UIC_Leaflet_776-2

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 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.

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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)

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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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ISBN 978-2-7461-0951-4 (French version) ISBN 978-2-7461-0952-2 (German version) ISBN 978-2-7461-0953-0 (English version)

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