UIC CODE 7 7 6 - 2

R

2nd edition, June 2009

Translation

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 véhicule-voie-pontAnforderungen 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 2009All 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

776-2R

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 analysisis 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 andcomprehensive rules of calculation already in existence. In contrast, serviceability limit states, notablydeformation 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 moreaccurately as trains travel at high, and very high speeds. The purpose of this leaflet is to specify thedesign requirements for rail-bridges as regards train/track/bridge interaction phenomena and inparticular speed, thereby taking into account bridge resonance phenomena. It outlines thecorresponding draft criteria and provides information on the phenomena to be controlled as well as theappropriate procedures for verifying the structures.

This leaflet should be used in conjunction with UIC Leaflet 776-1.

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 oftraffic in the same conditions of safety and comfort as on normal tracks and at any traffic speed up tothe crossing speed limit defined for this bridge and for all types of traffic scheduled to cross thestructure.

1.2 - Purpose of this leaflet

The procedures for verifying the strength of railway bridges are covered by detailed andcomprehensive rules of calculation already in existence. In contrast, serviceability limit states, notablydeformation 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 moreaccurately as trains travel at high, and very high speeds. The purpose of this leaflet is to specify thedesign requirements for rail-bridges as regards train/track/bridge interaction phenomena and inparticular speed, thereby taking into account bridge resonance phenomena. It outlines thecorresponding draft criteria and provides information on the phenomena to be controlled as well as theappropriate 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 andsecondary suspensions of the vehicles as well as the associated masses, the behavioural effects ofthe track and the deformability of the bridge deck and its supports.

This leaflet also contains more simple alternative methods giving acceptable results using commoncalculations.

Aside from its vertical component which is the most critical and which constitutes the greater part ofthis leaflet, train-track-bridge interaction also has a lateral component that has a bearing on lateralvehicle behaviour through the effects of the suspension, while also exercising an influence, albeit to alesser 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 theeffects of train-track-bridge interaction, irrespective of the proposed speed of traffic up to 350 km/hand 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 arenecessary

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 superiormaintenance)

Φ3 = Dynamic increment coefficient for rail bridges (tracks with standardmaintenance)

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 (ownweight + 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 thespanned rail tracks, the deck supports or even directly onto the deck. These deformations anddisplacements are described below.

2.2.1 - Static deformations

The vertical operating loads applied to the bridge cause the deck to bend, resulting in a verticaldisplacement of every point on the surface of the deck. In general, maximum displacement occurs atthe point in the middle of the deck, or at mid-span. This displacement is known as the deflection of thedeck. 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 orsuccession 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 therotation of each deck along a transversal axis or the total relative rotation between the adjacent endsof the deck. Static operating loads are used to define a rotation of θstat.

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 isimportant 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 observea horizontal rotation of the decks around a vertical axis at their ends. This has a bearing on thehorizontal geometry of the track.

Whenever the deck supports a non-centred track or several tracks, of which one is loaded, itundergoes torsion as a result of the operating loads. Distortion of the deck is measured along the axisof 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 distanceof 3 m (cf. Fig. 2).

Fig. 2 - Definition of deck distortion

θ 1

θ 2

θ 3

3 m

s

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As shown in Fig. 3 - page 6, deflection of the deck under operating loads causes the end of the deckbehind the support structures to lift. There is also a longitudinal displacement of the ends of the uppersurface 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 thebridge. Displacement covers the entire bridge in case of a single deck but it is relative in case of aseries of decks.

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 showdifferent values under dynamic loads (in general, all the more higher if train crossing speeds aregreater), whether they are vertical or horizontal deflections under operating loads, vertical andhorizontal 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 isexpressed as follows: dynamic distortion tdyn.

In general, the value retained is the maximum value obtained for a given speed.

δ H2δ H1

fixed support mobile support

neutral axis

α

δH1: End displacement of the fixed support structuresδH2: End displacement of the mobile support structures

structures structures

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3 - Requirements for train traffic safety

3.1 - Phenomena

3.1.1 - Quality of the wheel-rail contact

Excessive deformation of the bridge can jeopardise train traffic safety by causing unacceptablechanges in the vertical and horizontal geometry of the track, excessive rail stress and excessivevibrations in the bridge support structures. In the case of ballasted bridges, excessive vibrations coulddestabilise 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 effectsof 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 railduring compression (risk of buckling of the track, especially at bridge ends) or traction (risk of railbreakage). It is also important to minimise the forces lifting the rail fastening systems (verticaldisplacement at deck ends), as well as horizontal displacements (under braking/starting) which couldweaken the ballast and destabilise the track. It is also essential to limit angular discontinuty atexpansion joints and at points and switches in order to reduce any risk of derailment.

3.2 - Criteria

3.2.1 - Distortion

Distortion of the deck is calculated with the characteristic value of load model UIC 71 and with loaddiagrams SW/0 or SW/2 as necessary multiplied by Φ and α or the high-speed load diagram, includingthe effects of centrifugal force. Limit values of distortion as described before are described in Table 1.

