Calculation of dynamic impact loads for railway bridges using a direct ...

Calculation of dynamic impact loads for railwaybridges using a direct integration methodG Gu1∗, A Kapoor2, and D M Lilley3

1Bridges Department, Mott MacDonald, Croydon, UK2Faculty of Engineering and Industrial Sciences, Hawthorn Campus, Swinburne University of Technology,Victoria, Australia3School of Civil Engineering and Geosciences, The University of Newcastle upon Tyne, UK

The manuscript was received on 26 October 2007 and was accepted after revision for publication on 8 May 2008.

DOI: 10.1243/09544097JRRT189

Abstract: This paper introduces railway bridge assessment in the UK and the concept of dynamicimpact loads. The dynamic impact loads demonstrated in the codes of practice in several othercountries have been reviewed and have been found to be significantly different in the differentcodes. A technique for calculating dynamic impact load using a direct integration method hasbeen developed. In many cases, the mass of the train may be similar to the mass of the bridgein heavy railway bridges but the mass of the vehicle is normally neglected in an analysis due tocomplexities in computation. Time-varying non-linear mass models are employed to reflect theeffect of moving vehicle mass. An existing prototype bridge has been selected to compare resultsfrom the codes of practice and the technique developed in this study. The correlation betweenvehicle speed, axle load, and dynamic impact load has been investigated.

Keywords: railway bridge assessment, dynamic impact load, railway bridge dynamic, directintegration method, non-linear mass, moving mass, bridge vehicle interaction

1 INTRODUCTION

1.1 Background

According to the results of bridge assessments madeusing the current UIC (International Union of Rail-ways) and UK codes of practice, impact loads inducedby moving vehicles over short-span railway bridgesrange typically between 20 per cent and 50 per centof a static live load. As a result, in the UK where alarge-scale project of assessment of existing railwaybridges is in progress, the safety rating of the bridgerelies heavily on the dynamic impact load. In particu-lar, special attention is needed where dynamic impactloads for a bridge whose assessed capacity is closeto its required capacity because a below-standardbridge often demands expensive strengthening workor reduction of the maximum train speed. However,

∗Corresponding author: Bridges Department, Mott MacDonald,

Mott MacDonald House, Sydenham Road, Croydon CR0 2EE, UK.

email: Gunmo.gu@mottmac.com

dynamic impact loads computed from generic formu-lae stated within a code of practice are unlikely to besufficient to represent closely the real loads acting ona particular bridge. Therefore, a more refined and eco-nomical method of analysis for calculation of dynamicimpact load is required by industry. This paper reviewsthe current method based on codes of practice andintroduces a technique using a direct integrationmethod for economic calculation of dynamic impactloads for railway bridges [1–3].

1.2 Railway bridges

Although the appearance of railway bridges is similarto that of highway bridges, there are several significantdifferences between aspects relating to their struc-tural engineering and management. Static live loadson railway bridges are significantly higher than thoseof similarly sized highway bridges. The ratio of live loadto dead load and dynamic impact load (which is ofprime concern in this paper) are also much higher inrailway bridges. The greater ratio of live load to dead

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386 G Gu, A Kapoor, and D M Lilley

load raises issues in modelling moving vehicles as themasses of vehicles positioned on a bridge affect thedynamic response of the bridge. There are significantdifferences in fatigue behaviour, bridge-track interac-tion, seismic performance, and service life betweenrailway and highway bridges.

1.3 Terminology

Static load is defined as ‘weight of vehicles’. It isa product of the mass of vehicles and acceleration dueto gravity g (9.81 m/s2). Dead load such as weight ofthe structure itself, track, ballast etc. are not includedin this definition.

Dynamic load is defined as ‘load applied to thestructure whose values vary with time’. This genericdefinition is used for most issues relating to struc-tural dynamics, such as seismicity, shedding of windvortices, and blast effects. However, a more refineddefinition is required in order to discuss traffic-induced dynamic loads, whose mechanisms are quitedifferent from wind- and seismic-induced dynamicproblems.

