Φ and α or the high-speed load diagram, including the effects of centrifugal force. Limit values of distortion as described before are described in Table 1. Table 1 : Limit values of deck distortion Speed domain V (km/h)
Maximum distortion t (mm/3m)
V ≤ 120
t ≤ 4,5
120 < V ≤ 200
t ≤ 3,0
V > 200
t ≤ 1,5
Total distortion caused by distortion of the track when the bridge is not loaded (for example in a transition curve), and distortion due to total deformation of the bridge, must not exceed 7,5 mm/3m.
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3.2.2 -
Horizontal and vertical displacements
If continuous tracks are used, longitudinal horizontal displacements under the vertical effects of operating loads must remain below 10 mm. If continuous tracks are used, longitudinal horizontal displacements under the effects of braking/ starting must remain below 5 mm. They should be limited to 30 mm if the track has continuous welded rails and is fitted with an expansion joint at the end of the bridge, or if the track is fitted with scarfed joints. Vertical displacements at the ends of the deck should remain below 3 mm if the track is ballasted and 1,5 mm if the track is laid directly.
3.2.3 -
Acceleration of the deck
The risk of excessive vibrations of the deck corresponds to its levels of acceleration and consequently of the spanned track, and this should be verified. Deck acceleration should be considered a serviceability limit state as far as operating safety is concerned. In cases where the bridges have ballasted tracks, intense accelerations of the deck create the risk of destabilising the ballast. For this reason, it is important to ensure that maximum acceleration of the deck remains below 0,35g for frequencies up to 30 Hz. When verifying the acceleration of a deck with dual tracks in both running directions, it is assumed that only one track is loaded. In the case of bridges with slab tracks, the acceleration limit value is set at 0,5g for frequencies below 30 Hz. Dynamic analysis using the modal superposition method should take on board at least 3 modes as well as frequency vibration modes up to 1.5 times the frequency of the first mode.
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4 - Requirements for structural strength 4.1 -
Phenomena
4.1.1 -
Strength
This involves checking the ability of a structure, an element or a structural component, or a transverse section of an element or structural component to withstand actions without mechanical deterioration, for example bending strength and tensile strength also under dynamic effects. The strength calculation value of the structure or its elements must be greater than the calculation value of the corresponding action effects.
4.1.2 -
Fatigue
Fatigue describes the progressive damage to structures subjected to fluctuating or repeated stress, caused by the development of cracks that may eventually lead to their destruction. Fatigue increases with the number and the weight of trains, as well as with their speed. Fatigue service life should be sufficient to avoid any risk of cracking during the expected service life of the structure (usually, a minimum of 100 years).
4.1.3 -
Durability
The structure must be designed in such a way that its deterioration, during the period of use of the construction, does not jeopardise its durability or performance within its environment and in relation to the projected level of maintenance. Adequate measures are specified in order to limit deterioration on the basis of certain factors (such as properties of the soil, of the materials, foreseen maintenance during the life cycle of the structure, etc…).
4.2 4.2.1 -
Criteria Dynamic increment coefficient
When a dynamic analysis of the structure needs to be carried out (see Appendix A - page 16), with the relevant load models or real trains, it is important to determine the following dynamic increment coefficient: ϕ'dyn = max [ γ dyn / γ stat ] – 1 where γ dyn represents the dynamic deflection of the deck under the high-speed load diagram or real trains and γ stat represents the static deflection of the deck.
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The bridge is assessed using the logic diagram shown in Fig. 4.
(1 + ϕ’ dyn + 0,5 ϕ’’) x (load model for HS or real train) < Φ . (LM71+SW/0) NO
YES
Load model for HS or real trains with ϕ’dyn is decisive for the project
Φ . (LM71+SW/0) is decisive for the project
Fig. 4 - Logic diagram determining the loads to be taken into account for calculating bridge strength
4.2.2 -
ELU constraints
The resistance criterion involves checking that the calculation constraint of the effect of the actions is lower than or equal to the corresponding resistance constraint and remains so within the framework of the verification of the resistance limit-status.
4.2.3 -
ELS constraints
The non-cracking and reversibility criteria, part of the ELS verifications, involve checking the material stresses to ensure that the materials do not present a risk of developing irreversible deformations. The limit values with regard to constraints are given in the Eurocodes. They also involve for stressed concrete structures checking the limitation of crack openings. Such verifications may require making minimal reinforcements in the concrete.
4.2.4 -
Fatigue damage
Fatigue damage is a quantitative notion defined by a value between 0 and 1, and used to assess the relative evolution of cracking. The value is 0 if there is no damage and 1 if propagation is such that it destroys the structural element. Damage is determined by taking into account the successive loading of the component, which must remain at a permissible level for the lifetime of the structure. Fatigue dimensioning must be done to allow for the most unfavourable fatigue load conditions.
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5 - Requirements for passenger comfort 5.1 5.1.1 -
Physical phenomena Train-track interaction
Track levelling and lining variations generate vehicle movement that can affect passenger comfort and train safety. Almost every vehicle is mounted onto bogies. The movements that have a bearing on the vehicles are due to track levelling and lining defects (or track irregularities), the natural hunting movements of the axles and, when crossing bridges, the deformation of the bridge which modifies the path of the bogies. The running gear and suspensions generate rail vehicle body movement which affects passenger comfort and stresses which influence the vehicle running safety. The vehicle integrates primary and secondary suspensions (springs and dampers) as well as sprung and unsprung masses (masses, rotating masses inertia) that have an impact on this phenomenon. In order to separate the movements of the bogie from those of the body, the greatest possible vertical and transversal flexibility is required for secondary suspension. The required natural suspension frequencies are about 0,7 Hz (at present, 1 Hz is usually obtained but this can vary between 1 and 2 Hz).
5.1.2 -
Passenger comfort in vehicles when crossing bridges
In order to establish a maximum value that effectively translates the accelerations within the vehicle, it is important to know how vibrations impact passenger well-being. A certain number of physiological criteria linked to frequency, intensity of acceleration, steering relative to the spinal column and time of exposure (duration of vibrations) make it possible to assess vibrations and their influence on individuals. The limit exposure time to reduced comfort represents the limit of comfort adopted. This paragraph characterises the flexibility of bridges with regard to comfort. With knowledge of the dynamic deflection under a real train at mid-span on a civil engineering structure, it is possible to give an approximation of the path of a bogie during its passage over the structure. Knowing the transfer function that makes it possible to move from the path of the bogie to that of the body, it is possible to calculate vehicle acceleration. The acceleration limits inside the vehicles depend on the desired level of comfort and make it possible to limit the deflection of the structure.
5.1.3 -
Physiological fatigue of passengers
The preceding notion of comfort must nevertheless be reviewed whenever structures are very long, making for lengthy crossing times.
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5.2 -
Criteria to verify
Vertical acceleration in the vehicles Passenger comfort depends on vertical acceleration bv in the vehicle during the journey. The levels of comfort and the limit values associated with vertical acceleration in the vehicle are outlined in table 2. Table 2 : Indicative levels of comfort Level of comfort
Vertical acceleration bv (m/s2)
Very good
1,0
Good
1,3
Acceptable
2,0
The criteria to verify to guarantee passenger comfort relate to the vertical deflection of the decks and are listed here: In order to limit vertical acceleration in the vehicles, certain values will be given later to illustrate the maximum permissible vertical deflection δ along the centre of the track of railway bridges in relation to: -
span L [M]
-
train speed V [km/h]
-
number of span sections and
-
bridge configuration (isostatic beam, continuous beam).