Total distortion caused by distortion of the track when the bridge is not loaded (for example in atransition curve), and distortion due to total deformation of the bridge, must not exceed 7,5 mm/3m.

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

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3.2.2 - Horizontal and vertical displacements

If continuous tracks are used, longitudinal horizontal displacements under the vertical effects ofoperating 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 weldedrails and is fitted with an expansion joint at the end of the bridge, or if the track is fitted with scarfedjoints.

Vertical displacements at the ends of the deck should remain below 3 mm if the track is ballasted and1,5 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 consequentlyof the spanned track, and this should be verified. Deck acceleration should be considered aserviceability limit state as far as operating safety is concerned. In cases where the bridges haveballasted tracks, intense accelerations of the deck create the risk of destabilising the ballast. For thisreason, it is important to ensure that maximum acceleration of the deck remains below 0,35g forfrequencies up to 30 Hz.

When verifying the acceleration of a deck with dual tracks in both running directions, it is assumed thatonly one track is loaded.

In the case of bridges with slab tracks, the acceleration limit value is set at 0,5g for frequencies below30 Hz.

Dynamic analysis using the modal superposition method should take on board at least 3 modes aswell 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 transversesection 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 calculationvalue 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 lifeof 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 theconstruction, does not jeopardise its durability or performance within its environment and in relation tothe projected level of maintenance. Adequate measures are specified in order to limit deterioration onthe basis of certain factors (such as properties of the soil, of the materials, foreseen maintenanceduring the life cycle of the structure, etc…).

4.2 - Criteria

4.2.1 - Dynamic increment coefficient

When a dynamic analysis of the structure needs to be carried out (see Appendix A - page 16), withthe relevant load models or real trains, it is important to determine the following dynamic incrementcoefficient:

where represents the dynamic deflection of the deck under the high-speed load diagram or realtrains and represents the static deflection of the deck.

ϕ'dyn max γdyn / γstat[ ] 1–=

γdynγstat

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The bridge is assessed using the logic diagram shown in Fig. 4.

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 islower than or equal to the corresponding resistance constraint and remains so within the frameworkof 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 materialstresses to ensure that the materials do not present a risk of developing irreversible deformations. Thelimit values with regard to constraints are given in the Eurocodes.

They also involve for stressed concrete structures checking the limitation of crack openings. Suchverifications 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 therelative evolution of cracking. The value is 0 if there is no damage and 1 if propagation is such that itdestroys the structural element. Damage is determined by taking into account the successive loadingof the component, which must remain at a permissible level for the lifetime of the structure. Fatiguedimensioning must be done to allow for the most unfavourable fatigue load conditions.

(1 + ϕ’ dyn + 0,5 ϕ’’) x (load model for HS or real train) < Φ . (LM71+SW/0)

YESNO

Load model for HS or real trains with ϕ’dyn is decisive for the project

Φ . (LM71+SW/0) isdecisive for the project

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5 - Requirements for passenger comfort

5.1 - Physical phenomena

5.1.1 - Train-track interaction

Track levelling and lining variations generate vehicle movement that can affect passenger comfort andtrain 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 (ortrack irregularities), the natural hunting movements of the axles and, when crossing bridges, thedeformation of the bridge which modifies the path of the bogies.

The running gear and suspensions generate rail vehicle body movement which affects passengercomfort and stresses which influence the vehicle running safety.

The vehicle integrates primary and secondary suspensions (springs and dampers) as well as sprungand unsprung masses (masses, rotating masses inertia) that have an impact on this phenomenon. Inorder to separate the movements of the bogie from those of the body, the greatest possible verticaland transversal flexibility is required for secondary suspension. The required natural suspensionfrequencies are about 0,7 Hz (at present, 1 Hz is usually obtained but this can vary between 1 and2 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 physiologicalcriteria linked to frequency, intensity of acceleration, steering relative to the spinal column and time ofexposure (duration of vibrations) make it possible to assess vibrations and their influence onindividuals. The limit exposure time to reduced comfort represents the limit of comfort adopted. Thisparagraph 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 engineeringstructure, it is possible to give an approximation of the path of a bogie during its passage over thestructure. Knowing the transfer function that makes it possible to move from the path of the bogie tothat of the body, it is possible to calculate vehicle acceleration. The acceleration limits inside thevehicles depend on the desired level of comfort and make it possible to limit the deflection of thestructure.

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 areoutlined in table 2.

The criteria to verify to guarantee passenger comfort relate to the vertical deflection of the decks andare listed here:

In order to limit vertical acceleration in the vehicles, certain values will be given later to illustrate themaximum 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/bridgeinteraction.

Aside from other factors, the following behaviour elements are taken into consideration whencalculating 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 inthe 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.

