J. Cent. South Univ. (2013) 20: 3314−3323 DOI: 10.1007/s11771­013­1855­6

Sensitive factors research for track­bridge interaction of Long­span X­style steel­box arch bridge on high­speed railway

LIU Wen­shuo(刘文硕), DAI Gong­lian(戴公连), HE Xu­hui(何旭辉)

School of Civil Engineering, Central South University, Changsha 410075, China

© Central South University Press and Springer­Verlag Berlin Heidelberg 2013

Abstract: X­style arch bridge on high­speed railways (HSR) is one kind of complicated long­span structure, and the track­bridge interaction is essential to ensure the safety and smoothness of HSR. Taking an X­style steel­box arch bridge with a main span of 450 m on HSR under construction for example, a new integrative mechanic model of rail­stringer­cross beam­suspender­ pier­foundation coupling system was established, adopting the nonlinear spring element simulating the longitudinal resistance between track and bridge. The transmission law of continuous welded rail (CWR) on the X­style arch bridge was researched, and comparative study was carried out to discuss the influence of several sensitive factors, such as the temperature load case, the longitudinal resistance model, the scheme of longitudinal restraint conditions, the introverted inclination of arch rib, the stiffness of pier and abutment and the location of the rail expansion device. Calculating results indicate that the longitudinal resistance has a significant impact upon the longitudinal forces of CWR on this kind of bridge, while the arch rib’s inclination has little effect. Besides, temperature variation of arch ribs and suspenders should be taken into account in the calculation. Selecting the restraint system without longitudinally­fixed bearing and setting the rail expansion devices on both ends are more reasonable.

Key words: high­speed railway; track­bridge interaction; X­style steel­box arch bridge; continuous welded rail

1 Introduction

Because of the elimination of rail joints in the continuous welded rail, the smoothness and the stiffness of railway lines get increased generally. Meanwhile, the attrition rate and maintenance work of rails are vastly reduced [1]. Hence, continuous welded rail (CWR) has been applied widely in high­speed (HSR) throughout the world, and appreciated by various countries, since it was laid in Tokaido Shinkansen in 1964. Compared with the conventional railways, the bridges occupy a significantly higher ratio in the total mileage of HSR [2], so the problem of interaction between CWR and bridges of HSR has become a research focus. In recent decades, theoretical research on track­bridge system has been carried out by scholars, and a large number of field testing and laboratory experiments have been conducted by research institutes [1, 3−8]. To calculate the forces and displacements related to the track­bridge interaction phenomena, related codes were set successively in different countries, such as the “Special Procedures on Railway Shinkansen Bridge (DS899/59)” [9] formulated in Germany in 1985, the “Track­bridge Interaction

Recommendations for Calculations (UIC774­3)” [10] enacted in Europe in 1991 and the “Code for Design of Railway CWR (draft)” [11] promulgated in China in 2008. However, most of the previous studies were concerned on concrete bridges with small or medium span, while such study on large­span steel bridges is extremely rare and insufficient.

The X­style steel­box arch bridge is one kind of statically determinate structure, with beautiful appearance and strong spanning capacity. Compared with the parallel arch bridge, this kind of bridge has better stability, larger transverse stiffness, and integral deck system which could make a better adaptability to the need of the track. So, they are appropriate to be constructed in HSR to span rivers, existing lines and other obstacles. As a new type of bridge structure, with a short application time of just more than ten years [12], related studies of X­style steel­box arch bridges are generally focused on their stability and dynamic performances while few on the track­bridge interaction [13]. To ensure the running safety, in­depth sensitive parameter analysis of the interaction between the CWR and this kind of bridge is necessary and of practical significance.

Foundation item: Projects(51378503, 51178471) supported by the National Natural Science Foundation of China Received date: 2012−06−17; Accepted date: 2012−10−12 Corresponding author: DAI Gong­lian, Professor; Tel: +86−18073187230; E­mail: daigong@vip.sina.com

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2 Establishing track­bridge coupling model of steel arch bridges with stringer and cross beam

Let Δ be the displacement of the bridge, and y be the displacement of rail. According to Hooke’s law and the equilibrium of rail free body with the length of dx, we obtain the differential equation of relative displacement between the track and the bridge.

2 2

2 2 d ( ) d d d y N z

EA x x ∆

= − (1)

where z is the track­bridge relative displacement, z=y−Δ; N(z) is the longitudinal resistance between the track and the bridge.