Impulsive load is a relatively large magnitude forcethat acts over a very short but finite-time interval. Withregard to a dynamic system comprising vehicles mov-ing at speed over a bridge, as the vehicle live load actsfor a very short time on a particular finite-mass ele-ment of the bridge, the motion of the mass element ofthe railway bridge is similar to the motion of a dynamicsystem under an impulsive force.

Dynamic impact load acting on a railway bridgeis now defined as an impulsive force applied to thebridge, which is generated by moving vehicles. Inpractice, if the frequency of the impulsive force on arailway bridge is generated by regular wheel spacingsand is not close to a natural frequency of the bridge,then the cyclical effect of multiple impulsive loadscan be neglected. In summary, dynamic impact loadsinduced by moving vehicles do not normally causeresonant phenomena within the structures of railwaybridges.

Dynamic increment is equivalent to the dynamicmagnification factor or system response ratio andare common terms in structural dynamics. In thisstudy, the minimum and maximum amplitudes of themotion of vibration are substituted by the static anddynamic displacements of a particular mass elementor point in a structural member. Therefore, dynamicincrement can be determined by equation (1).

Dynamic increment

=Maximum displacement

induced by moving vehicle

Maximum displacementinduced by static vehicle

− 1 (1)

Displacement of a structural member is generallymeasured at elements, which govern the entire designof the member. In simply supported girder-typebridges, this is usually at the mid-span position wherethe largest vertical bending deflection occurs. If thestructural member is continuous in terms of struc-tural behaviour, the dynamic increment investigatedat a point of interest represents the dynamic incre-ment at all other points on the continuous member.For example, in 2- or 3-span continuous bridges, avalue of dynamic increment obtained at a partic-ular section can be used for assessments at otherpositions.

It should be noted that dynamic increment relatesonly to the displacement of the bridge. A number ofresearch papers discuss in depth the velocities andaccelerations of bridges. Accelerations of parts of abridge are not considered in this paper in determiningdynamic impact load, although they are recognizedas important with regard to track safety. For instance,Eurocode 1 suggests that the maximum peak val-ues of bridge deck acceleration calculated along eachtrack should not exceed 3.5 m/s2 for ballasted trackand 5 m/s2 for direct fastened tracks where trackand structural elements are designed for high speedtraffic [4].

Influence lines for displacement obtained forpseudo-static loading representing combined staticand dynamic loads, particularly vertical displacement,provides key information in design and assessmentof a particular structure. As velocity and accelera-tion can readily be derived after the computation ofdisplacement at each time step using available com-puter software, many researchers studying dynamicimpact loads determine values of velocities and accel-erations but few papers demonstrate their practicalmeaning. This paper does not discuss velocities andaccelerations in detail. [5–10]

Dynamic impact, dynamic increment, dynamicimpact factor are needed to determine values ofdynamic impact load for a particular bridge. The cor-relation between static load, dynamic impact load,dynamic increment, and dynamic impact factor areexpressed as follows

Dynamic impact

= Static live load × Dynamic increment

Dynamic impact factor = 1 + Dynamic increment

Dynamic impact load

= Static live load × Dynamic impact factor

Dynamic impact factor is the application factor repre-senting the dynamic amplification of the static loadeffect. With regard to dynamic impact, historically,there were two opinions about trains travelling over

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Calculation of dynamic impact loads 387

bridges. The first opinion is that the running vehicleon the bridge generates a dynamic impact in addi-tion to the static live load, and the second is that thestructure would not have sufficient time to attain fulldeflection in the time when the vehicle traverses thebridge. This means the speed of the vehicle ‘releases’static load to the structure rather than creating extraforce in addition to train static load. As far as engi-neers (not physicists) are concerned, dynamic impactfactor is always greater than unity in real bridge situa-tions. According to the UIC code of practice, dynamicimpact factor is governed by two independent fac-tors: structural response and track irregularities. For abridge supporting trains travelling at speeds between90 and 200 km/h, the dynamic impact factors are givenby equation (2) [1, 2, 11]

ϕb = 1 + ϕI + ϕII (2)

where ϕb is the dynamic impact factor for bendingfor track with standard maintenance, ϕI the dynamicincrement due to structural response, and ϕII thedynamic increment due to track irregularities.