Another possibility involves determining vertical acceleration bv by dynamic analysis of the train/bridge interaction. Aside from other factors, the following behaviour elements are taken into consideration when calculating dynamic analysis: -
the dynamic interaction of the mass between the vehicles of a given train and the structure,
-
the damping characteristics and suspension stiffness of the vehicles,
-
an adequate number of vehicles to produce the maximum load effects in the longest span section,
-
an adequate number of span sections in a multi-span structure to generate resonance effects in the vehicle's suspension.
Vertical deflections δ are determined using load model 71 multiplied by coefficient Φ. In the case of bridges with double tracks or more, only one track is loaded.
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For exceptional structures such as continuous beams with a large variation in span lengths or spans with many different stiffness levels, it is important to do a specific dynamic calculation.
3 000 V=
2 500
L/δ
2 000 1 500 V=
1 000
V=
350
V= V = 300 280 V= 250 V= 220 V= 200 160
120
500 0
0
10
20
30
40
50
60 L [m]
70
80
90
100
110
120
Fig. 5 - Maximum permissible vertical deflection δ for rail bridges corresponding to a permissible vertical acceleration of bv = 1/ms2 in the coach NB :
The figure is available for a succession of isostatic spans with three or more decks
New lines generally satisfy the primary level of comfort ("very good" and bv = 1,0 m/s2 ). The limit values L/δ for this level of comfort are given in Fig.5. For the other levels of comfort and the related maximum permissible vertical accelerations b’v, the values L/δ given in Fig. 5 may be divided by b’v [m/s2]. The values L/δ given in Fig. 5 are indicated for a succession of isostatic beams with three spans or more. For a bridge with a single span or a succession of two isostatic beams or two continuous spans, the values L/δ given in the diagram should be multiplied by 0,7. For continuous beams with three spans or more, the values L/δ given in Fig. 5 should be multiplied by 0,9. The values L/δ given in Fig. 5 are valid for spans up to 120 m. A specific analysis should be done for longer span lengths.
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6 - Regulatory provisions: summary 6.1 -
Static verifications
This type of verification is done systematically under load model 71 incremented by the relevant dynamic coefficient. Table 3 : Static limit values Criterion verified
Description of verification
Comfort
Vertical deflections
cf. Fig. 5 - page 13
Track stability
Expandable lengths
Lt = expandable length
Track stability
-
continuous track (no AD) : Lt ≤ 60 m (metal)
-
track with AD
Lt ≤ 90 m (concrete/ mixed)
-
non-ballasted track
project specified special study
Track stability
Calculation under incremented LM 71 1 loaded track
longitudinal displacements under the effects of vertical track loads -
Track stability
Limit value
with CWR
10 mm
2 loaded tracks a
longitudinal displacements under braking/starting for track: -
with CWR
5,0 mm
-
with AD or for jointed track 30 mm
2 loaded tracks a
vertical displacements at end of decks with: -
ballasted track
3,0 mm
-
slab track
1,5 mm
14
2 loaded tracks a
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Table 3 : Static limit values Criterion verified
Description of verification
Wheel/rail contact
deck distortion
Limit value
-
V ≤ 120 km/h
distort ≤ 4,5 mm/3m
-
120 ≤ V ≤ 200 km/h
distort ≤ 3,0 mm/3m
-
V > 200 km/h
distort ≤ 1,5 mm/3m
Calculation under incremented LM 71
1 loaded track
rotations due to horizontal deflections -
V ≤ 120 km/h
θ ≤ ⋅ 0035rd
-
120 ≤ V ≤ 200 km/h
θ ≤ ⋅ 0020rd
-
V > 200 km/h
θ ≤ ⋅ 0015rd
2 loaded tracksa
a. bridge with two tracks or more
6.2 -
Additional dynamic verifications
This type of dynamic verification is always carried out under real trains or under a universal dynamic loaded train (HSLM) incremented by the corresponding dynamic coefficient. Table 4 : Dynamic limit values Criterion verified
Description of verification
Track stability and wheel/rail contact
Vertical accelerations
Limit value
-
ballasted track
0,35 g (3,43 m/s2)
-
slab track
0,5 g (4,91 m/s2)
Comfort and strength of the structure
Vertical deflections
L/600 or L/800
Track stability
Longitudinal displacements at deck ends
Verification done in point 6.1 - page 14
Wheel/rail contact
Distortion
Distortion
Lateral stiffness
V > 200 km/h
t dyn ≤ 1, 2 mm / 3m
Horizontal deflections
Verification done in point 6.1 - page 14
First natural frequency of lateral vibration
≤ 1,2 Hz
15
Loading (real trains or HSLM)
1 loaded track 1 loaded track
1 loaded track
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Appendices
Appendix A - Verification procedures for dynamic calculation A.1 - General Dynamic phenomena When a train crosses a bridge at a certain speed, the deck will deform as a result of excitation generated by the moving axle loads. At low speeds, structural deformation is similar to that corresponding to the equivalent static load case. At higher speeds, deformation of the deck exceeds the equivalent static values due to the bridge inertia forces and the effects of track defects and vehicle running defects. The increase in deformation is also due to the regular excitation generated by evenly spaced axle loads and by the succession of reduced inter-axles and inter-bogie spacing. Risk of resonance A risk of resonance exists when the excitation frequency (or a multiple of the excitation frequency) coincides with the natural frequency of the structure (or a multiple thereof). When this happens, structural deformation and acceleration show rapid increase (especially for low damping values of the structure) and may cause: -
loss of wheel/rail contact
-
destabilisation of the ballast
In such situations, train traffic safety on the bridge is compromised. This may occur at critical speeds, represented approximately by values obtained for isostatic bridges and using the following formulae: nj Lc v crit = ----------i
j = 1, 2, 3, …, i = 1, 2, 3, …, 1/2, 1/3, 1/4, …
Resonance phenomena are unlikely to occur in rail bridges if speeds remain under 200 km/h and if the different conditions outlined in the following paragraphs are met. Importance of dynamic calculation In view of the potential risk outlined earlier and the tendency for speeds to increase, calculations need to be done to determine the extent of deformations which, at resonance, may lead to a dynamic load that is greater than UIC load model 71 incremented by the dynamic coefficient Φ2. Furthermore, accelerations of the structure cannot be determined by static analysis, one reason for justifying dynamic analysis. Even though deck accelerations are low at low speeds, they can reach unacceptable values at higher speeds. In practice, the acceleration criterion will, in most cases, be the decisive factor.
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Appendices A.2 - Conditions dictating dynamic calculations A.2.1 -
Parameters
The dynamic behaviour of a bridge depends on: -
the traffic speed across the bridge,
-
the span L of the bridge and its structural configuration,
-
the mass of the structure,
-
the number of axles, their loads and distribution,
-
the natural frequencies of the entire structure,
-
the suspension characteristics of the vehicle,
-
the damping of the structure,
-
the regularly spaced supports of the deck slabs and of the construction,
-
the wheel defects (flats, out-of-roundness, etc.)