Reasonable simulation of the longitudinal resistance between the track and the bridge is the key to ensure the accuracy of the CWR calculation. Considering the nonlinear characteristic, nonlinear spring element is adopted to simulate the longitudinal resistance. This simulating method is recommended as the computing model in the UIC code [10] and has already been verified by domestic and foreign scholars [3−8]. According to the calculation of the simply­supported deck bridge whose span is 75 m in the Appendix C.1 of UIC774­3 using this method by the common finite element software ANSYS, the calculating results are quite close to that obtained by simplified algorithm in UIC code (see Table 1), which proves the correctness of using nonlinear springs in the FEM calculation of the track­bridge interaction.

In the traditional analysis of the additional longitudinal force of concrete bridges, single beam model is widely used to simulate the bridge [4−8]. However, for long­span steel bridge with stringers and cross beams, whose load transmitting mechanism is more complicated, the traditional model is not suitable [14]. In order to make the force transmission law conforms to the reality strictly, an integrative spatial finite element model of rail­stringer­cross beam­suspender­pier­foundation coupling system was established (see Fig. 1), in which the locations and correlations of the stringers, cross beams, arch ribs, suspenders, supports and the substructures were realistically simulated by the spatial finite elements. Nonlinear springs were adopted to simulate the longitudinal resistance, meanwhile, spatial beam elements were used for rails, stringers, cross beams

and arch ribs, and spatial link elements were applied for suspenders. The upper edge and the neutral axis were connected by rigid beam elements to simulate the deflection of the bridges under vertical train load. Besides, degrees of freedom (DOF) of the supports were coupled according to the actual arrangement, piers and abutments were simulated in accordance with actual dimensions, and the foundation’s stiffness was achieved by establishing multidirectional linear spring elements at the bottom of piers or abutments.

3 Basic assumptions

Based on the principle of the track­bridge interaction as well as the characteristics of half­through X­style arch bridge, assumptions used in calculations are listed as follows.

1) The fixed supports can prevent the beam’s expansion completely, while ignoring the resistance between the movable bearing and the bridge.

2) The degrees of freedom at the bottom of the arch springing were fully­constrained, and the foundation displacement was not considered.

3) Only the girder’s temperature rise or decline was chosen as the temperature load while calculating the additional expansion force, ignoring the alternation and the superposition of the both.

4) In order to analyze the influence of each factor, the additional expansion force, the additional bending force and the rail­broken additional force were calculated separately without adding them up. When the condition that the train brakes on the bridge should be considered, the additional force caused by the vertical load and horizontal braking force simultaneously (bending­ braking force for short) was taken as the analysis index.

4 Case study

4.1 Project introduction An X­style steel­box arch bridge with a main span

of 450 m, located on a certain high­speed railway line under construction in China, is analyzed in this work. With the design speed of 200 km/h and reserved speed of 250 km/h, adopting ballast deck and carrying two tracks with the spacing of 4.6 m, this arch bridge is the first half­through X­style steel­box railway arch bridge in China. Moreover, its main span is currently the longest

Table1Model verification Expansion force Bending force Braking force

Content UIC Article Article /UIC UIC Article Article /UIC UIC Article Article /UIC

Maximum tension 11 11.98 1.089 32.5 32.49 1.000 36.5 37.16 1.018

Maximum pressure −34 −34.31 1.009 −32.5 −31.72 0.976 −13 −13.20 1.015

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Fig. 1 Track­stringer­cross beam­suspender­pier­foundation integrative model

among all the railway arch bridges. In bridge designing, variable steel­box section is

used in arch rib, with the rise­span ratio of 1/4, the arch axial coefficient of 1.8, and the introverted inclination of arch rib of 4.8°. Composite deck system, which consisted of steel stringers, crossbeams and reinforced concrete slab, is selected, with the distance of 76.6 m between the deck and arch crown. The center­to­center spacing of two main stringers which use steel­box section is 20 m, and six I­shaped longitudinal secondary beams are set between two stringers. Besides, CHN60­type rails are applied on this bridge.

There are four skewbacks which adopt spread foundation in the whole bridge. Consolidated with skewbacks, arch ribs are connected with main stringers

through the suspenders. Meanwhile, several cross beams are located at the junctions of arch ribs and floor system, and bearings are installed to connect the cross beams and stringers of the floor system. No fixed bearings are arranged in the longitudinal direction, and the span layout is shown in Fig. 2.