In theory, dynamic increment due to structuralresponse is dependent on train speed, train type,span length, stiffness, and mass of the bridge. Mostdesign codes propose the use of simple formulaeincluding the effects of structural response and trackirregularities. According to design codes used in theUK, US, UIC, Korea, and Spain, only span length isconsidered as a variable, with some other consid-erations. Dynamic increments in design codes aredetermined by empirical methods and have been usedin the design and assessment of railway bridges inthe UK and US. Dynamic analysis for the purpose ofdetermining dynamic increment is not a mandatoryrequirement in most codes except Eurocode 1, whichdoes require dynamic analysis for certain situations.According to EN 1991-2:2003, which will soon replace

BS 5400 part 2 in the UK, dynamic analysis is generallyrequired for bridges with spans less than 40 m carry-ing trains at speeds over 200 km/h. Eurocode 1 is notyet formally adopted due to unavailability of NationalAnnexes [1, 2, 4, 12–14].

1.4 RA1 train

In the UK, an assumed train (called RA1) is commonlyused in bridge assessment comprising 12 axle loadsfollowed by a uniformly distributed load. The originof the RA1 train is not certain, but it is believed tohave originated from early trains of railway vehicles,which comprised a steam locomotive and a series offreight wagons or passenger coaches. The RA1 train isfrequently used in assessing existing bridges. Designsof new bridges use an assumed train load (called RU),which is slightly heavier than RA1. Real trains withaccurate axle spacing and loads are used when a moredetailed analysis is required (such as calculation ofdesign fatigue life or assessment of a bridge whichis below-standard for safety). The arrangement of theaxles of RA1 train is given in Fig. 1 [2, 3].

1.5 Assessment

In practical terms, the assessment of railway bridgesis commonly referred to as ‘rating the railway bridges’.In the UK, the results of rating can be either ‘pass’ or‘fail’ for a certain train running at a certain speed orroute availability (RA) number also at a certain speed.The RA rating of heavy railway bridges ranges fromRA0 to RA15. RA1 train and RA should not be con-fused by readers. It is a coincidence that there is asimilarity with ‘RA’. However the two terms have com-pletely different meanings. The former refers to a traintype and the latter refers to a magnitude of load on abridge or a capacity of a bridge followed by a number.Calculations for RA number are demonstrated below.

Fig. 1 Steam locomotive and RA1 train load [2, 15]

1.5.1 Computation of the capacity of structuralmembers within a bridge

Regardless of the loading on a bridge, structural capac-ities of its individual structural elements are com-puted. The structural capacities can relate to bendingmoments, shear forces, or axial forces depending onthe element’s role in a bridge and its potential mode offailure.

1.5.2 Forces under dead load

Without a vehicle on the bridge, forces induced byself-weight of structural members, ballast, track etc.are computed by manual calculation or appropriatecomputer software.

1.5.3 Forces in bridge under RA1 train

Another name for RA1 train loading is 20 BSU (BritishStandard Units). RA1 train and BSU loadings weredefined previously in BS153:1971. This documentwas withdrawn but the descriptions of the RA1 train(BSU) are referred to in the latest Network Rail codeRT/CE/C/025 and are still in common use in indus-try. If all values of force shown in Fig. 1 are dividedby a factor of 20, then the result will be 1 BSU load.This 1 BSU load is meant to be imposed on the bridgeand resulting forces in the structural members arecomputed [2, 16].

1.5.4 Number of static BSU

If trains crossing a bridge are moving slowly enoughnot to generate dynamic effects, the number of BSUthat the bridge can support safely can be calculated.For instance, if the answer is 20, this means that thebridge can support the RA1 train, which is 20 BSU.If the answer is 15, the bridge can only support atrain with 75 per cent of the weight of the RA1 train.This is illustrated in Fig. 2 and can be calculated

Fig. 2 Calculations of static BSU

using equation (3).