-
the vertical track defects,
-
the dynamic characteristics of the track.
A.2.2 -
Logic diagram
The logic diagram in Fig.1 - page 18 is used to determine whether dynamic analysis is necessary. This is valid for the isostatic structures which behave in identical fashion to a linear beam. Tables 8 - page 38 and 9 - page 39 are represented in Appendix B. The validity limits for these tables are indicated in the notes after the tables. Independently from the logic diagram in Fig. 1, a dynamic analysis is necessary if the frequent working speed of a regular train is equal to a speed of resonance of the structure.
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Appendices V L n0
= traffic speed in [km/h] = span in [m] = first natural bending frequency of the unloaded bridge in [Hz]
nT
= first natural torsion frequency of the unloaded bridge in [Hz]
V lim ⁄ n0 et ( V ⁄ no )
are defined in Appendix B - page 38.
lim
START
yes
V ≤ 200 Km/h no no
Continuous bridge
Simple structure
yes
no
yes yes
L ≥ 40 m no
no
no
yes
nT > 1,2 no
For the dynamic analysis use the natural modes for torsion and for bending natural modes for bending sufficient
yes
no within limits of figure A4
Use Tables 8 and 9
no
Vlim/n0 ≤ (V/n0)lim
yes
Dynamic analysis not required Verification with regard to the acceleration not required at resonance. Use Φ with static analysis in accordance
Dynamic analysis required Calculate bridge deck acceleration and ϕ’dyn etc. or modify the structure and verify
Fig. 1 - Logic diagram to determine wether a specific dynamic analysis is required
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Appendices A.3 - Fundamental hypotheses for dynamic calculation relating to the bridge A.3.1 -
Material characteristics
Young's modulus for structural steel is 210 kN/mm2 for both static and dynamic behaviour. The dynamic value Edyn of Young’s modulus must be used in dynamic calculations. It depends on the static secant modulus and the speed of concrete deformation. Young's modulus for compressed concrete increase with stress and strain. Stress levels impact Edyn less in traction than in compression. Table 1 gives the values of the secant modulus of elasticity for concrete aged 28 days. Fig. 2 and 3 - page 20 show the relationship between the static modulus and the dynamic modulus of elasticity in both cases. Table 1 : Ecm values for concrete of different strengths f ck [kN/mm2 E cm [kN/mm2
20
25
30
35
40
45
50
29
30,5
32
33,5
35
36
37
The value of Poisson's coefficient ν, for steel is 0,3, whereas for concrete, it is 0,2. The shear modulus for structural steel is taken equally at 80 kN/mm2 for both static and dynamic behaviour. The shear modulus G for concrete can be calculated from the equation: E dyn G dyn = --------------------2(1 + υ )
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Appendices fdyn fstat
4
Edyn,εu,dyn Estat εu,stat
4
Compression
3 2.5
2,5
f
2 1,8 1,6 1,5 1,4 1,3 1,2 1,1 1 0,9
3
50
fcm = 20
2
1
f
3 E
1,5
α
1
εμ 1
0.1
101
1
102
103
104
105
106
107
.
σ [N/mm2s]
3 .10-5 10-4
-5
10
108
-2
10-3
0,1
10
102
10
1
3
. 10 ε [S-1]
Fig. 2 - Influence of the stress/strain relationship on the E values for concrete in compression
4
fdyn fstat
Edyn,εu,dyn Estat εu,stat
4
Traction
3
fcm = 20
3
50
2,5
2,5
2 1,8 1,6 1,5 1,4 1,3 1,2 1,1 1 0,9
f
1
0.1
101
1
102
103
104
105
10-4
2
3
δ
106
.
εμ
1,5
Et
1
107
108
σ [N/mm2s]
3 .10-5 10-5
1
f
10-3
10-2
0,1
1
10
102
3
. 10 ε [S-1]
Fig. 3 - Influence of the stress/strain relationship on the E values for concrete in traction
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Appendices A.3.2 -
Damping coefficient
Structural damping is a key parameter in dynamic analysis. The magnitude of the vibrations depends heavily on structural damping, especially in proximity to resonance. Although it is unfortunately not possible to predict the exact value in the case of new bridges, for existing bridges the damping values can be easily deduced by calculating the logarithmic decrement from the free vibration measurements. Table 2 gives the lower limits of the percentage values of critical damping ζ [%] based on a certain number of past measurements. Table 2 : Percentage values of critical damping ζ [%] for different bridge types and span lengths L Type of bridge
Lower limit of the percentage of critical damping ζ [%] Span length L < 20 m
Span length L ≥ 20 m
Metal and mixed
ζ = 0,5 + 0,125 (20 - L)
ζ = 0,5
Encased steel girders and reinforced concrete
ζ = 1,5 + 0,07 (20 - L)
ζ = 1,5
Pre-stressed concrete
ζ = 1,0 + 0,07 (20 - L)
ζ = 1,0
A.3.3 -
Mass
Maximum dynamic effects occur at resonance peaks, where a multiple of the load frequency coincides with the natural frequency of the structure. Underrating the mass will lead to overestimation of the natural frequency of the structure and of the speed at which resonance occurs. At resonance, the maximum acceleration of a structure is inversely proportional to the distributed mass of the structure. Two special cases must be considered for the mass of the structure, including the ballast: 1. a lower limit of the mass of the deck to obtain maximum accelerations; 2. an upper limit of the mass of the structure to obtain the lowest speeds at which effects of resonance will occur. The mass of the ballast on the bridge is calculated for two specific cases: 1. minimum density of the clean ballast and minimum thickness; 2. maximum saturated density of the ballast with slack, taking into account possible future lifting of the track.
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Appendices A.3.4 -
Rigidity (cracked sections, coefficient of equivalence,...)
For the same reasons as mentionned in point A.3.3 - page 21 (first alinea), it is best to use only the lower value within the stiffness range. The stiffness and mass of a bridge deck vary throughout the lifetime of the structure and impacts its dynamic behaviour. The stiffness range mentioned earlier corresponds to the two extreme values, on the one hand for sections free of cracks and without any reduction in stiffness, and on the other hand cracked sections and any effect leading to a reduction in stiffness such as the effect of differential settlement, contraction and temperature. Bending and torsional stiffness should take account of the impact of tensile stiffening onto the behaviour of reinforced concrete subjected to bending and torsion. Surveys carried out show that the Branson method to determine the equivalent bending stiffness of reinforced concrete can be used. The average value of the effective inertia along the entire length of an evenly loaded element is obtained by: 4 4 ⎛ M cr⎞ ⎛ M cr⎞ I c = ⎜ ---------⎟ ⋅ I G + 1 – ⎜ ---------⎟ ⋅ I cr ⎝ MA ⎠ ⎝ MA ⎠
The inertia for specific sections found along the length of the element is calculated by using the following expression: 3 3 ⎛ M cr⎞ ⎛ M cr⎞ I c = ⎜ ---------⎟ ⋅ I G + 1 – ⎜ ---------⎟ ⋅ I cr ⎝ MA ⎠ ⎝ MA ⎠
The coefficient of equivalence is given in the Eurocodes.