4.2 Finite element model of track­bridge coupling system In accordance with the actual span arrangement, the

main bridge, one 32 m standard simply­supported box girder on the right side and subgrade at both ends were simulated in the track­bridge coupling model. Selecting the starting point of the arch bridge’s left end as the origin of horizontal axis, the fixed bearing of the 32 m simply­supported girder was located on the junction pier (3# pier). To minimize the influence of boundary conditions, 200 m rails on the subgrade on each beam end were established [11]. The sketch drawing of the finite element calculating model is shown in Fig. 3.

The live load used in the analysis was ZK load (0.8 UIC), and two­line simplified uniform live load acted on the length of the whole bridge when the additional bending force was discussed [15−16]. As to the additional braking force, 0.164 was taken as the braking force ratio [7, 17] and the driving direction was assumed to be from left to right. The rail­broken force was calculated according to Ref. [17], assuming temperature variation of the bridge was −45 °C and only single rail of one railway line broke at the position of 0# abutment

Fig. 2 Span arrangement of main bridge (Unit: cm)

Fig. 3 Finite element model of track­bridge coupling system

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where the maximum additional expansion force occurred.

5 Influence of sensitive factors on track­ bridge interaction

There are lots of components in the complicated track­bridge system. For long­span X­style arch bridge, the additional longitudinal forces may be mainly affected by several factors, including the temperature load case, the longitudinal resistance model, the longitudinal restraint schemes, the introverted inclination of arch rib, the stiffness of pier and abutment, the location of the rail expansion device and so on.

5.1 Influence of temperature load case The temperature variation of bridges is the basic

reason for the generating of the additional expansion force, so the calculating result is determined by the selection of the temperature load cases. In Ref. [11], the temperature rising or dropping 25 °C of the beam is advised for steel bridges. However, up to now, no specifications on the temperature variation value of arch ribs and suspenders are involved in the related codes of various countries around the world [18]. Based on the structural characteristics and material properties of X­style arch bridge, taking the experimental data of ORE [3] as reference, the additional expansion force under five different load cases were analyzed and listed in Table 2. In all the following tables, Frail represents the additional force of rails, Fpier is the additional force of piers and abutments, and Fas is the horizontal force of the arch springing.

As shown in Fig. 4, the regularity of additional expansion force under five different load cases is consistent, and the peak values appear at the junction of the bridge and subgrade where rails are under pressure, while the rails in the middle part of the bridges are under uniform tension. Results under Case 2 are much larger than those of Case 1, which shows the great impact of the temperature change of the main beam. When the arch rib’s temperature rising 25 °C and 40 °C are included in

the load case, the additional forces of rails and piers only have a slight increase while the horizontal force of the arch springing changes greatly, increasing from 9.8 kN to 1 342.4 kN and 2 141.9 kN, respectively. The comparison between Case 4 and Case 5 indicates that the temperature rising of suspenders can decrease the additional expansion force and beam displacement. Therefore, the load Case 5 which takes into full consideration of the temperature variation of the main beam, arch ribs, and suspenders should be selected.

5.2 Influence of longitudinal resistance model The longitudinal resistance is the critical part in the

CWR calculation, but currently the models and values of longitudinal resistance specified in codes of different countries differ greatly from each other [9−11]. The ideal elastic­plastic model is adopted in UIC code 774­3, while model with constant small resistance is used in Japan. Based on the Ref. [17], combined with results of a great deal of model experiments and field tests, the calculating model and value of longitudinal resistance of ballast track and ballastless track are also stated in “2008 China Code for Design of Railway CWR (draft)” [11]. In order to compare the differences caused by selecting different resistance models, the additional forces using four different models (as listed in Table 3 and shown in Fig. 5) were calculated.

According to Fig. 6, when different resistance models are adopted, the additional forces of rails vary greatly at the beam ends where peak values occur, though the forces are very close in the middle area of the bridge. As the resistance value in the model A is the same with that of the model B when the track is loaded, so the additional bending force is basically the same, but the additional expansion force of model B is just equal to 80% of model A’s result and the rail gap is about 144% larger. The model with constant resistance with small value (5 kN/m/rail) is generally applied in the Shinkansen in Japan, and the additional bending force is just 30.8% (pressure) and 47% (tension) of that calculated by model A, the additional expansion force is approximately 83%, but the rail gap was up to 129%. In