Number of static BSU

= capacity of element − force due to dead loadforce due to 1 static BSU load

1.5.5 Dynamic BSU and RA evaluations

In order to compute the number of dynamic BSU,‘force due to 1 static BSU load’ in equation (3) isreplaced by ‘force due to 1 dynamic BSU’. DynamicBSU is related to static BSU and dynamic impact factorand given by equation (4).

Number of dynamic BSU = Number of static BSUDynamic impact factor

Finally, RA is derived by equations (5) and (6).

Static RA = Number of static BSU − 10, (5)

Dynamic RA = Number of dynamic BSU − 10, (6)

RA is the integers and the minimum is 0 and maximumis 15.

2 CALCULATION OF DYNAMIC IMPACT FACTORUSING CODES OF PRACTICE

2.1 Review of codes of practice

Unless a bridge supports high-speed trains, in mostcases the dynamic impact factor of a bridge is calcu-lated in accordance with the methods described incodes of practice. Several countries have their ownstandards for railway bridges. A review of existing stan-dards has shown that most countries do not take trackirregularities into account separately in their designcodes, but generally contain formulae, which incorpo-rate impact factors that combine effects of structuralresponse and track irregularities. In contrast, the UICcode suggests a separate formula for impact factorinduced by track irregularities for real train analysis.The UK assessment code suggests formulae, whichare also recommended by the UIC code; a summaryof the application of the UK code is presented inFig. 3 [1–3, 12, 16–18].

Values of dynamic increments used in other coun-tries have been investigated and are presented in Fig. 4.Spain, Germany, and the UK employ the UIC recom-mendations, and values from Korea and USA showcommonality in the pattern of the data. As shown inFig. 4, depending on the code of practice, similar-sizedbridges can have a wide range of dynamic impact load.Taking as an example a ballasted railway bridge with a

Fig. 3 Summary of the UK assessment method

Fig. 4 Dynamic increment in different countries

span of 10 m, the US and UIC codes suggest dynamicincrements of 0.342 and 0.585, respectively. This indi-cates that design of this bridge using the UIC codeconsiders 18 per cent more live load (1.585/1.342 =1.18) than the US code.

2.2 Dynamic impact load of an existing bridgeusing code of practice

A bridge over the M25 (Structure ID: VTB2 93B) nearLondon is a two-span half-through deck-type railwaybridge. Each span consists of composite cross gird-ers and two main girders. The main girders are simplysupported on each span. At the intermediate support,

the ‘country’ side girders sit on top of the ‘London’side girder in a halving-joint arrangement. A view ofthe bridge and key structural information are given inFigs 5 and 6.

The bridge has been analysed using simple hand cal-culations; in addition, eigenvalue buckling and non-linear analysis using a finite-element method havealso been performed in order to identify any poten-tial structural problems. The effects of dynamic impactload on the bridge have also been analysed in orderto compare results from the current code and themethod developed in this study. For vehicle speedsof 145 km/h (90 m/h), the computed impact factor is1.289, when the effect of track irregularities is included;this reduces to 1.211 if track irregularities are ignored.The results are summarized in Table 1.

3 ANALYSIS TECHNIQUE USING DIRECTINTEGRATION METHOD

In the very early stage of this study, an analysis inaccordance with the theory of a continuous systemwas attempted. This theory was found to be unsuit-able for practical applications but more for theoreticalreview owing to computational complexity in solv-ing second-order partial differential equations. Themethod of mode of superposition (which is a com-monly used technique in structural dynamics) wasconsidered but also found to be unsuitable for thisstudy. This is because the technique is only valid for

Fig. 5 Bridge over M25: structure ID: VTB2 93B [19]

Fig. 6 Cross-section of M25 bridge [19]

Table 1 Dynamic impact factor computed using codeof practice [19]