A.3.5 -
Natural frequencies
Fig. 4 - page 23 shows the limits of domain N of the natural frequencies n0 in [Hz] as a function of the span length L in [m] on the deck.
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Appendices
150 100 80 60
n0 [Hz]
40
20 15
Natural frequency upper limit
10 8 6 4 Natural frequency lower limit 2 1,5 1,0
2
4
6
8 10
15 20 L (m)
40
60 80 100
n is the first natural frequency of an unloaded bridge 0 L is the span for an isostatic bridge or L φ for other types of bridge
Fig. 4 - Limits of natural frequencies n0 en [Hz] in relation to the span length L [in m] The upper limit of n0 (N) is expressed by: n 0 = 94 ,76 × L
– 0 ,748
The lower limit of n0 (N) is expressed by: for 4m ≤ L ≤ 20m
n 0 = 80 ⁄ L n 0 = 23 ,58 × L
– 0, 592
for 20m < L ≤ 100m
(Range N is defined within these limits). There is no need for dynamic calculation if the speed of the line Vline is less than or equal to 200 km/ h and if the first natural bending frequency is within the limits of domain N in Fig. 4. Otherwise, an additional verification must be done. If the natural frequency is above the upper limit of domain N in Fig. 4, dynamic analysis is necessary.
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Appendices If the deck cannot be considered as a beam or a slab or if the first natural frequency in torsion nT lies within the domain (0,8n0, 1,2n0) where n0 is the first natural bending frequency, then dynamic analysis is necessary. If the deck can be considered as a beam or a slab and if the first natural frequency in torsion lies outside the domain (0,8n0, 1,2n0), additional verification is needed if the ratio V/n0 does not comply with the limits laid down in Appendix B - page 38. The natural frequency is given by the following general formula, which makes a clear statement of the importance of an accurate assessment of the product Eidyn and of the deck mass per unit length. 2
EI λj fj = ------------- ⎛ ------⎞ 1 ⁄ 2 2⎝ μ ⎠ 2πL The natural frequency of an isostatic beam can be calculated or estimated using the following simple formula: 17,753 n 0 = -----------------δstat This equation, where δstat in (mm) is calculated with the short term modulus, only refers to isostatic beams.
A.4 - Fundamental hypotheses relating to vehicles (excitation) The tools most commonly used for dynamic calculations do not take account of interaction phenomena. Train-bridge interaction modelling is described in point A.6 - page 31. The effect of trainbridge interaction can be integrated to the conventional mobile load diagram in point A.6 by adding adequate damping to the bridge damping. The following formulae can be used to calculate additional damping as a function of the length of the span: 2
a1 L + a2 L Δζ = ---------------------------------------------------------2 3 1 + b1L + b2L + b3L Coefficients a1, a2, b1, b2, b3 are determined for the ICE 2 and the Eurostar and for L/f = 1 000, 1 500, 2 000. Only ζ = 0,005 was considered because the effect of additional damping is greater with low structural damping. For different damping values, the coefficients calculated for ζ = 0,005 can be used, seeing that ζ has minimal effect on Δζ. The coefficient values are given in Table 3 - page 25 for the ICE 2 and in Table 4 - page 25 for the Eurostar.
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Appendices
Table 3 : Coefficients for calculating Δζ under ICE 2 L/f
a1
a2
b1
b2
b3
(l/m)
(l/m2)
(l/m)
(l/m2)
(l/m2)
1 000
1,3254x10-2
-5,9x10-5
5,5226
-0,7095
2,64x10-2
1 500
3,6965x10-4
-1,2006x10-5
-0,15345
1,03806x10-2
-2,075x10-4
2 000
5,5653x10-4
2,31x10-6
3,3321x10-2
-8,87x10-3
3,88x10-4
Table 4 : Coefficients for calculating Δζ under Eurostar L/f
a1
a2
b1
b2
b3
(l/m)
(l/m2)
(l/m)
(l/m2)
(l/m2)
1 000
7,1513x10-3
-9,29x10-5
5,40433
-0,75612
2,860x10-2
1 500
3,08531x10-4
-1,0377x10-5
-6,13910x10-2
7,86x10-5
7,03x10-5
2 000
4,79510x10-4
7,391x10-6
0,3591085
-4,11551x10-2
1,2771x10-3
These formulae are valid only for 5 < L < 30 m and 1000 < L/f < 2000. For the L/f values lying between those in the tables, a linear interpolation can be done.
A.4.1 A.4.1.1 -
Train models Hypotheses relating to vehicles
Current and future high-speed trains can be classed into three major categories, as indicated below in Fig. 5, 6 and 7 - page 26 :
D O/ BA
Fig. 5 - Articulated train
D O/ BS
O/ BA
Fig. 6 - Conventional train
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Appendices
D O/ BA
D
D IC
O/BA
ec
Fig. 7 - Train with equally-spaced axle (e.g. Talgo) A.4.1.2 -
Interoperability
High-speed trains now run on international lines in different countries and their numbers will most probably increase in the future. It is therefore essential to establish minimum technical specifications for projects relating to bridges and rolling stock so as to allow high-speed trains to travel throughout the European network in safety. The Technical Specifications for Interoperability relating to rolling stock can be outlined as follows: In order to ensure that high-speed trains crossing bridges or viaducts do not generate effects (stresses, deformations) incompatible with their dimensioning - whether they are strength characteristics or operating criteria - these trains should be designed to comply with the criteria listed in the right-side column in Table 5 - page 26: Tableau 5 : Technical Specifications for Interoperability of rolling stock Trains with equally-spaced axle Type TALGO
10 m ≤ D ≤ 14 m
P ≤ 170 kN
7 m ≤ e c ≤ 10 m
8 ≤ D 1C ≤ 11m where
D 1C = coupling distance between power car and coach E c = coupling distance between 2 trainsets
Articulated trains
18 m ≤ D ≤ 27 m
Type EUROSTAR, TGV
2 ,5 m ≤ d BA ≤ 3 ,5 m
Conventional trains
18 m ≤ D ≤ 27 m et P < 170 kN or values translating the inequality below
Type ICE, ETR, VIRGIN
All types
P ≤ 170 kN
πd BA πd BS πd HSLMA 4P cos -------------- cos -------------- ≤ 2P HSLMA cos -----------------------D D D HSLMA L ≤ 400 m
Σ P ≤ 10 000 kN
Note: -
D, D1C, dBA, dBS and ec are defined for articulated, conventional and trains with equally-spaced axle in Fig. 5, 6 and 7 above
-
P is the axle load.