Table2 Temperature load cases and calculating results Load case

ΔT of beam/°C

ΔT of arch ribs/ °C

ΔT of suspenders/ °C

Max of Frail * (pressure/tension)/kN

Max of Fpier/ kN

Max of relative displacement/mm Fas **/kN

1 +25 0 0 −2 131.6/844.6 905.7 48.5 9.8

2 +30 0 0 −2 309.2/1021.5 984.1 60.6 15.7

3 +25 +25 0 −2 157.2/927.3 922.6 47.6 1 342.4

4 +25 +40 0 −2 172.2 /976.9 932.5 47.3 2 141.9

5 +25 +40 +40 −2 165.4 /965.0 928.1 47.3 2 137.6 *—The resistance model used in this table is the resistance of ballast track in Ref. [8]; **—The value of Fas listed in this table is the force of one springing, similarly hereinafter.

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Fig. 4 Additional expansion force comparison between different temperature load cases: (a) Additional expansion force of rails; (b) Relative displacement between track and bridge

general, with the reduction of the track’s resistance, various additional forces of rails and piers decrease obviously, the gap at the rail­broken position will increase gradually, but the force of arch springing changes little.

5.3 Influence of longitudinal restraint scheme For long­span bridges, how to choose the

longitudinal restraint system will affect the load transmission of the structure, and then influence structural static and dynamic responses. Similarly, its influence on track­bridge interaction should also be paid

Table3 Longitudinal resistance models

Model Name Value of resistance kN/m/line

A

Model of ballast track in China code for

design of railway CWR (draft)

> = ≤ = mm 2 60, mm 2 , 30

loaded u r u u r

> = ≤ = mm 2 30, mm 2 , 5 1

unloaded u r u u r

B

Small resistance model of in China code for

design of railway CWR (draft)

> = ≤ =

mm 5 . 0 , 6 2 mm 5 . 0 , 2 5

loaded u r u u r

> = ≤ =

mm 5 . 0 , 3 1 mm 5 . 0 , 26

unloaded u r u u r

C Ballast track model

in UIC774­3

> = ≤ = mm 2 , 60 mm 2 , 30

loaded u r u u r

> = ≤ = mm 2 , 20 mm 2 , 10

unloaded u r u u r

D Resistance model in Shinkansen in Japan

r=10

Fig. 5 Resistance­displacement curve of four resistance models

attention to. To explore the effects of longitudinal restraint conditions, four schemes proposed in Table 5 were calculated and compared, and the results are shown in Fig. 7.

Adopting four constraint schemes, the results of additional bending force of rails are quite close to each other, with the maximum deviation of 6.2%. As to the expansion force, the maximum value at the beam ends of Scheme 1 is about 2 165 kN, and it would be reduced by

Table 4 Additional force comparison between different longitudinal resistance models Additional expansion force* Additional bending force Rail­broken additional force

Model Max of Frail (pressure/tension)/kN

Max of Fpier/kN

Fas/kN Max of Frail

(pressure/tension)/kN Max of Fpier/kN

Fas/kN Max of Frail (tension)/ kN

Max of Fpier/kN

Fas/kN Rail

gap/cm A −2 165.4/965.0 928.1 2 137.6 −693.9/948.7 144.1 14 527.3 847.2 150.8 17.3 3.9

B −1 323.2/994.2 553.4 2 151.4 −330.5/568.4 85.3 14 528.4 847.2 169.6 18.1 7.9

C −1 719.3/976.8 748.6 2 144.5 −700.0/948.1 144.4 14 527.3 847.2 174.0 18.9 5.6

D −1 098.6/1 000.3 444.6 2 153.3 −124.7/264.5 71.0 14 529.6 847.2 185.2 15.8 10.2 *—The temperature variation used in this part is load case 5 stated in Section 5.1, similarly hereinafter.

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Fig. 6 Influence of longitudinal resistance models on additional longitudinal force: (a) Additional expansion force of rails; (b) Additional bending force of rails; (c) Rail­broken additional force of rails

Table 5 Longitudinal restraint schemes Scheme Longitudinal restraint arrangement

1 No fixed bearings in longitudinal direction along whole bridge

2 One fixed bearing is set at junction position of arch ribs and floor system on left end

3 One fixed bearing is set at junction position of arch ribs and floor system of right end

4 Arch ribs and floor system are consolidated on both ends

25% if the arch ribs and the floor system are consolidated as Scheme 4, but the temperature force within the arch springing would be 8 times larger. When Scheme 2 or Scheme 3 is applied, the peak value at the position of fixed bearing will decrease evidently, but increase as high as 35% on the other beam end. In addition, the 3# pier would bear a larger additional force if Scheme 2 is used. Moreover, the rail­broken additional force and the rail gap calculated under four schemes are approximate, which indicates that the longitudinal restraint has little effect on them. Comparative analysis above shows that the calculating result of additional forces is significantly affected by the longitudinal restraints scheme, and the Scheme 1 is the optimal one for this type of bridge.