Table 2 : Indicative levels of comfort

Level of comfort Vertical acceleration bv

(m/s2)

Very good 1,0

Good 1,3

Acceptable 2,0

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For exceptional structures such as continuous beams with a large variation in span lengths or spanswith many different stiffness levels, it is important to do a specific dynamic calculation.

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 limitvalues 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, thevalues 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 ormore.

For a bridge with a single span or a succession of two isostatic beams or two continuous spans, thevalues 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 by0,9.

The values L/δ given in Fig. 5 are valid for spans up to 120 m. A specific analysis should be done forlonger span lengths.

V = 350

V = 300V = 280V = 250V = 220V = 200V = 160

V = 120

3 000

2 500

2 000

1 500

1 000

500

00 10 20 30 40 50 60 70 80 90 100 110 120

L/δ

L [m]

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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 relevantdynamic coefficient.

Table 3 : Static limit values

Criterion verified Description of verification Limit value Calculation under incremented LM 71

Comfort Vertical deflections cf. Fig. 5 - page 13 1 loaded track

Track stability Expandable lengths Lt = expandable length

- continuous track (no AD) : Lt ≤ 60 m (metal)

- track with AD Lt ≤ 90 m (concrete/mixed)

- non-ballasted track project specifiedspecial study

Track stability longitudinal displacements under the effects of vertical track loads

- with CWR 10 mm 2 loaded tracks a

Track stability longitudinal displacements under braking/starting for track:

2 loaded tracks a- with CWR 5,0 mm

- with AD or for jointed track 30 mm

Track stability vertical displacements at end of decks with:

- ballasted track 3,0 mm 2 loaded tracks a

- slab track 1,5 mm

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6.2 - Additional dynamic verifications

This type of dynamic verification is always carried out under real trains or under a universal dynamicloaded train (HSLM) incremented by the corresponding dynamic coefficient.

Wheel/rail contact deck distortion

- V ≤ 120 km/h distort ≤ 4,5 mm/3m

- 120 ≤ V ≤ 200 km/h distort ≤ 3,0 mm/3m 1 loaded track

- V > 200 km/h distort ≤ 1,5 mm/3m

rotations due to horizontaldeflections

- V ≤ 120 km/h

- 120 ≤ V ≤ 200 km/h 2 loaded tracksa

- V > 200 km/h

a. bridge with two tracks or more

Table 4 : Dynamic limit values

Criterion verified Description of verification Limit valueLoading

(real trains or HSLM)

Track stability and wheel/rail contact

Vertical accelerations

1 loaded track

- 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 1 loaded track

Track stability Longitudinal displacements at deck ends

Verification done in point 6.1 - page 14

Wheel/rail contact Distortion Distortion

V > 200 km/h 1 loaded track

Lateral stiffness Horizontal deflections Verification done in point 6.1 - page 14

First natural frequency of lateral vibration

≤ 1,2 Hz

Table 3 : Static limit values

Criterion verified Description of verification Limit value Calculation under incremented LM 71

θ ≤ 0035rd⋅

θ ≤ 0020rd⋅

θ ≤ 0015rd⋅

tdyn 1 2 mm / 3m,≤

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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 excitationgenerated by the moving axle loads. At low speeds, structural deformation is similar to thatcorresponding to the equivalent static load case. At higher speeds, deformation of the deck exceedsthe equivalent static values due to the bridge inertia forces and the effects of track defects and vehiclerunning defects. The increase in deformation is also due to the regular excitation generated by evenlyspaced 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 thestructure) 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:

Resonance phenomena are unlikely to occur in rail bridges if speeds remain under 200 km/h and ifthe 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 needto be done to determine the extent of deformations which, at resonance, may lead to a dynamic loadthat 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 forjustifying dynamic analysis.

Even though deck accelerations are low at low speeds, they can reach unacceptable values at higherspeeds. In practice, the acceleration criterion will, in most cases, be the decisive factor.

j = 1, 2, 3, …, i = 1, 2, 3, …, 1/2, 1/3, 1/4, …vcrit

njLci

-----------=

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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 tablesare indicated in the notes after the tables.

Independently from the logic diagram in Fig. 1, a dynamic analysis is necessary if the frequent workingspeed of a regular train is equal to a speed of resonance of the structure.

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Appendices

Fig. 1 - Logic diagram to determine wether a specific dynamic analysis is required

V = traffic speed in [km/h]L = span in [m]n0 = first natural bending frequency of the unloaded bridge in [Hz]

nT = first natural torsion frequency of the unloaded bridge in [Hz]

are defined in Appendix B - page 38.Vlim n0⁄ et V no⁄( )lim

START

V ≤ 200 Km/h

Continuousbridge

Simple structure

L ≥ 40 m

nT > 1,2 no

Use Tables 8 and 9For the dynamicanalysis use thenatural modes for torsion and for bending

natural modesfor bendingsufficient

Vlim/n0 ≤ (V/n0)lim

Dynamic analysis requiredCalculate bridge deckacceleration and ϕ’dyn etc. ormodify the structure and verify

Dynamic analysis not requiredVerification with regard to the acceleration not required atresonance. Use Φ with staticanalysis in accordance

no yes

yes

yes

yes

yes

yesno

nono

no

nono no within limits offigure A4

yes

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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 thestatic secant modulus and the speed of concrete deformation. Young's modulus for compressedconcrete 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 ofelasticity in both cases.