Dynamic increment due tostructural response

Dynamic increment due to trackirregularities

Dynamic Imapct factor includingtrack irregularities

1 + 0.211 + 0.078 = 1.289

linear systems with simple external force patterns,which can be represented by either sinusoidal-typeformulae or a constant value. As material non-linearityand complex loading are required for this study, ithas been concluded that the method of direct inte-gration, which has virtually no limitations (apart fromhardware capacity and computation time) is the onlyviable technique. Another remarkable advantage inusing direct integration in this study is that it is veryeconomical in enabling a parametric study of the rele-vant factors relating to the vibration of a bridge. Resultsfor various types of bridges can be produced rapidly ina similar manner to that of a spread-sheet calculation.The authors have developed software for calculatingdynamic impact factors based on a method of directintegration [11, 20].

3.1 Modelling

In general, bridge design is governed by vertical forces.Calculation of transverse dynamic impact loads on

railway bridges is not common in design and assess-ment. Therefore, two-dimensional models using sin-gle degree of freedom for each mass element of abridge are sufficient unless there are more specificrequirements. A main girder of the M25 bridge canbe idealized as a simply supported beam, as shownin Fig. 7.

Fig. 7 Modelling of the bridge

In reality, trains are supported by numerous axles orbogies but at this early stage of the research, a simpli-fied model has been studied, where only a single axle isassumed to support the vehicle body mass. The overalleffect of a single (very common) locomotive or coachcomprising two bogies each with two axles may not betoo different from the simple model. This view is sup-ported by the fact that distances between axles of theAssessment Load Wagon in the UK is 1.829 m, whichis significantly smaller than the span of 21.26 m in thisbridge [2].

Prime interest lies in the worst condition of staticloading combined with dynamic effects. In addition,the distance between axles has a minor effect on thedynamic impact factor although it is one of the param-eters potentially governing resonance induced by thepassage of successive wheel loads.

3.2 Condensation of stiffness matrix

In developing the simple model, the first step is toconstruct the global stiffness matrix, as shown inequation (7). This typical stiffness matrix includesrotational freedom at each node. However, the rota-tional inertia in the analysis of the beam has a minoreffect and, therefore, elimination of rotational degreesof freedom using a technique of matrix condensa-tion is performed so as to improve computationalefficiency.

Prior to elimination of rotational freedoms in theglobal stiffness matrix, individual elements in thematrix are rearranged by separating displacement androtational terms. A modified stiffness matrix is shownin equation (8).

[K] =[[Kyy] [Kyθ ][Kθy] [Kθθ ]

] [[y][θ ]

Using the condensation formula given in equation (9)the stiffness matrix including all degrees of freedom isdownsized to the condensed matrix Ky(q×q) for effectivedynamic analysis. This condensed stiffness matrix isgiven in equation (10).

[Ky] = [Kyy] − [Kyθ ][Kθθ ]−1[Kθy] (9)

Ky(q×q)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

ky(1,1) ky(1,2) − − − − ky(1,q)

ky(2,1) k(2,2) − − − − −− − − − − − −− − − − − − −− − − − − − −− − − − − − −

ky(q,1) − − − − − ky(q,q)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

−−−

yq−1

The relatively small size of condensed stiffness matrixKy (lesser than 25 per cent of the size of K ) will greatlyincrease the computational efficiency during an anal-ysis of a bridge with a large number of degrees offreedom. An important point is that this procedureneed only be performed once prior to the followingstep-by-step calculations. As rotational stiffnesses areeliminated, rotational inertias of mass elements do notneed to be calculated. This method is highly recom-mended as an aid in developing other models usingthis form of analysis.

