COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Research on Rigidity Limits of Bridge with Conventional Spans for
Chinese High-speed Railway
M. M. Gao *, J. Y. Pan, Y. Q. Yang
China Academy of Railway Science, No.2 daliushu Road, Haidian district, Beijing, 100081 China
Email: [email protected], [email protected], [email protected]
Abstract In this paper , an efficient dynamic model of coupling vibration analysis for train-bridge system is given.
Aiming at whole train passing through the multi-span bridge with ballasted track, a model of bridge with various finite
elements is established. The motion of a four-axle car with two stage suspension is modeled based on multibody
dynamic theory with linear springs and dampers. According to the assumptions that both wheel and rail are rigid bodies
and keep contact to each other in vertical direction, the vehicle spacial vibration model has 27 degree of freedom.
Meanwhile, proceeding from microcosmic analysis of wheel/rail relationship, describing wheel/rail interaction and
displacement coordination with wheel/rail contact creep theory in horizontal direction, regarding track irregularity as
the excitation source, and adopting time domain method, an overall analysis of the procedure from train’s entering into
to exiting from the bridge has been made. To the train- bridge coupling vibration, implicit integration method is used,
and thus the limitation of time step can be avoided in explicit integration method which is caused by high frequency of
bridge, at the same time the accuracy and rapidity of analysis can both be achieved and well effects are obtained.. A
computer program TYCHE has been developed for the dynamic analysis of train-bridge system. The above procedure
has been used to make train-bridge coupling vibration analysis for single track prestressed concrete box girder bridges
with 32m span respectively. And German ICE3 within speed of 250-350km/h has been considered. By analyzing the
disciplinarians between beam rigidity and the index of running safety and riding comfort, the vertical rigidity limits of
bridge with 32m spans aimed at high-speed are proposed .
Key words: train-bridge, coupling vibration, car acceleration;rigidity limits, riding comfort, running safety
INTRODUCTION
With the development of high-speed railway, the train’s dynamics effects on line and bridge has become increasingly
significant. The vibration of the bridge superstructure triggered by train may cause fatigue of the components and
weaken its strength and stability; while excessive vibration of the bridge will affect the train’s running safety and
stability. When the train’s dynamic frequency is equivalent or close to the self-excited vibration frequency of the
bridge, the resonance caused will strengthen the dynamic response between the train and the bridge, thus leading to
unexpected damages.
The extensive application of bridge in high-speed railway makes it necessary to conduct deeper research on the
dynamic interaction between train and bridge. The thesis has taken into account the common interaction between the
train, railway line and structure and defined a reasonable bridge rigidity under the prerequisite of running safety and
comfort so as to instruct the design of the bridge.
COMPARISON OF THE SPECIFICATIONS AT HOME AND ABROAD
1. Vertical rigidity limits in Chinese Specifications Currently the design of high-speed railways is mainly in
accordance with the “Temporary Provisional Regulations on design of newly built railway lines for passenger traffic
of 300-350km/h ”[5]. Designed live load for bridge is ZK load, as shown in Fig. 1.
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Figure 1: Legend of ZK standard live load
The regulations also stipulates that the ZK load should be taken in account in every track for both single-track and
double-track bridge design, and meanwhile, under the static force of ZK live load, vertical deflection of the beam shall
not exceed that list in the Table 1[5].
Table 1 Vertical Deflection Limits of the Beam
Span
L≤24m
24m<L≤80m
L>80m
Single span
L/1300
L/1000
L/1000
Multi span
L/1800
L/1500
L/1000
Item
According to the regulations, besides that the static analysis should be in line with the relative prescriptions, the
train-bridge coupling vibration should also be analyzed according to the actual trains running through the bridge.
2. Vertical rigidity limits in European Specifications Since the Chinese specification for design live load and certain
limits has mainly referred to that of the Europe, we will focus to review the corresponding clauses in the European
specifications.
The maximum speed in deflction/span ratio limit curve in Eurocode specifications is 350km/h,and live load for bridge
design is UIC71 live load,as shown in Fig. 2.
Figure 2: Legend of UIC71 standard live load
UIC71 load should be taken into account in only one track for both single-track and double-track bridge when
examining deflection limits, and impact influence should be considered at the same time.