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 dynamicbehaviour.

The shear modulus G for concrete can be calculated from the equation:

Table 1 : Ecm values for concrete of different strengths

[kN/mm220 25 30 35 40 45 50

[kN/mm229 30,5 32 33,5 35 36 37

fck

Ecm

Gdyn

Edyn

2 1 υ+( )---------------------=

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Appendices

Fig. 2 - Influence of the stress/strain relationship on the E values for concrete in compression

Fig. 3 - Influence of the stress/strain relationship on the E values for concrete in traction

4

3

2.5

1,8

1,41,3

1,1

0,9

2

EdynEstat

fdynfstat

εu,dynεu,stat,

1

1,2

1,51,6

Compression

1

1,5

2

2,5

3

4

0.1 1 101 102 103 104 105 106 108

10-5 10-4 10-3 10-2 0,1 1 10 102 103.ε [S-1]

σ [N/mm2s].

3 .10-5

107

fcm = 20 50

f

E1

3f

α1

εμ

4

3

2,5

1,81,61,5

1,1

0,9

2

1,4

Traction

1

1,5

2

2,5

3

EdynEstat

fdynfstat

εu,dynεu,stat,

4

0.1 1 101 102 103 104 105 106 108

10-5 10-3 10-2 0,1 1 10 102 103.ε [S-1]

σ [N/mm2s].

107

10-4

3 .10-5

fcm = 20 50

f f1

δ1

Et

εμ

3

1,31,2

1

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20

Appendices

A.3.2 - Damping coefficient

Structural damping is a key parameter in dynamic analysis. The magnitude of the vibrations dependsheavily on structural damping, especially in proximity to resonance. Although it is unfortunately notpossible to predict the exact value in the case of new bridges, for existing bridges the damping valuescan 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 certainnumber of past measurements.

A.3.3 - Mass

Maximum dynamic effects occur at resonance peaks, where a multiple of the load frequency coincideswith the natural frequency of the structure. Underrating the mass will lead to overestimation of thenatural 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 massof 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 ofresonance 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 ofthe track.

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-stressedconcrete

ζ = 1,0 + 0,07 (20 - L) ζ = 1,0

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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 thelower value within the stiffness range.

The stiffness and mass of a bridge deck vary throughout the lifetime of the structure and impacts itsdynamic behaviour. The stiffness range mentioned earlier corresponds to the two extreme values, onthe one hand for sections free of cracks and without any reduction in stiffness, and on the other handcracked sections and any effect leading to a reduction in stiffness such as the effect of differentialsettlement, contraction and temperature. Bending and torsional stiffness should take account of theimpact 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 ofreinforced concrete can be used. The average value of the effective inertia along the entire length ofan evenly loaded element is obtained by:

The inertia for specific sections found along the length of the element is calculated by using thefollowing expression:

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 thespan length L in [m] on the deck.

IcMcrMA---------⎝ ⎠⎜ ⎟⎛ ⎞

=4

IG 1McrMA---------⎝ ⎠⎜ ⎟⎛ ⎞4

Icr⋅–+⋅

IcMcrMA---------⎝ ⎠⎜ ⎟⎛ ⎞

=3

IG 1McrMA---------⎝ ⎠⎜ ⎟⎛ ⎞3

Icr⋅–+⋅

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22

Appendices

Fig. 4 - Limits of natural frequencies n0 en [Hz] in relation to the span length L [in m]

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, anadditional verification must be done. If the natural frequency is above the upper limit of domain N inFig. 4, dynamic analysis is necessary.

The upper limit of n0 (N) is expressed by:

The lower limit of n0 (N) is expressed by:

for

for

(Range N is defined within these limits).

150

100

8060

40

2015

10

8

6

4

2

1,5

1,02 4 6 8 10 15 20 40 60 80 100

L (m)

n 0 [H

z]

Natural frequencyupper limit

Natural frequencylower limit

n is the first natural frequency of an unloaded bridge0L is the span for an isostatic bridge or L for other types of bridgeφ

n0 94 76 L0 748,–

×,=

n0 80 L⁄= 4m L 20m≤ ≤

n0 23 58 L0 592,–

×,= 20m L< 100m≤

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23

Appendices

If the deck cannot be considered as a beam or a slab or if the first natural frequency in torsion nT lieswithin the domain (0,8n0, 1,2n0) where n0 is the first natural bending frequency, then dynamic analysisis necessary.

If the deck can be considered as a beam or a slab and if the first natural frequency in torsion liesoutside the domain (0,8n0, 1,2n0), additional verification is needed if the ratio V/n0 does not complywith 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 theimportance of an accurate assessment of the product Eidyn and of the deck mass per unit length.