3.3 Eigenvalue analysis and damping matrix

Eigensolutions and predicted values of resonant fre-quencies are not required or produced using themethod of direct integration. However, eigenvalueextraction to determine natural frequency is very use-ful in estimating an appropriate time-step interval toensure the stability, accuracy, and reasonable cost ofcalculations in the method of direct integration. Eigen-values for first- and second-modes are computed bymeans of inversed vector iteration and Gram–Schmidtorthogonalization. Computed eigenvalues can be con-firmed by the results of finite-element analysis wherethe entire bridge structure (comprising main girders,cross girders, concrete deck, sleepers, rails, and bal-last) is modelled in detail. The natural frequency fromfinite-element analysis is 6.427 Hz and the value fromthe idealized beam model is 5.542 Hz, i.e. there is a dif-ference of 0.885 Hz or 13.8 per cent between the twovalues. The difference is likely to be the result of ignor-ing stiffness of the decking in the simple beam analysis.This variation does not significantly affect values ofdynamic impact factors as train speed and bridgespan have much greater influence on dynamic impactfactor than natural frequency. In addition, the differ-ence between predicted values of natural frequencywould not justify the significant increase in costs, iffinite-element analysis were to be used, and it can beconcluded that sufficiently accurate eigenvalues forthis type of bridge can be obtained using a simplebeam model. A finite-element model of the bridge isshown in Fig. 8. The first- and second-predicted modesof vibration are related to lateral movement. The thirdmode of vibration is related to vertical movement and

Fig. 8 Finite-element model of the M25 bridge

Fig. 9 First mode of vibration

is of prime interest in this study. The first and thirdmodes are shown in Figs 9 and 10 [6, 20, 21].

As stated earlier, only vertical deflection is of inter-est in this study and so lateral and torsional vibrationmodes obtained from the finite-element analysis aredisregarded. Vibration of local elements and highermodes of vibration are also ignored. Another advan-tage of eigenvalue extraction is that it allows compu-tation of a modal damping matrix. Computation of amodal damping matrix requires the second mode ofvibration to be obtained from eigenvalue extraction inaddition to the first mode, although the second modeis less likely to be produced in a real simply supportedbeam. Construction of the damping matrix requiresa value for the second-natural frequency as shown inequations (11) and (12) [3, 20, 21].

[c] = aq[k] +(

2 ξ̂1 �1

)[m] {�}1 [[m] {�}1]T (11)

aq = 2 ξ2

�2ξ̂1 = ξ1 − ξ2

Fig. 10 Third mode of vibration (vertical movementdominant)

The expanded damping matrix is given inequation (12)

[c]q×q

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

cy(1,1) cy(1,2) − − − − cy(1,q)

cy(2,1) cy(2,2) −− − −− − −− − −− − −

cy(q,1) − − − − − cy(q,q)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

−−−

yq−1

where [c] is the damping matrix for the first mode,�1 the natural circular frequency for first mode, �2

the natural circular frequency for second mode, ξ1 thedamping ratio for first mode, ξ2 the damping ratio forsecond mode, {�}1 modal matrix for first mode, andM1 modal mass for the first mode.

It is difficult to evaluate the magnitude of damp-ing in a bridge structure as it is dependent on severalparameters, and therefore, it is often ignored whendeveloping a safe and conservative approach to designand assessment. In theory, damping in these systemsshould be non-linear in situations where non-linearproperties relating to mass are assumed. However,consideration of non-linear damping will result inunnecessary use of computational resource, as damp-ing ratios are only estimated values. In this study, lineardamping has been assumed with a damping ratio of0.5 per cent of critical damping in accordance with therecommendation in Eurocode 1 [4].

3.4 Nodal forces and time-varying mass

Time-varying values of nodal forces and non-linearmass are required to be computed. Non-linear massmodels are not normally considered in the dynamics of

railway bridges due to the high level of complexity, theyintroduce into the calculations. In a railway bridge,the ratio of live load to dead load is much greaterthan that of a building or a highway bridge, and there-fore, the effects of moving masses need to be includedwhen considering railway bridges. The concepts ofnodal forces and time-varying masses are illustratedin Fig. 11. In theory, formation of the effective massmatrix (and its inverse) at every time step dramati-cally increases the computational effort required. Inthis study, a simple model with 11 degrees of freedomis used and requires only a few additional seconds ofcomputing time.