Deflection/span ration δ / L (corresponding to the comfort level of “Excellent”) limits can be seen in the Fig. 3, or
Table 2.
Table 2 Deflection/span ratio δ / L limits for simply-supported bridge including 3 or more spans
Span L (m)
Train Speed
V(km/h)
L≤15
15<L≤30
30<L≤50
50<L≤90
90<L≤120
V≤120
1/800
1/900
1/800
1/600
1/600
120<V≤160
1/900
1/1200
1/1200
1/800
1/600
160<V≤200
1/1000
1/1400
1/1500
1/1300
1/600
200<V≤280
1/1200
1/1500
1/2100
1/2100
1/1400
280<V≤350
1/1500
1/1600
1/2100
1/2400
1/2000
Note: Deflection/span ration cannot be greater than L/600 under any circumstance.
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Figure 3: Bridge Rigidity Required in the Eurocode Specifications according to Comfort Degree
3. Comparison of two kinds of specifications Take 32m double-track simply supported box girder bridge, popular
for high-peed railways, as an example, deflection limit in Chinese specification and European code can be compared.
In order to distinguish rigidity limits in two specifications under the same conditions, the corresponding limits in
Chinese specification can be modified into single-track UIC71 live load and the deflection/span ratio turned into
1/2400. If the influence of impact factor on limit in the European specification is gotten rid of, then the limit for tracks
with standard maintenance shall be 1/2364, close to 1/2400 in Chinese code. In this case, it can be considered that
regulations on vertical rigidity of bridges are basically same in the two kinds of specifications.
RESEARCH ON VERTICLE RIGIDITY OF BRIDGE WITH DYNAMIC EFFECT TAKING INTO
ACCOUNT
1. Model of train-bridge dynamic analysis
1) Bridge model Taking the 32m span double-track simply supported box girder bridge for Beijing-Shanghai
High-speed railway as an example, keeping the thickness of top, bottom and web plate unchanged, computational work
has been done with different cross-section characteristics (detailed values can be found in Table 3) corresponding to
different section depth.
Using two-joint space straight beam with uniform cross-section as basic element, with 3 linear and 3 angular degrees of
freedom at each node, the whole element has 12 degrees of freedom. In order to guarantee adequate incitation of
vibration, the bridge model of 10-span single supported beam has been established. While the influences of pier and
foundation rigidity are not taken into account. The second-phase dead load of double-track bridge is calculated as
18.5t/m.
Table 3 Characteristics of midspan section of 32m double-track simply supported box beam
2.6750
Flexual Inertia
Moment along the
Bridge
Direction(m4)
3.1697
Flexual Inertia
Moment at
Cross-bridge
Direction(m4)
80.7386
Depth between
Gravity Center
and Beam
Top(m)
0.6481
8.2636
3.1375
4.1305
82.1662
0.7156
2.2
8.4316
3.5790
5.2354
83.5938
0.7842
2.4
8.5996
4.0471
6.4885
85.0214
0.8538
2.6
8.7676
4.5083
7.8937
86.4490
0.9244
2.8
8.9356
4.8798
9.4549
87.8766
0.9960
3.0
9.1036
5.4544
11.1762
89.3043
1.0684
Beam Depth
(m)
Cross-section
Area (m2)
Twist Inertia
Moment
(m4)
1.8
8.0956
2.0
The rayleigh damping is adopted in this paper. Damping ratio is usually between 2-5% for concrete bridge, and 2% is
been taken for safety reason.
2) Train Model and Track Irregularity Train model is composed of several locomotives and rolling stocks. All rigid
motions of car elements are considered for defining the degree of freedom, that is: carbody and bogie have 5 degree of
freedom respectively, including bounce, lateral movement, roll, pitch and yaw. Each wheel-set has 2 degree of freedom
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as lateral movement and yaw. As for 4-axle train, each has 23 degree of freedom, for 6-axle one, each 27[2]. See Fig. 4
for vehicle computation model.