5.4 Influence of inclination of arch rib Inclination of arch rib is an important design

parameter of the X­style arch bridge, with a significant effect on the structural rigidity and the mechanical performance [13]. Compared with the parallel arch (0°) , the out­of­plane fundamental frequency increases respectively by 17.2%, 29.4% and 37.1% corresponding to the inclinations of 3°, 4.8° and 6°. It is demonstrated that the structural out­of­plane stiffness and stability will be obviously improved with the increase of inclination angle. At present, the effect of the arch rib’s inclination on the CWR calculation is still not involved in the existing literature. To optimize the structure design, various additional forces of rails and pier were calculated and compared when the inclinations were 0°, 3°, 4.8° and

Table 6 Additional force comparison between different restraint schemes Additional expansion force Additional bending force Rail­broken additional force

Scheme Max of Frail (pressure/ tension)/kN

Max of Fpier/ kN

Fas (left/right)/ kN

Max of Frail (pressure/ tension)/kN

Max of Fpier/kN

Fas (left/right)/ kN

Max of Frail

(tension)/ kN

Max of Fpier/ kN

Fas/ kN

Rail gap/ cm

1 −2 165.4/965.0 928.1 2 137.6/2137.6 −693.9/948.7 144.1 14 527.3/14527.3 847.2 150.8 17.3 3.91 2 −2 918.6/937.8 1 436.1 3 871.8/2021.3 −670.7/945.9 164.2 14 385.5/14387.0 847.2 11.4 347.1 3.23 3 −2 968.6/936.0 204.7 2 013.5/4028.1 −715.2/951.0 128.2 14 532.8/14446.6 847.2 11.6 35.4 3.36 4 −1 614.5/587.5 490.1 16 793/16947 −715.0/955.4 134.2 14 128.5/14175.7 847.2 4.2 295.1 3.23

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6°. As listed in Table 7, along with the increase of the inclination of arch ribs, the additional expansion force, additional bending force, and additional bending­braking force increase gradually but the change is minor.

5.5 Influence of longitudinal stiffness of piers At present, the pier’s longitudinal stiffness of

simply­supported bridges with span of less than or equal to 48 m is limited in China’s railway codes [11, 16], but as to the continuous beams and special bridge structures, there are no related provisions. Studies indicate that longitudinal stiffness of piers has an obvious influence on the braking force and rail gap value of continuous beams [17]. In order to explore the influence of longi tudinal sti ffness of piers on the addit ional

longitudinal forces in X­style arch bridge, the stiffness coefficient α=KCalculated/KActual was introduced based on the coupling model mentioned above, and the additional longitudinal forces were calculated and compared respectively when α was equal to 0.1, 0.5, 1, 2 and 10.

The analysis results show that both the rails’ additional pressure maximum and the additional force of piers get larger with the increase of longitudinal stiffness of piers, and besides, the latter changes more significantly. On the contrary, the broken gap value decreases with the increase of longitudinal stiffness of piers gradually, and the gap values are 3.97 cm, 3.93 cm, 3.91 cm, 3.90 cm and 3.89 cm respectively under five stiffness coefficient conditions, as shown in Fig. 8.

Fig. 7 Influence of longitudinal restraint schemes on additional longitudinal force: (a) Additional expansion force of rails; (b) Additional bending force of rails; (c) Rail­broken additional force of rails; (d) Additional longitudinal force of braking pier

Table 7 Additional force comparison between different inclinations of arch rib Additional

expansion force Additional

bending force Additional

bending­braking force Rail­broken

additional force Inclination of arch rib/

(°) Max of Frail (pressure

/tension)/kN

Max of Fpier/kN

Max of Frail (pressure/ Tension)/kN

Max of Fpier/kN

Max of Frail (pressure/tension)/

kN

Max of Fpier/kN

Max of Frail (pressure/ Tension)/kN

Max of Fpier/kN

Rail gap/ cm

0 −2 163.7/964.8 927.06 −656.2/933.4 144.10 −1 578.7/1762.8 639.39 847.2 150.8 3.91