[m]q×q

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

m1(t) 0 − − − − 00 m2(t) −− − −− − −− − −− mq−1(t) 00 − − − − 0 mq(t)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

[F ]t =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

f1(t)f2(t)−−−

fq−1(t)fq (t)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

3.5 Train speed and load

Bridge assessments are often made using an assumedfreight train commonly referred to as the ‘AssessmentLoad Wagon’ with a bogie vertical load of 500 kN pro-ducing the worst load condition on the bridge. Themaximum permitted train speed is 145 km/h (90 m/h)in this case. However, it is very unlikely that a fullyloaded freight train will travel at the maximum speedof 90 m/h at the M25 Bridge, and it has been assumedthat the worst loading condition will be produced by apassenger train with a bogie load of 300 kN [2].

3.6 Step-by-step integration

The equation of motion given in equation (15) isdirectly integrated using a numerical step-by-stepprocedure

[m]{ y ′′} + [c]{ y ′} + [k]{ y} = {F (t)} (15)

In this ordinary differential equation, the first- andsecond-differential terms represent velocity and accel-eration, respectively, of the mass and can be replacedby displacement terms as illustrated in Fig. 12 and

Fig. 12 Integration of time steps [22]

Fig. 11 Computation of time-varying nodal force and mass

equations (16) and (17) [22].

υt+0.5�t = yt+�t − yt

�t, υt−0.5�t = yt − yt−�t

�t(16)

αt = υt+0.5�t − υt−0.5�t

�t= yt+�t − 2yt + yt−�t

(�t)2(17)

Acceleration and velocity in equation (15) are substi-tuted by equations (16) and (17) and it gives

(�t)2[m]t+�t + 1

2�t[c]

t+�t

= {F }t −[[

k] + 2

(�t)2[m]t

1(�t)2

[m]t−�t − 12�t

t−�t(18)

The final stage has been to write software to calcu-late successive values of {y} at different time intervalsusing equation (18). This software has been used toexamine predicted values of vertical displacement anddynamic increment and their sensitivity when using

linear or non-linear models for mass, and a limitedrange of different train speeds and different axle loads.

3.7 Results

The value of the Dynamic Increment predicted by thenon-linear mass model is approximately 65% of thatobtained from the linear mass model (see Fig. 13 andTable 2). The structural response of the non-linearmass model along the girder is given in Fig. 14. Asmight be expected, the value of Dynamic Incrementis predicted to increase as a function of train speed, asshown in Fig. 15 and Table 3. Values of Dynamic Incre-ment are also dependent on the magnitude of appliedload as indicated in Fig. 16 and Table 4.

3.8 Post-loading free oscillation and fatigue

It is observed from Fig. 15 that the amplitude ofvibration of the bridge after the axle force has leftthe bridge is not proportional to the speed of thetrain. This is thought to be caused by the combined

Fig. 13 Vertical displacements at mid-span assuming static load, dynamic load with linear massand dynamic load with non-linear mass, v = 145 km/h, F = 300 kN

Table 2 Maximum values of vertical displacements at mid-span anddynamic increments using different mass models (from Fig. 13)

Dynamic loading Dynamic loadingStatic loading with linear mass with non-linear mass

Deflection 2.415 mm 2.707 mm 2.605 mmDynamic increment – 0.1209 0.0787

Fig. 14 Vertical displacements at nodes along the girder = 145 km/h, F = 300 kN, non-linear mass

Fig. 15 Vertical displacements at mid-span at different train speeds, F = 300 kN, non-linear mass

Table 3 Maximum values of vertical displacementsat mid-span and dynamic increments atdifferent train speeds (from Fig. 15)

Dynamicvertical Dynamic

Speed displacement increment

96.6 km/h (60 m/h) 2.539 mm 0.0513145 km/h (90 m/h) 2.605 mm 0.0787193 km/h (120 m/h) 2.819 mm 0.1673241 km/h (150 m/h) 3.366 mm 0.3938

effects of the non-linear mass and damping. Whenthe axle force leaves the bridge the entire dynamicsystem loses a large part of its mass and energy asso-ciated with the moving mass. The structure will alsobe freely vibrating where the motion is controlled bythe initial conditions, structural, and material prop-erties including damping. The greater vehicle speedproduces greater speed of the mass of the bridge,and thus, generates greater damping forces to restrainthe free movement of the structure. Dynamic impact