Figure 4: vehicle computation model
In order to simplify the analysis, we made the following assumptions for the train model:
(1) Car body, bogie and wheel-set are regarded as rigid elements connected to each other by linear springs and viscous
dashpots. Car body is symmetrical about center of mass in both left and right as well as front and rear;
(2) The train makes uniform motion along longitudinal axis of the bridge , disregarding effect of longitudinal dynamic
forces , wheel and rail keep in contact to each other.
Due to the fact that there is not high-speed lines in China until now, this paper uses German low-interference track
spectra to conduct simulation calculation. The generated irregularity sample has wavelength from 1m to 80m, and with
level irregularity altitude of 7.59mm, align irregularity altitude of 5.5mm, cross-level irregularity altitude of 3.95mm.
Conditions of train forming and speed as well as track irregularity for train-bridge coupling vibration analysis are
shown in Table 4.
Table 4 Conditions of Train-bridge Coupling Vibration Analysis
Train Type
Configuration
Calculated
Speed (km/h)
German ICE3 High-speed
Train with Power
Distributed
Configuration of
16 cars
250,280,300
(3M+1T)×4
,320,350
Rail Irregularity
sample generated by German low
interference spectrum, with dot distance of
0.25m and cut-off wavelength of 80m
3) Model of Wheel-Rail and Train-Bridge Coupling Vibration Analysis The paper adopts step-by-step integral method
to analyze train-bridge coupling vibration. Under the prerequisite that wheelset does not jump off the rail, wheelset
motion equation is established based on wheel-rail geometry theories and creep theory of wheel-rail contact. The
convergence condition is that relative error, produced after two iterative results of acting force between wheelsets and
rail, is less than allowable error[4]. This method can be used to solve non-linear dynamic problem and the dynamic
response problem generated when train goes on and leaves the bridge car by car.
2. Evaluation standard for train-bridge dynamic analysis Since dynamic analysis is indispensable for
determination of rigidity limit, assessment standard becomes extremely important. Assessment standards applicable to
train and bridge respectively are proposed on the basis of referring to current specifications and past main research
achievements.
1) Evaluation Standard For Running Safety and Riding Comfort This paper mainly discusses vertical dynamic
response, thus, only evaluation standard for vertical dynamic performances of bridge and train is determined.
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(1) Evaluation Index For Running Safety: This paper combines Chinese specification GB5599-85[3] standard and
relevant research achievements, and adopts safety evaluation index for load reduction rate of single wheel as:
ΔP / P0 ≤ 0.6
where P0 =static wheel load.
(2) Evaluation Index For Carbody Acceleration. Referring to literature [1] and past research achievements, this paper
defines assessment index of passenger train acceleration as: (a) Routine maintenance standard:
Vertical carbody acceleration aV ≤ 0.10 g
(b) Comfort management standard:
Vertical carbody acceleration aV ≤ 0.13g
(3) Evaluation Index For Riding Comfort. In light of “Railway Vehicles-Specification for Evaluation of the Dynamic
Performance and Accreditation Test GB5599-85”[3], stability index W is depended on to classify running stability
level of passenger train:
W≤2.5
excellent
2.5<W≤2.75
good
2.75<W≤3.0
qualified
2) Evaluation Response Limit of Bridge
(1) Acceleration Limit of Bridge. Referring to requirements on vibration acceleration of bridge in European
specifications, applied limit of dynamic bridge response is: As for single ballasted track, maximum vertical
acceleration amax ≤ 0.35g ; as for ballastless track, the value amax ≤ 0.5g
(2) Limit of Bridge Deformation. Although “Code for rating existing railway bridges”[6] does not define limit of
bridge deformation under high-speed running condition, in view of the fact that bridge deformation is reflected as
irregularity of rail finally, maintenance target value of track irregularity could be regarded as basic limit of bridge
deformation. Maintenance target value of Japanese Shinkansen is shown in Table 5.
Table 5 Maintenance target value of Japanese Shinkansen (1996) (mm)
JR-RTRI(draft)
Chord length
measured
Type of track
irregularity
240km/h
300km/h
40mchord
profile
10
versine
align
10
JR-east
JR-west
240km/h
JR-Tokaido
270km/h
270km/h
7
10
7
8
7
7
6
7
The table indicates that allowable limit of profile track irregularity is 7mm at 300km/h, since irregularity of track on
bridge includes deformations of deck and track structure, moreover, deformation of track structure is unavoidable,
thus, regarding bridges with a span no more than 40m, basic limit of vertical bridge deformation is assumed as 5mm,
based on which train-bridge dynamic analysis is conducted to determine final limit of vertical bridge rigidity.