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Appendices When relating to infrastructure (bridges), the Technical Specifications for Interoperability are as follows: In order to ensure that they deliver dynamic behaviour with regard to current and future train traffic, bridges should be calculated using the high speed load model (HSLM) consisting of the HSLM-A (for the definition of train A, set of 10 reference trains A1 to A10 (see Fig. 8 - page 27 and Table 6 page 28) and HSLM-B (cf. Fig. 9 and 10 - page 29). In order to apply HSLM-A and B, refer to Table 7 - page 29. The verifications of the various parameters indicated in this leaflet must be done within a speed range of 0 km/h and 1,2 V km/h, V being the potential speed of the line. Methods can also be developed to designate the most aggressive of these trains within the speed range in question and for a given structure. This is essentially the case of isostatic structures, where the train to designate may be determined by the aggressivity method devised by the ERRI Committee D 214-2 (see Bibliography - page 43). The HSLM-A consists of 10 trains defined as follows:
D
4xP (1)
11
2xP (3)
3xP (2) d
9
NxD
d
D
2xP (3)
(3) d
3
(3)
2xP (3)
3xP (2)
4xP (1)
d D
3,525
Fig. 8 - Layout of the universal dynamic train A (1) Power car (identical leading and trailing power cars) (2) End coaches (identical leading and trailing coaches) (3) Intermediate coaches
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Appendices Table 6 : Definition of the 10 trains of the universal dynamic train A Number of intermediate coaches
Length of coach
Axle spacing in the bogie
Localised force
N
D [m]
d [m]
P [kN]
A1
18
18
2,0
170
A2
17
19
3,5
200
A3
16
20
2,0
180
A4
15
21
3,0
190
A5
14
22
2,0
170
A6
13
23
2,0
180
A7
13
24
2,0
190
A8
12
25
2,5
190
A9
11
26
2,0
210
A10
11
27
2,0
210
Universal train
The HSLM-B consists of a number N of localised forces of 170 kN with a regular spacing d [m] where N and d are defined in Fig. 9 and 10 - page 29.
N x 170 kN
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
Fig. 9 - Diagram of universal dynamic train B The values of d and N are determined using Fig. 10:
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Appendices 20
6 5,5
15
4,5 10
4
N
d [m]
5
3,5 5
3 2,5 6,5
5,8
L[m]
5,5
4,8
4,5
4,2
3,8
3,5
3,2
2,8
2,5
0 1,6
1
2
Fig. 10 - Universal dynamic train B Where L is the span of the bridge in [m]. The next table illustrates how HSLM-A and HSLM-B are applied and indicates the trains to be used for dynamic bridge calculations. Table 7 : Application of HSLM-A and HSLM-B Length of span
Structural configuration of bridge
L<7m
L ≥7m
Isostatic bridgea
HSLM-B
HSLM-Ab
Continuous structure
HSLM-A
HSLM-A
or
Trains A1 to A10
Trains A1 to A10
Complex structurec a. Valid for bridges whose behaviour is limited to that of a line beam (longitudinal direction) or a slab, on fixed supports with minimal bias effects. b. For isostatic beams with a 7 m span or more, a single model HSLM-A may be used for dynamic analysis, under the aggressivity method defined in ERRI report D214-2. Alternatively all 10 models HSLM-A-1 to HSLM-A-10 may be used. c. Model HSLM-B should also be used.
A.4.2 -
Load distribution
In the live load model, each axle is represented by a constant and concentrated load that moves across the bridge. When taking into account the elastic properties of the upper track structure, it is clear that the reactions under the rail are diffused. This means that the bridge underneath is not loaded by the concentrated loads but rather by the distributed loads in the direction of the track.
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Appendices A diagram (Fig. 16 - page 41) is appended and gives the reduction coefficient to be applied to the acceleration obtained under concentrated loads. This reflects the dynamic effects of axle loads distributed lengthwise over 2,5 and 3,0 m as determined by the lowest speed/frequency ratio to be taken into account under the dynamic effects of axle loads.
A.4.3 -
Dynamic signature
The dynamic signature of a train is obtained by breaking down the load diagram of a train in Fourier series and by extrapolating it to the natural modes. It represents the dynamic excitation features of the train and is independent of the characteristics of the structure. The signature depends on axle spacing and loads only. The following is the relevant formula, where So λ is the dynamic signature (λ is the wave length = v/n0).
d ,i 2 d ,i 2 ΣPi sin ⎛ 2π ------⎞ + ΣPi cos ⎛ 2π ------⎞ ⎝ ⎝ λ⎠ λ⎠
S0 ( λ ) =
Bridges whose dynamic behaviour is calculated using the load diagram specific to high-speed trains as defined in point A.4.1 - page 25 need not be calculated under the real load of current high-speed trains or new trains whose dynamic signature falls within the envelope of dynamic signatures of load diagrams specific to high-speed trains. For each train, it is possible to determine an excitation spectrum that takes account of the composition of excitation produced by the train at a given point and a given speed. The dynamic signature is a useful method of producing a quick comparison of the effects of different trains. For example, if the magnitude of the dynamic signature of a new train is lower than that of existing trains working a specific line, that very line may be used by the new train without the need for doing a dynamic verification of the structures on the line. The diagram below shows the dynamic signatures of certain well-known high-speed trains (TGV Eurostar - ICE2 - ETR - Virgin and Shinkansen). Train signatures
6 000
TGVA Eurostar Thalys2 ICE2 ETR-Y Shinkansen IC225_normal Virgin eurotrain
5 000 s0
(kN)
4 000 3 000 2 000 1 000 0
0
5
10
15 λ (m)
20
25
30
Fig. 11 - Dynamic signatures of several high-speed trains
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Appendices A.5 - Fundamental hypotheses relating to the track A.5.1 -
Track irregularities
It is important to take account of the effect of track irregularities on the dynamic behaviour of a bridge to be dimensioned for high speeds. The increase in dynamic loading as opposed to static loading depends on the speed of the train as it passes over track irregularities while crossing the bridge and is inversely proportional to the length of the span. The value obtained, without taking track irregularities into account, should be multiplied by 1 + ϕ"/2 in order to dimension the bridge of a well-maintained track for track irregularities, or multiplied by 1 + ϕ" for a track receiving standard maintenance. The previous coefficient takes also into account the irregularities concerning the vehicles.
A.5.2 -
Vertical stiffness of the track
Vertical stiffness of the track is comprised of the stiffness levels of different phases of materials: rail pads - base plates - sleepers and ballast - sometimes the anti-vibration mat under the ballast. These materials show variable stiffness and the resulting track stiffness may then be represented by an average value that is contingent on the composition of the track. In the case of normal ballasted track, this value may be taken to be 500 MN/m for rail pad stiffness, 538 MN/m for sleeper/ballast interface stiffness and finally, 1000 MN/m for the ballast/deck.
A.6 - Calculations A.6.1 -
Models
The bridge/track/train system must be modelled as accurately as possible to obtain the accelerations and deformations of a bridge crossed by a train. Models of varying degrees of complexity are possible for the train as well as the track. Generally speaking, the live load diagram described and represented in point A.4.1 - page 25 an example for which is given in Fig. 12 should be used with loads represented by a series of constant local forces. The other two models possible are described below in point A.6.1.2 - page 33.