Fig. 16 Vertical displacements at mid-span predicted using different axle loads, v = 145 km/h,non-linear mass

Table 4 Maximum values of vertical displacements atmid-span and dynamic increments for differentaxle loads (from Fig. 16)

Static Dynamic DynamicAxle load deflection deflection increment

100 kN 0.8049 mm 0.8894 mm 0.1050300 kN 2.415 mm 2.605 mm 0.0787500 kN 4.025 mm 4.322 mm 0.0738

factor is not affected by post-loading free oscillation,but the oscillation is important in fatigue analysis,where the ranges of stress are governing parame-ters. Codes of practices do not address the effect ofadverse deflection induced by dynamic behaviour ofbridges on fatigue life, and it is not usually consid-ered in design. This issue needs to be investigatedfurther.

3.9 Assessment using a revised dynamic impactfactor

Vertical deflection of a girder is mostly caused by bend-ing. Therefore, the dynamic impact factor relating todeflection is related to the bending capacity of thegirder and the magnitude of applied load. According tothe results of the assessment using the UK code, Girder2 (which is one of the four main girders of the M25bridge and analysed in this paper) is RA10 in bend-ing at a train speed of 145 km/h (90 m/h). This girderhas been re-assessed using the method describedabove and found to be RA12 in bending at the sametrain speed. The two assessments are summarized inTable 5 [2].

Table 5 Comparison of assessments of Girder 2

DirectCode integration

Bending capacity 16 600 kNmBending moment produced by

dead load7002 kNm

Available moment capacity 9598 kNmBending moment produced by

20 unit RA1 load7310 kNm

Bending moment produced by 1unit static load

365.5 kNm

Static BSU 26.26Static RA RA15Dynamic increment due to track

irregularities0.078

Dynamic increment due tostructural response

0.211 0.0787

Dynamic impact factor 1.289 1.157Bending moment produced by 1

unit dynamic live load471.1 kNm 422.9 kNm

Dynamic BSU 20.37 22.70Dynamic RA RA10 RA12

4 CONCLUSIONS AND RECOMMENDATIONS

The method of analysis has shown that the new ratingof RA12 for Girder 2 is higher than its previous ratingof RA10. The technique demonstrated in this paper isnot only economical in terms of computational effortbut uses a more detailed method of analysis than thecurrent code of practice, and it shows the structuralresponse more clearly. The dynamic impact load com-puted by this technique is likely to be more reliablethan that predicted by the code of practice, as theresults are derived from the predicted motion of thestructure. However, this reliability needs to be veri-fied by comparison with results of more sophisticatednumerical models and physical measurements.

Track irregularities often have significant effect onthe dynamic impact load that must be considered fora particular bridge structure, but quantitative assess-ment of certain types and magnitudes of track irregu-larities is not yet available to industry. Real train loads,two vehicles approaching on adjacent tracks fromopposite directions, bridges with continuous spans orvarying section properties will all influence dynamicimpact loads. In order to improve the accuracy of themethod of analysis, the authors are currently involvedin research into these issues. It is felt that there is astrong likelihood that, in many cases, a more refinedanalysis will avoid the need for expensive strengthen-ing work to bridges. It is therefore recommended that,in future, values of dynamic impact factor are calcu-lated using the technique demonstrated in this paperand compared with values obtained from other meth-ods (if available) for bridges where the RA number islower than the required target number.

ACKNOWLEDGEMENTS

This research is funded by Mott Macdonald UK whowas commissioned to undertake an analysis of thebridge described in this paper by Network Rail UK.The authors would like jointly to acknowledge thesupport of Mott MacDonald, Network Rail, and theSchools of Mechanical and Systems Engineering, andCivil Engineering and Geosciences at the University ofNewcastle upon Tyne.

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