The natural frequency of an isostatic beam can be calculated or estimated using the following simpleformula:

This equation, where δstat in (mm) is calculated with the short term modulus, only refers to isostaticbeams.

A.4 - Fundamental hypotheses relating to vehicles (excitation)

The tools most commonly used for dynamic calculations do not take account of interactionphenomena. Train-bridge interaction modelling is described in point A.6 - page 31. The effect of train-bridge interaction can be integrated to the conventional mobile load diagram in point A.6 by addingadequate damping to the bridge damping.

The following formulae can be used to calculate additional damping as a function of the length of thespan:

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 lowstructural damping. For different damping values, the coefficients calculated for ζ = 0,005 can beused, seeing that ζ has minimal effect on Δζ. The coefficient values are given in Table 3 - page 25 forthe ICE 2 and in Table 4 - page 25 for the Eurostar.

fj λj2

2πL2

------------- EIμ------⎝ ⎠⎛ ⎞ 1 2⁄=

n017 753,

δstat------------------=

ζa1L a2L

2+

1 b1L b2L2

b3L3

++ +----------------------------------------------------------=Δ

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24

Appendices

These formulae are valid only for 5 < L < 30 m and 1000 < L/f < 2000. For the L/f values lying betweenthose in the tables, a linear interpolation can be done.

A.4.1 - Train models

A.4.1.1 - Hypotheses relating to vehicles

Current and future high-speed trains can be classed into three major categories, as indicated belowin Fig. 5, 6 and 7 - page 26 :

Fig. 5 - Articulated train

Fig. 6 - Conventional train

Table 3 : Coefficients for calculating Δζ under ICE 2

L/fa1 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/fa1 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

DO/

BA

O/BA

DO/BS

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25

Appendices

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 mostprobably increase in the future. It is therefore essential to establish minimum technical specificationsfor projects relating to bridges and rolling stock so as to allow high-speed trains to travel throughoutthe 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 strengthcharacteristics or operating criteria - these trains should be designed to comply with the criteria listedin 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 ≤ ≤ 10 m

= coupling distance between power car and coach

= coupling distance between 2 trainsets

Articulated trains

Type EUROSTAR, TGV

18 m ≤ D ≤ 27 m P ≤ 170 kN

Conventional trains

Type ICE, ETR, VIRGIN

18 m ≤ D ≤ 27 m et P < 170 kN or values translating theinequality below

All types 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.

O/BAD

IC

D

O/BA

Dec

ec 8 D1C 11m where≤ ≤

D1C

Ec

2 5 m dBA 3 5 m,≤ ≤,

4PπdBS

D--------------

πdBAD

-------------- 2PHSLMA

πdHSLMADHSLMA-------------------------cos≤coscos

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26

Appendices

When relating to infrastructure (bridges), the Technical Specifications for Interoperability are asfollows:

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 (forthe 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 rangeof 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 speedrange in question and for a given structure. This is essentially the case of isostatic structures, wherethe train to designate may be determined by the aggressivity method devised by the ERRI CommitteeD 214-2 (see Bibliography - page 43).

The HSLM-A consists of 10 trains defined as follows:

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

D NxD D

(1)3xP

(3) (3) (3) (3) (3) (2) (1)

9 11 3

3,525

d d d

D

4xP(2)

2xP 2xP 2xP 3xP 4xP

d

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27

Appendices

The HSLM-B consists of a number N of localised forces of 170 kN with a regular spacing d [m] whereN and d are defined in Fig. 9 and 10 - page 29.

Fig. 9 - Diagram of universal dynamic train B

The values of d and N are determined using Fig. 10:

Table 6 : Definition of the 10 trains of the universal dynamic train A

Universal trainNumber 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

N x 170 kN

d d d d d d d d d d d d d d d

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28

Appendices

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 usedfor dynamic bridge calculations.

A.4.2 - Load distribution

In the live load model, each axle is represented by a constant and concentrated load that movesacross the bridge. When taking into account the elastic properties of the upper track structure, it isclear that the reactions under the rail are diffused.

This means that the bridge underneath is not loaded by the concentrated loads but rather by thedistributed loads in the direction of the track.

Table 7 : Application of HSLM-A and HSLM-B

Structural configuration of bridgeLength of span

L < 7 m L ≥ 7 m

Isostatic bridgea

a. Valid for bridges whose behaviour is limited to that of a line beam (longitudinal direction) or a slab, on fixedsupports with minimal bias effects.

HSLM-B HSLM-Ab

b. For isostatic beams with a 7 m span or more, a single model HSLM-A may be used for dynamic analysis, underthe aggressivity method defined in ERRI report D214-2. Alternatively all 10 models HSLM-A-1 to HSLM-A-10may be used.

Continuous structure HSLM-A HSLM-A

or Trains A1 to A10 Trains A1 to A10

Complex structurec

c. Model HSLM-B should also be used.