3 −2 164.8/964.9 927.77 −654.8/940.5 144.10 −1 579.7/1762.8 639.43 847.2 150.8 3.91

4.8 −2 165.4/965.0 928.13 −653.9/946.0 144.11 −1 580.3/1762.8 639.42 847.2 150.8 3.91

6 −2 165.8/965.2 928.34 −326.7/474.2 144.12 −1 580.6/762.8 639.42 847.2 150.8 3.91

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5.6 Influence of location of rail expansion device The rails and piers are subject to larger additional

expansion forces because of the large expansion length. As the foregoing calculations show that the rail gap would be too large if small resistance fasteners are used (see Table 4), so the rail expansion device (RED for short) should be located to reduce it. In this work, the RED was set in four different locations to compare their impacts: 1) In the middle of the span; 2) At the left end of the bridge; 3) At the right end of the bridge; 4) At both ends of the bridge. The results of the additional forces are given in Table 8 and Fig. 9.

When the RED is arranged in the middle of the span, the additional expansion force of rail in that location will be released, but there is no effect on decreasing the Frail

at the beam ends or the Fpier, so the maximum of Frail is up to 2 167.2 kN. If the RED is only set at the left (or right) end of the bridge, the maximum of Frail occurred at the other end of the bridge is about 0.647 (0.668) times as it is set in the middle, the additional force of braking pier is down to 48.9% (32.7%) while the stroke of the RED is up to 182% (193%). When the RED is set at both ends of the bridge, the additional expansion force of rails along the whole bridge is significantly reduced, the maximum values of Frail and Fpier decline to 29.5% and 26.5% of the result when RED is set in the middle of the span, while the stroke of the RED is up to 129%.

There is no obvious difference in the Frail and Fpier under the vertical load, when the RED is located in different manners, but the variation law of the stroke

Fig. 8 Influence of longitudinal stiffness of piers on additional longitudinal force: (a) Additional expansion force of rails; (b) Additional bending­braking force of rails; (c) Displacement of rails in the case of rail breaking; (d) Additional longitudinal force of braking pier

Table 8 Additional force comparison between different arrangements of rail expansion device Additional expansion force Additional bending force

Position of rail expansion device Max of Frail

(pressure/tension)/kN Max of Fpier/

kN Rail gap/

cm Max of Frail

(pressure/tension)/kN Max of Fpier/

kN Rail gap/

cm

In middle of span −2 167.2/900.2 929.3 5.86 −693.7/948.7 144.2 4.36

At left end of bridge −1 401.3/989.2 454.5 10.68 −735.1/942.2 112.4 6.26

At left end of bridge −1 447.9/987.8 304.2 11.28 −903.0/944.3 112.8 6.43

At both ends of bridge −640.4 /997.6 245.8 7.54 −903.1/940.2 96.3 5.00

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Fig. 9 Influence of the location position of rail expansion device on additional longitudinal force: (a) Additional expansion force of rails; (b) Additional longitudinal force of braking pier

under the vertical load is the same as that under temperature load: the smallest value occurs when the RED is located in the middle of the span, the largest value appears when the RED is located at only one end of the bridge. Based on the analysis above, setting the rail expansion device at both ends of the bridge is more reasonable.

6 Conclusions

1) With regard to such special structure of X­style steel­box arch bridge, the track­bridge interaction analysis plays a significant role during the process of structural design. Among the factors discussed, the longitudinal resistance model, the longitudinal restraint conditions and the location of rail expansion device have larger effect on longitudinal forces, while the inclination has a little influence.

2) The temperature rising of the arch rib can increase the additional longitudinal forces of rails and piers, while the temperature increase of suspenders will decrease them. Meanwhile, the horizontal forces of arch springing change greatly with the arch rib’s temperature variation. Therefore, the temperature variation of the main beam, arch ribs, and suspenders should be taken into full consideration if CWR is laid on X­style arch bridge.

3) With the reduction of the resistance, various additional forces of rails and piers decrease obviously in general, while the rail gap will increase largely. By increasing the longitudinal stiffness, the additional forces of rails and piers will increase to some extent, but the gap of rail will be reduced.

4) Calculated result of additional expansion force is closely related to the restraint arrangement and location of rail expansion device. The maximum of rail’s

additional expansion force declines by at least 25% when the scheme without fixed bearings is adopted, and the additional expansion force of rails along the whole bridge will be reduced generally and stay uniform when the rail expansion devices are arranged on both ends.

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(Edited by HE Yun­bin)