3
1 Axle 2 number 3,52 3
5,02
11
14,00
4
5
25 27 26 28
23 24
7 8
6
3,275 3 3
15,70
3 8 x 15,70 3 + 7 x 3,00 237,59
15,70
6,275
18,70
8 x 18,70
18,70
3,275 3 3
29 30 11
6,275 14,00
3,52 3
5,02
Normal load: all axles = 17 t P = 17 x 30 = 510 t
Fig. 12 - Diagram of dynamic train-bridge model
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Appendices In the load diagram used for dynamic calculations, the train is represented by axle forces and spacing, and crosses the bridge at constant speed. This model is adequate for dynamic calculations. The sections below make a case for improving such calculations and the model described above does not take account of the dynamic behaviour of the train and the track. Similarly, the distribution of axle forces lengthwise through the rails is not taken into consideration. Studies carried out by the ERRI Committee D214 (see Bibliography - page 43), clearly show that the live load diagram using a series of constant local forces produces the highest deformations and accelerations at resonance.
A.6.1.1 -
Bridge model
Modelling methods that use beam elements are the most appropriate to quantify the behaviour of bridges and structures essentially composed of bars. In order to ensure that the equivalent standard beam gives a reliable representation of the overall dynamic behaviour of the structure, the modelling method should integrate the correct mechanical and mass characteristics of the structure, including the real support features. The problem resides in translating the dynamic physical properties to a digital model. The findings obtained from the initial calculations: -
pulse, frequency and period of natural modes,
-
deformed modes.
give a fair idea of the behaviour of the structure under dynamic stresses. The structure must be modelled as accurately as possible and this could be done using two- or threedimensional elements. The modelling elements used for bridges may be slabs. The method makes it possible to examine longitudinal and transversal modes, using slabs that may be orthotropic or skew plates. An example of the three-dimensional bridge model is given below.
Fig. 13 - Example of three-dimensional bridge model
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Appendices A.6.1.2 -
Vehicle model
The train consists of a series of vehicles represented by their masses, moments of inertia and suspension characteristics. The vehicles making up the train are represented by the body, two bogies and four axles. The primary and secondary suspensions are represented by parallel spring-damper systems. Live load diagrams are the simplest form of load models and give less accurate results than the more complex models. In addition to the primary and secondary suspensions lengthwise along the bodies, the articulated vehicles are mounted on two viscous, non-linear dampers. A very stiff spring and a viscous damper constitute the suspension, positioned vertically between the two bodies. Figure 14 - page 33 gives a complete model for a set of articulated vehicles.
Mc Ic
z1 z8(rotation) 2xD1 Cs
Ks Mb Ib Kp
z2 z9(rotation)
2xD2
z3 z10(rotation)
Cp z4
Me
z5
z6
z7
Ke
Fig. 14 - Example of complete models for a conventional train extract A.6.1.3 -
Track model
As with the bridge, the track is represented by Timoshenko beam elements for the rails and takes account of the rail/sleeper fastening characteristics as well as the ballast (if one exists). A sleeper is generally represented by two beam elements, with two covering the rail and one used for the deck. Sleepers and ballast are modelled as concentrated masses. They are linked to the nodes of the rail and the bridge by a parallel spring and damper system. The track can be modelled to any length on both sides of the bridge. This latest model gives more accurate results especially for short bridges, where the stiffening effect of the bridge has to be taken into account. The effects of track distribution are not considered. Each vehicle is able to absorb the kinetic energy of the bridge and it is for this reason that, at resonance, the deflections and accelerations of the bridge obtained with this model are lower than those obtained with a live load diagram. The most complete model for analysing train/track/bridge interaction is shown in Fig 15 - page 34.
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Appendices
Fig. 15 - Diagram of the dynamic train-track-bridge model
A.6.2 -
Methods
This leaflet only provides information on the new methods used in dynamic calculations for old and new rail bridge decks crossed by trains running at speeds under 1,2 Vdes. or under 420 km/h. In all cases, calculations must be done for speeds up to 1,2 Vligne. ERRI report D 214/RP9, (see Bibliography - page 43) presents a number of calculation methods with differing levels of accuracy to analyse and check the criteria outlined in point A.3.2 - page 21. An approximate method and two simplified methods can be used to determine bridge deck deflection and acceleration (cf ERRI D 214/RP6 - see Bibliography page 43). Dimensioning diagrams for bending and torsion can also be used to determine the maximum acceleration amax, and the maximum deflection dmax of a structure (cf also ERRI D214/RP6). Various programs are available and details can be found in ERRI report D 214/RP7 (see Bibliography - page 43); they can be used to calculate the dynamic response under live train loads, of isostatic bridges, series of isostatic decks, continuous bridges using the beam theory, the dynamic response of plates and by taking into account the two longitudinal and transversal modes. They can also run calculations for orthotropic square plates and skew plates. With regard to beams, the effects of bending and shear are taken into account (Timoshenko or EulerBernouilli beam) as well as torsion (Saint-Venant or Vlasov). Two types of analyses can be carried out: with or without interaction with the train. The most problematic cases, for example special structures (bridges with long spans such as bowstring bridges), have to be solved using generic finite element programs. As with the finite element methods (FEM), the different programs can be used to determine the successive natural modes of the structure, then to calculate the response of the structure by modal superposition with the train speeds that correspond to the resonance situations.
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Appendices Various programs such as ANSYS, NASTRAN, ABAQUS, SAP, FASTRUDL and so on, can be used to obtain the modal responses of bridge decks. Modelling can be done with beam models using torsional characteristics if the bridge is not a skew bridge and the structure is not a special case (see above). However, spatial modelling is necessary in such cases. Dynamic analysis of a structure can be used to resolve a system of differential equations of lesser importance. Two fundamental approaches may be implemented: one method consists in solving the system of equation by direct integration, whereas the other defines the solution based on the natural modes of vibration of the structure. This is known as modal superposition. A concise description follows in the next two paragraphs. A.6.2.1 -
Modal analysis
Modal analysis is used to calculate the natural modes and frequencies of the model, as well as the resulting variables (participation factors, effective modal masses). For undamped, free vibrations, the equation of movement without a second element is reduced to: 2
[ K ] – ω [ M ] [ Φi ] = 0 where Φ represents the circular frequency vector (= pulse) and [Φ] is the modal crossing matrix consisting of natural orthonorm modal vectors [Φi] in relation to [M] or [K]. In principle, all the modes with natural frequencies lower than the cut-off frequency should be retained; in practice, the modes retained are often those making an important contribution to the response (criterion of the sum of effective modal masses of the modes retained and which should be slightly different from the total mass of the structure). When the natural vectors are calculated, the modal matrix is formed [Φ] after which the ωi can be deduced. A.6.2.2 -
Analysis by modal superposition
The fundamental equation of the dynamic approach represents a system of N simultaneous differential equations, where N is the number of degrees of freedom (ddl) of the structure. If threedimensional modelling is used, this number N is equal to six times the number of nodes less the number of ddl blocked at the support. When the number increases to a value that is very high for large models, the size of the problem needs to be reduced by transformation techniques. Solving the differential equations then becomes faster and is more accurate. The integration method used is now as follows: for each mode i, the resulting equation gives an evaluation introduced by the Duhamel integral or the Fourier transform. The sum of the solutions gives the full response. The integration approach for mobile loads is a slow process. Modal superposition is used to accurately quantify the respective contributions of each mode to the total dynamic response and to identify the risks of resonance and dynamic amplification of some types of stresses.