1

6

5,5

5

4,5

4

3,5

3

2,5

2

1,6

2,5

2,8

3,2

3,5

3,8

4,2

4,5

4,8

5,5

5,8

6,5

20

15

10

5

0

d [

m]

L[m]

N

776-2R

29

Appendices

A diagram (Fig. 16 - page 41) is appended and gives the reduction coefficient to be applied to theacceleration obtained under concentrated loads. This reflects the dynamic effects of axle loadsdistributed lengthwise over 2,5 and 3,0 m as determined by the lowest speed/frequency ratio to betaken 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 Fourierseries and by extrapolating it to the natural modes. It represents the dynamic excitation features of thetrain and is independent of the characteristics of the structure. The signature depends on axle spacingand loads only.

The following is the relevant formula, where So λ is the dynamic signature (λ is the wave length = v/n0).

Bridges whose dynamic behaviour is calculated using the load diagram specific to high-speed trainsas defined in point A.4.1 - page 25 need not be calculated under the real load of current high-speedtrains or new trains whose dynamic signature falls within the envelope of dynamic signatures of loaddiagrams specific to high-speed trains. For each train, it is possible to determine an excitation spec-trum that takes account of the composition of excitation produced by the train at a given point and agiven speed.

The dynamic signature is a useful method of producing a quick comparison of the effects of differenttrains. For example, if the magnitude of the dynamic signature of a new train is lower than that ofexisting trains working a specific line, that very line may be used by the new train without the need fordoing 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).

Fig. 11 - Dynamic signatures of several high-speed trains

S0 λ( ) ΣPi 2πd i,λ------⎝ ⎠

⎛ ⎞sin2

ΣPi 2πd i,λ------⎝ ⎠

⎛ ⎞cos+2

=

TGVAEurostarThalys2ICE2ETR-YShinkansenIC225_normalVirgineurotrain

0

1 000

2 000

3 000

4 000

5 000

6 000

0 5 10 15 25 30

s0(kN)

λ (m)20

Train signatures

776-2R

30

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 bridgeto be dimensioned for high speeds.

The increase in dynamic loading as opposed to static loading depends on the speed of the train as itpasses over track irregularities while crossing the bridge and is inversely proportional to the length ofthe span.

The value obtained, without taking track irregularities into account, should be multiplied by 1 + ϕ"/2 inorder 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 theirregularities 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. Thesematerials show variable stiffness and the resulting track stiffness may then be represented by anaverage 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 interfacestiffness 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 accelerationsand deformations of a bridge crossed by a train. Models of varying degrees of complexity are possiblefor the train as well as the track.

Generally speaking, the live load diagram described and represented in point A.4.1 - page 25 anexample for which is given in Fig. 12 should be used with loads represented by a series of constantlocal forces. The other two models possible are described below in point A.6.1.2 - page 33.

Fig. 12 - Diagram of dynamic train-bridge model

Axlenumber

1 2 34

56 7 8 23

2425

2627

2829

30

3,523 11 3 3

3,27515,70 3 3 3 3 38 x 15,70 15,70

3,275 3,52

+ 7 x 3,00

5,02 14,00 6,275 18,70 8 x 18,70 18,70 6,275 14,00 5,02

11

237,59

Normal load: all axles = 17 tP = 17 x 30 = 510 t

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31

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. Thesections below make a case for improving such calculations and the model described above does nottake account of the dynamic behaviour of the train and the track. Similarly, the distribution of axleforces lengthwise through the rails is not taken into consideration.

Studies carried out by the ERRI Committee D214 (see Bibliography - page 43), clearly show that thelive load diagram using a series of constant local forces produces the highest deformations andaccelerations at resonance.

A.6.1.1 - Bridge model

Modelling methods that use beam elements are the most appropriate to quantify the behaviour ofbridges and structures essentially composed of bars.

In order to ensure that the equivalent standard beam gives a reliable representation of the overalldynamic behaviour of the structure, the modelling method should integrate the correct mechanical andmass characteristics of the structure, including the real support features. The problem resides intranslating the dynamic physical properties to a digital model. The findings obtained from the initialcalculations:

- 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 three-dimensional elements.

The modelling elements used for bridges may be slabs. The method makes it possible to examinelongitudinal and transversal modes, using slabs that may be orthotropic or skew plates. An exampleof the three-dimensional bridge model is given below.

Fig. 13 - Example of three-dimensional bridge model

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32

Appendices

A.6.1.2 - Vehicle model

The train consists of a series of vehicles represented by their masses, moments of inertia andsuspension characteristics. The vehicles making up the train are represented by the body, two bogiesand four axles. The primary and secondary suspensions are represented by parallel spring-dampersystems.

Live load diagrams are the simplest form of load models and give less accurate results than the morecomplex models.