3. Analysis on calculating results Table 6 and table 7 show train-bridge dynamic analyzing results for a 10-span 32m
simple supported beam that is passed by German ICE3 high-speed train.
It could be seen from Table 6 that all maximum dynamic deflections corresponding to beam depth ranging from 1.8m
to 2.4m exceed 7mm, vertical acceleration values approach or exceed limit value of 0.35g；maximum dynamic
deflection in solution with beam depth of 2.6m exceeds 5mm, impact coefficient reaches 2.150, these indicate presence
of strong vibration. So solutions with depths of 2.8m and 3.0m satisfy requirements of high-speed running in terms of
bridge dynamic response. Because type and parameter of China railway’s high-speed train are not determined yet, and
difference between selected irregularity spectra and future practical track condition is possible, therefore, vertical
bridge rigidity shall reserve appropriate margin for concern of safety. As for current solution with beam depth of 3.0m,
flexural moment of inertia in mid-span section is 120% of the value with depth of 2.8m. The solution with depth of
3.0m is reasonable with proper margin.
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Table 6 Bridge Response with A 32m Simple Supported Beam with Different Depth
Beam depth (m)
1.8
2.0
2.2
2.4
2.6
2.8
3.0
Train speed
(km/h)
Impact
coefficient
Dynamic
deflection(mm)
Vertical
acceleration (m/s2)
250
2.387
15.822
3.473
280
1.475
9.774
2.124
300
1.192
7.901
1.489
320
1.085
7.195
1.200
350
1.018
6.748
0.907
250
1.727
8.786
1.991
280
2.720
13.835
4.040
300
1.787
9.091
2.525
320
1.455
7.403
1.984
350
1.142
5.810
1.233
250
1.296
5.200
1.317
280
1.752
7.031
1.981
300
2.902
11.647
4.110
320
2.341
9.396
3.340
350
1.564
6.277
2.071
250
1.227
3.972
1.223
280
1.307
4.233
1.287
300
1.564
5.065
1.599
320
2.165
7.010
2.589
350
2.596
8.407
3.699
250
1.215
3.235
1.098
280
1.250
3.326
1.306
300
1.291
3.438
1.231
320
1.419
3.778
1.347
350
2.150
5.723
2.411
250
1.156
2.568
0.864
280
1.221
2.712
1.015
300
1.237
2.750
1.208
320
1.279
2.843
1.176
350
1.447
3.216
1.296
250
1.185
2.228
0.701
280
1.195
2.247
1.008
300
1.221
2.296
0.970
320
1.223
2.299
1.124
350
1.291
2.426
1.086
Table 7 indicates that as for solutions with beam depth from 1.8m to 2.4m, deformation of beam leads to greater
deformation of rail, wheel load reduction rate exceeds limit value of 0.6, running safety could not be met; in addition,
with regard to solutions with depth from 1.8m to 2.2m, vertical carbody acceleration exceeds limit of 0.13g, comfort
index is close to upper limit of 3.0 and exceeds the limit in some conditions, thus, it is concluded that vertical bridge
rigidity could not meet requirements of running safety and riding comfort.