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Appendices A.6.2.3 -
Analysis by direct integration
When the analysis uses numerical methods to directly integrate the dynamic equation, the loads become the dynamic systems in the case of vehicles and their internal behaviour impacts the response from the structure. -
the two systems can be considered separate systems,
-
the vehicle can be considered a finite element.
This last method takes track profile defects into account and deduces the forces of interaction between the structure and the vehicle as well as the internal forces in the dynamic system that is built. In this method, the equation of the dynamic is solved, with or without prior transformation, by using the conventional algorithms for numerical resolution of second-degree differential equations. These numerical methods calculate the response to regularly spaced time intervals (in general). The selected time pitch determines the accuracy of the results and has a bearing on the length of computer calculations. Numerical integration methods are all based on the search for balanced solutions of the dynamic equation at regular time intervals. A.6.2.4 -
Filtering
Acceleration, primarily at mid-span on the deck, impacts the behaviour of the ballast and consequently the track as well as passenger comfort. Acceleration contains significant high-frequency components thus resulting in very high, and effective levels of acceleration. It is clear that the ballast and track act as a low-pass filter, with only the level of acceleration within a certain frequency bandwidth affecting the stability of the ballast. Evidently, it is important to have identical filtration for both measurements and calculation models. The ideal low-pass filter transmits all the signal components between 0 Hz and a cut-off frequency fc without attenuation or phase shift. A filter is characterised by its time-based pulse response and its frequency response. The latter is the Fournier transform of the former. The bandwidth is the frequency bandwidth in which the filter gain is between two set values. It indicates how the filtered signal spectrum will be deformed. The steeper the filter gradient, the more the upper frequencies of the cut-off frequency will be efficiently scattered. Many attempts have been made at calculation/measurement correlations. In most measurement cases, it has been demonstrated that a larger number of modes than the primary mode of the structure is excited. Filtering should have cut-off frequencies 30% greater than the frequency of the last mode of interest. Filtering at 30% gives acceleration levels that are significantly higher than filtering at 20%.
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Appendices A.6.3 -
Torsion and combined torsion and bending
Torsion does not have to be considered on decks with single tracks, but it does need to be examined in decks with two tracks or more. In theory, bending calculations transpose easily in case of torsion. Nevertheless, to use the formulae and diagrams properly, only one deformation mode should be considered. As it is, the primary mode is always the bending mode. Therefore, if more than one mode is to be examined in torsion-bending cases, a program has to be used. The natural torsional frequencies can be obtained from the following formula: i Fi = ------- GIω ----------2L ρJp where
L Iω G ρ Jp
= span of the slab section = torsional rigidity = shear modulus = density = mass rigidity in torsion
Calculation charts can be established by transposing the formulae and charts from pure bending to pure torsion. The question of combined bending and torsion cannot be covered by simplified methods. It is indeed possible to obtain the respective response of each of the two effects, but both responses cannot be added together. In fact, these simplified methods directly determine the maximum effects, without giving a temporal response. To give rules of addition, the respective moments of the two elementary maxima must be known (torsion, bending) and above all if there is any possibility to achieve simultaneousness. It is therefore important to use programs that give time-based answers. In this case, the elementary temporal responses are added and the maximum temporal answer is read. The DIA and CEDYPIA software programs operate in this manner.
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Appendices
Appendix B - Criteria to be satisfied in the case where a dynamic analysis is not required Appendix B is not valid for Load Model HSLM.
NB :
The two criteria mentioned in point 3.2.3 - page 8 (amax < 0,35 g ou 0,5 g depending on track installation and δdyn < Φ2 δUIC) are always respected, (in which case, there is no need for dynamic calculation), when the Vlim/n0 ratio is lower than the values in Tables 8 or 9 - page 39 (depending on the limit acceleration to be considered) as a function of the span interval in [m], the deck mass interval per linear metre in [t/m] and the damping considered. Table 8 : Maximum value (v/n0)lim for an isostatic beam or plate and a maximum permissible acceleration amax < 3,50 m/s2 Mass m
≥ 5,0 ≥ 7,0 ≥ 9,0 ≥ 10,0 ≥ 13,0 ≥ 15,0 ≥ 18,0 ≥ 20,0 ≥ 25,0 ≥ 30,0 ≥ 40,0 ≥ 50,0
103 kg/m
< 7,0
< 9,0 < 10,0 < 13,0
ζ
v/no
v/no
v/no
%
m
m
2
1,71
4
< 15,0
< 18,0
< 20,0
< 25,0
< 30,0
< 40,0
< 50,0
-
v/no
v/no
v/no
v/no
v/no
v/no
v/no
v/no
v/no
m
m
m
m
m
m
m
m
m
m
1,78
1,88
1,88
1,93
1,93
2,13
2,13
3,08
3,08
3,54
3,59
1,71
1,83
1,93
1,93
2,13
2,24
3,03
3,08
3,38
3,54
4,31
4,31
2
1,94
2,08
2,64
2,64
2,77
2,77
3,06
5,00
5,14
5,20
5,35
5,42
4
2,15
2,64
2,77
2,98
4,93
5,00
5,14
5,21
5,35
5,62
6,39
6,53
1
2,40
2,50
2,50
2,50
2,71
6,15
6,25
6,36
6,36
6,45
6,45
6,57
2
2,50
2,71
2,71
5,83
6,15
6,25
6,36
6,36
6,45
6,45
7,19
7,29
1
2,50
2,50
3,58
3,58
5,24
5,24
5,36
5,36
7,86
9,14
9,14
9,14
2
3,45
5,12
5,24
5,24
5,36
5,36
7,86
8,22
9,53
9,76
10,36
10,48
1
3,00
5,33
5,33
5,33
6,33
6,33
6,50
6,50
6,50
7,80
7,80
7,80
2
5,33
5,33
6,33
6,33
6,50
6,50
10,17
10,33
10,33
10,50
10,67
12,40
[17,5 ; 20,0)
1
3,50
6,33
6,33
6,33
6,50
6,50
7,17
7,17
10,67
12,80
12,80
12,80
[20,0 ; 25,0)
1
5,21
5,21
5,42
7,08
7,50
7,50
13,54
13,54
13,96
14,17
14,38
14,38
[25,0 ; 30,0)
1
6,25
6,46
6,46
10,21
10,21
10,21
10,63
10,63
12,75
12,75
12,75
12,75
[30,0 ; 40,0)
1
10,56
18,33
18,33
18,61
18,61
18,89
19,17
19,17
19,17
≥ 40,0
1
14,73
15,00
15,56
15,56
15,83
18,33
18,33
18,33
18,33
Span
L∈
ma
[5,00 ; 7,50)
[7,50 ; 10,0)
[10,0 ; 12,5)
[12,5 ; 15,0)
[15,0 ; 17,5)
a. L ∈
[a, b) means a ≤ L < b
Nota : -
Table 8 includes the safety coefficient of 1,2 on (v/n0)lim for acceleration criteria, deformation and strength and a safety coefficient of 1,0 on (v/n0)lim for fatigue;
-
Table 8 takes into account track irregularities with (1+ ϕ′′/ 2).