In addition to the primary and secondary suspensions lengthwise along the bodies, the articulatedvehicles are mounted on two viscous, non-linear dampers. A very stiff spring and a viscous damperconstitute the suspension, positioned vertically between the two bodies. Figure 14 - page 33 gives acomplete model for a set of articulated vehicles.

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 takesaccount 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 forthe deck. Sleepers and ballast are modelled as concentrated masses. They are linked to the nodes ofthe rail and the bridge by a parallel spring and damper system. The track can be modelled to anylength on both sides of the bridge. This latest model gives more accurate results especially for shortbridges, where the stiffening effect of the bridge has to be taken into account. The effects of trackdistribution are not considered. Each vehicle is able to absorb the kinetic energy of the bridge and itis for this reason that, at resonance, the deflections and accelerations of the bridge obtained with thismodel 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.

McIc

2xD1

z1z8(rotation)

Ks Cs

MbIb

Kp Cp

Me

Ke

2xD2z2z9(rotation)

z3z10(rotation)

z4 z5 z6 z7

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33

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 andnew rail bridge decks crossed by trains running at speeds under 1,2 Vdes. or under 420 km/h. In allcases, 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 withdiffering 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 deflectionand acceleration (cf ERRI D 214/RP6 - see Bibliography page 43).

Dimensioning diagrams for bending and torsion can also be used to determine the maximumacceleration 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 isostaticbridges, series of isostatic decks, continuous bridges using the beam theory, the dynamic responseof plates and by taking into account the two longitudinal and transversal modes. They can also runcalculations for orthotropic square plates and skew plates.

With regard to beams, the effects of bending and shear are taken into account (Timoshenko or Euler-Bernouilli 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 asbowstring 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 thesuccessive natural modes of the structure, then to calculate the response of the structure by modalsuperposition with the train speeds that correspond to the resonance situations.

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34

Appendices

Various programs such as ANSYS, NASTRAN, ABAQUS, SAP, FASTRUDL and so on, can be usedto obtain the modal responses of bridge decks. Modelling can be done with beam models usingtorsional characteristics if the bridge is not a skew bridge and the structure is not a special case (seeabove). However, spatial modelling is necessary in such cases.

Dynamic analysis of a structure can be used to resolve a system of differential equations of lesserimportance. Two fundamental approaches may be implemented: one method consists in solving thesystem of equation by direct integration, whereas the other defines the solution based on the naturalmodes of vibration of the structure. This is known as modal superposition. A concise descriptionfollows 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 theresulting variables (participation factors, effective modal masses).

For undamped, free vibrations, the equation of movement without a second element is reduced to:

where Φ represents the circular frequency vector (= pulse) and [Φ] is the modal crossing matrixconsisting 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 slightlydifferent from the total mass of the structure). When the natural vectors are calculated, the modalmatrix 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 simultaneousdifferential equations, where N is the number of degrees of freedom (ddl) of the structure. If three-dimensional modelling is used, this number N is equal to six times the number of nodes less thenumber of ddl blocked at the support. When the number increases to a value that is very high for largemodels, the size of the problem needs to be reduced by transformation techniques. Solving thedifferential equations then becomes faster and is more accurate.

The integration method used is now as follows: for each mode i, the resulting equation gives anevaluation introduced by the Duhamel integral or the Fourier transform. The sum of the solutions givesthe 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 thetotal dynamic response and to identify the risks of resonance and dynamic amplification of some typesof stresses.

K[ ] ω2

M[ ] Φ i[ ] 0=–

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35

Appendices

A.6.2.3 - Analysis by direct integration

When the analysis uses numerical methods to directly integrate the dynamic equation, the loadsbecome the dynamic systems in the case of vehicles and their internal behaviour impacts theresponse 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 betweenthe 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 theconventional algorithms for numerical resolution of second-degree differential equations. Thesenumerical methods calculate the response to regularly spaced time intervals (in general). The selectedtime pitch determines the accuracy of the results and has a bearing on the length of computercalculations.

Numerical integration methods are all based on the search for balanced solutions of the dynamicequation at regular time intervals.

A.6.2.4 - Filtering

Acceleration, primarily at mid-span on the deck, impacts the behaviour of the ballast and consequentlythe track as well as passenger comfort. Acceleration contains significant high-frequency componentsthus resulting in very high, and effective levels of acceleration. It is clear that the ballast and track actas a low-pass filter, with only the level of acceleration within a certain frequency bandwidth affectingthe stability of the ballast.

Evidently, it is important to have identical filtration for both measurements and calculation models. Theideal low-pass filter transmits all the signal components between 0 Hz and a cut-off frequency fcwithout attenuation or phase shift.

A filter is characterised by its time-based pulse response and its frequency response. The latter is theFournier transform of the former. The bandwidth is the frequency bandwidth in which the filter gain isbetween two set values. It indicates how the filtered signal spectrum will be deformed. The steeper thefilter 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 measurementcases, it has been demonstrated that a larger number of modes than the primary mode of the structureis excited. Filtering should have cut-off frequencies 30% greater than the frequency of the last modeof interest. Filtering at 30% gives acceleration levels that are significantly higher than filtering at 20%.

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36

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 examinedin decks with two tracks or more.

In theory, bending calculations transpose easily in case of torsion. Nevertheless, to use the formulaeand diagrams properly, only one deformation mode should be considered. As it is, the primary modeis always the bending mode.

Therefore, if more than one mode is to be examined in torsion-bending cases, a program has to beused. The natural torsional frequencies can be obtained from the following formula:

Calculation charts can be established by transposing the formulae and charts from pure bending topure torsion.

The question of combined bending and torsion cannot be covered by simplified methods. It is indeedpossible to obtain the respective response of each of the two effects, but both responses cannot beadded together. In fact, these simplified methods directly determine the maximum effects, withoutgiving a temporal response. To give rules of addition, the respective moments of the two elementarymaxima must be known (torsion, bending) and above all if there is any possibility to achievesimultaneousness.

It is therefore important to use programs that give time-based answers. In this case, the elementarytemporal responses are added and the maximum temporal answer is read. The DIA and CEDYPIAsoftware programs operate in this manner.

where L = span of the slab sectionIω = torsional rigidityG = shear modulusρ = densityJp = mass rigidity in torsion

Fi i2L------- GIω

ρJp-----------=

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37

Appendices

Appendix B - Criteria to be satisfied in the case where a dynamic analysis is not required

NB : Appendix B is not valid for Load Model HSLM.

The two criteria mentioned in point 3.2.3 - page 8 (amax < 0,35 g ou 0,5 g depending on trackinstallation and δdyn < Φ2 δUIC) are always respected, (in which case, there is no need for dynamiccalculation), when the Vlim/n0 ratio is lower than the values in Tables 8 or 9 - page 39 (depending onthe limit acceleration to be considered) as a function of the span interval in [m], the deck mass intervalper linear metre in [t/m] and the damping considered.

Nota :

- Table 8 includes the safety coefficient of 1,2 on (v/n0)lim for acceleration criteria, deformation andstrength and a safety coefficient of 1,0 on (v/n0)lim for fatigue;

- Table 8 takes into account track irregularities with (1+ ϕ′′/ 2).

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 < 15,0 < 18,0 < 20,0 < 25,0 < 30,0 < 40,0 < 50,0 -

Spana

a. L ∈ [a, b) means a ≤ L < b

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

[5,00 ; 7,50) 2 1,71 1,78 1,88 1,88 1,93 1,93 2,13 2,13 3,08 3,08 3,54 3,59

4 1,71 1,83 1,93 1,93 2,13 2,24 3,03 3,08 3,38 3,54 4,31 4,31

[7,50 ; 10,0) 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

[10,0 ; 12,5) 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

[12,5 ; 15,0) 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

[15,0 ; 17,5) 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

L m∈

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Appendices

NB:

- Table 9 includes a safety coefficient of 1,2 on (v/n0)lim for the acceleration criteria, for thedeformation 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).

Table 9 : Maximum value of (v/n0)lim for an isostatic beam or a plate on simple supportsand 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

103 kg/m < 7,0 < 9,0 < 10,0 < 13,0 < 15,0 < 18,0 < 20,0 < 25,0 < 30,0 < 40,0 < 50,0 -

Spana

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

[5,00 ; 7,50) 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

[7,50 ; 10,0) 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

[10,0 ; 12,5) 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

[12,5 ; 15,0) 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

[15,0 ; 17,5) 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

a. L ∈ [a, b) means a ≤ L < b

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 lineat the considered point. For verifying the individual real trains, a reducedspeed can be used (maximum permissible speed of the trains) [m/s],

n0 first natural frequency of the span [Hz]

and defined in point 4.2.1 - page 9

L m∈

Φ2 ϕ″

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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 onrigid supports. Tables 8 and 9 are not applicable to half through and truss bridges with shallowfloors 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 issufficient 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 loadsthrough the track (rail-sleeper-ballast)

The diagram in Fig. 16 - page 41 gives the reduction coefficient to be applied to the accelerationobtained under concentrated loads of a train in order to take account of the dynamic effects of the axleloads lengthwise distributed over 2,5 m and 3,0 m through the track (rail-sleeper-ballast) and the deckdepending on the lowest speed/natural frequency.

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Appendices

Fig. 16 - Reduction coefficient R = Amax (with distribution) /Amax (without distribution)

v/n0

1,2

1

0,8

0,6

0,4

0,2

00

2 4 6 8 10 12

w = 2,5 m distribution of the loading w = 3,0 m distribution of the loading

v/no speed/lowest natural frequency

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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 RailwaysUIC 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-supportedbeams 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 structurefor 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 resultsof 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 ofresults of D 214.2 (final report), September 2000

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Printed by the International Union of Railways (UIC)16, rue Jean Rey 75015 Paris - France, June 2009Dépôt 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)

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