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Appendices Table 9 : Maximum value of (v/n0)lim for an isostatic beam or a plate on simple supports and a maximum permissible acceleration amax < 5,0 m/s2 Mass
≥ 5,0 ≥ 7,0 ≥ 9,0 ≥ 10,0 ≥ 13,0 ≥ 15,0 ≥ 18,0 ≥ 20,0 ≥ 25,0 ≥ 30,0 ≥ 40,0 ≥ 50,0 < 7,0
< 9,0 < 10,0 < 13,0
< 15,0
< 18,0
< 20,0
< 25,0
< 30,0
< 40,0
< 50,0
-
ζ
v/no
v/no
v/no
v/no
v/no
v/no
v/no
v/no
v/no
v/no
v/no
v/no
%
m
m
m
m
m
m
m
m
m
m
m
m
2
1,78
1,88
1,93
1,93
2,13
2,13
3,08
3,08
3,44
3,54
3,59
4,13
4
1,88
1,93
2,13
2,13
3,08
3,13
3,44
3,54
3,59
4,31
4,31
4,31
2
2,08
2,64
2,78
2,78
3,06
5,07
5,21
5,21
5,28
5,35
6,33
6,33
4
2,64
2,98
4,86
4,93
5,14
5,21
5,35
5,42
6,32
6,46
6,67
6,67
1
2,50
2,50
2,71
6,15
6,25
6,36
6,36
6,46
6,46
6,46
7,19
7,19
2
2,71
5,83
6,15
6,15
6,36
6,46
6,46
6,46
7,19
7,19
7,75
7,75
1
2,50
3,58
5,24
5,24
5,36
5,36
7,86
8,33
9,14
9,14
9,14
9,14
2
5,12
5,24
5,36
5,36
7,86
8,22
9,53
9,64
10,36
10,36
10,48
10,48
1
5,33
5,33
6,33
6,33
6,50
6,50
6,50
7,80
7,80
7,80
7,80
7,80
2
5,33
6,33
6,50
6,50
10,33
10,33
10,50
10,50
10,67
10,67
12,40
12,40
[17,5 ; 20,0)
1
6,33
6,33
6,50
6,50
7,17
10,67
10,67
12,80
12,80
12,80
12,80
12,80
[20,0 ; 25,0)
1
5,21
7,08
7,50
7,50
13,54
13,75
13,96
14,17
14,38
14,38
14,38
14,38
[25,0 ; 30,0)
1
6,46
10,20 10,42
10,42
10,63
10,63
12,75
12,75
12,75
12,75
12,75
12,75
[30,0 ; 40,0)
1
18,33
18,61
18,89
18,89
19,17
19,17
19,17
19,17
19,17
≥ 40,0
1
15,00
15,56
15,83
18,33
18,33
18,33
18,33
18,33
18,33
103 kg/m Span
L∈
ma
[5,00 ; 7,50)
[7,50 ; 10,0)
[10,0 ; 12,5)
[12,5 ; 15,0)
[15,0 ; 17,5)
a. L ∈
[a, b) means a ≤ L < b
NB: -
Table 9 includes a safety coefficient of 1,2 on (v/n0)lim for the acceleration criteria, for the deformation and strength and a safety coefficient of 1,0 on (v/n0)lim for the fatigue;
-
Table 9 take account of track irregularities with (1+ ϕ′′/ 2). with:
L
span of the bridge [m]
m
mass of the bridge [103 kg/m]
ζ
coefficient of critical damping [%]
ν
maximum nominal speed, generally equal to the maximum speed of the line at the considered point. For verifying the individual real trains, a reduced speed can be used (maximum permissible speed of the trains) [m/s],
n0
first natural frequency of the span [Hz]
Φ 2 and ϕ″
defined in point 4.2.1 - page 9
39
776-2 R
Appendices Tables 8 - page 38 and 9 - page 39 are valid for: -
simply supported bridges with insignificant skew that may be modelled as a line beam or slab on rigid supports. Tables 8 and 9 are not applicable to half through and truss bridges with shallow floors or other complex structures that may not be adequately represented by a line beam or slab,
-
bridges where the track and depth of the structure to the neutral axis from the top of the deck is sufficient to distribute point axle loads over a distance of at least 2,50 m,
-
typical trains,
-
structures designed for characteristic values of vertical loads or classified vertical loads, with α ≥ 1,
-
carefully maintained track,
-
spans with a natural frequency n0 less than the upper limit in Fig.4 - page 23,
-
structures with torsional frequencies nT satisfying: nT > 1,2 x n0.
Where the above criteria are not satisfied, a dynamic analysis should be carried out. Reduction coefficient for the acceleration under the distribution effect of the axle loads through the track (rail-sleeper-ballast) The diagram in Fig. 16 - page 41 gives the reduction coefficient to be applied to the acceleration obtained under concentrated loads of a train in order to take account of the dynamic effects of the axle loads lengthwise distributed over 2,5 m and 3,0 m through the track (rail-sleeper-ballast) and the deck depending on the lowest speed/natural frequency.
40
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Appendices 1,2
1
0,8
0,6
0,4
0,2
0 0
2
v/no speed/lowest natural frequency 10 8 6 12
v/n0 4
w = 2,5 m distribution of the loading
w = 3,0 m distribution of the loading
Fig. 16 - Reduction coefficient R = Amax (with distribution) /Amax (without distribution)
41
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List of abbreviations AD
(Appareil de dilatation) Expansion joint
CWR
Continuous welded rails
ELS
(Etat limite de service) Serviceability limit states
ELU
(Etat limite ultime) Ultimate limit states
HSLM
High speed load model
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Bibliography 1. UIC leaflets International Union of Railways UIC Leaflet 776-1: Loads to be considered in railway bridge design, 5th edition, August 2006
2. ERRI reports European Rail Research Institute (ERRI) ERRI D 214/RP 6: Rail bridges for speeds > 200 km/h - Calculation for bridges with simply-supported beams during the passage of a train, December 1999 ERRI D 214/RP 7: Rail bridges for speeds > 200 km/h - Calculation of bridges with a complex structure for the passage of traffic - Computer programs for dynamic calculations, December 1999 ERRI D 214/RP 9: Rail bridges for speeds > 200 km/h - Final Report - Part A: Synthesis of the results of D 214 research - Part B: Proposed UIC Leaflet, December 1999 ERRI D 214.2/RP1: Use of universal trains for the dynamic design of railway bridges - Summary of results of D 214.2 (final report), September 2000
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Warning No part of this publication may be copied, reproduced or distributed by any means whatsoever, including electronic, except for private and individual use, without the express permission of the International Union of Railways (UIC). The same applies for translation, adaptation or transformation, arrangement or reproduction by any method or procedure whatsoever. The sole exceptions - noting the author's name and the source - are "analyses and brief quotations justified by the critical, argumentative, educational, scientific or informative nature of the publication into which they are incorporated". (Articles L 122-4 and L122-5 of the French Intellectual Property Code). © International Union of Railways (UIC) - Paris, 2009 Printed by the International Union of Railways (UIC) 16, rue Jean Rey 75015 Paris - France, June 2009 Dépôt Légal June 2009
ISBN 978-2-7461-0951-4 (French version) ISBN 978-2-7461-0952-2 (German version) ISBN 978-2-7461-0953-0 (English version)
776-2 R