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Table 7 Locomotive & rolling stock response with 10 span 32m simple supported beam
Motor car
Beam
depth
(m)
1.8
2.0
2.2
2.4
2.6
2.8
3.0
Train speed
(km/h)
Wheel load
reduction
rate
Vertical
acceleratio
n(m/s2)
Vertical
comfort
Wheel load
reduction
rate
250
0.690
1.384
2.854
0.665
Trailer car
Vertical
acceleratio
n
(m/s2)
1.353
280
0.570
1.322
2.549
0.593
1.282
2.472
300
0.542
1.306
2.573
0.571
1.289
2.454
320
0.554
1.309
2.647
0.589
1.264
2.535
350
0.603
1.346
2.721
0.650
1.305
2.608
250
0.487
1.282
2.518
0.488
1.229
2.464
280
0.636
1.374
2.938
0.639
1.264
2.686
300
0.580
1.311
2.649
0.629
1.267
2.528
320
0.545
1.283
2.598
0.578
1.242
2.517
350
0.559
1.272
2.674
0.592
1.252
2.587
250
0.411
1.153
2.460
0.424
1.138
2.429
280
0.470
1.220
2.569
0.491
1.160
2.459
300
0.641
1.434
3.190
0.688
1.283
2.714
320
0.576
1.269
2.759
0.630
1.251
2.641
350
0.538
1.274
2.633
0.581
1.229
2.543
250
0.394
1.100
2.452
0.408
1.086
2.406
280
0.427
1.125
2.556
0.438
1.093
2.468
300
0.445
1.150
2.610
0.478
1.108
2.498
320
0.513
1.202
2.750
0.521
1.182
2.601
350
0.557
1.249
2.820
0.613
1.271
2.653
250
0.377
1.069
2.445
0.389
1.077
2.429
280
0.412
1.095
2.505
0.430
1.057
2.452
300
0.422
1.103
2.548
0.457
1.082
2.479
320
0.455
1.122
2.651
0.490
1.139
2.523
350
0.492
1.179
2.765
0.549
1.200
2.729
250
0.370
1.050
2.417
0.381
1.065
2.433
280
0.408
1.072
2.463
0.419
1.065
2.439
300
0.413
1.083
2.524
0.452
1.077
2.473
320
0.440
1.098
2.521
0.484
1.123
2.485
350
0.480
1.113
2.666
0.527
1.168
2.541
250
0.365
1.037
2.414
0.381
1.041
2.410
280
0.401
1.057
2.462
0.416
1.060
2.434
300
0.413
1.066
2.498
0.451
1.070
2.482
320
0.438
1.079
2.498
0.481
1.111
2.456
350
0.480
1.098
2.540
0.521
1.155
2.478
Vertical
comfort
2.674
Regarding solutions with depth from 2.6m to 3.0m for a 32m simple supported beam, wheel load reduction rate meet
requirements under every conditions, vertical carbody acceleration is less than limit of 0.13g. Riding comforts is
qualified in the solution with beam depth of 2.6m, and becomes good in solutions with depth of 2.8m and of 3.0m.
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Considering the fact that substructure shall meet requirements of running at 420km/h, comfort index shall reach
excellent or good in the range of design speed, so as to reserve proper margin for higher speed. These requirements
could be met by both solutions with depth of 2.8m and 3.0m. Similar to bridge response, the solution with depth of 3.0
m reserves proper margin and to be regarded as an reasonable one.
CONCLUSIONS
(1) In Chinese “Temporary Provisional Regulations on design of newly built railway lines for passenger traffic of
300-350km/h” and European specifications, limits of vertical rigidity for multi-span simply supported beam are
basically the same.
(2) Multi-span arrangement is widely used for bridges of high-speed lines. The probability of two train passing through
the bridge simultaneously is greater, thus, for Chinese condition to take ZK live load on each line as design load for
double track bridge and put forward the rigidity limit of relevant bridge are suitable.
(3) Since there are no completed high-speed lines on Chinese railway yet, and the actual rail irregularity data and train
type are lacking, therefore, this paper adopts German ICE3 and low-interference spectra to make train-bridge dynamic
analysis. Nevertheless, determining of vertical rigidity limit for bridges shall reserve proper margin to adapt to possible
track and train conditions in Chinese railway.
(4) Taking bridge deformation as special track irregularity, the deformation limit of bridge for dynamic analysis could
be defined by referring to maintenance target value of track irregularity. By referring to provisions of Japanese
Shinkansen, 5mm dynamic deflection limit is adopting for bridges with span no more than 40m, and based on it
train-bridge dynamic analysis is made to define the final vertical rigidity limit of the bridge.
(5) Train-bridge dynamic analysis indicates that: as for a 32m simply supported beam, 2.8m is the minimum depth that
could meet requirements on limits of bridge deformation and acceleration, as well as running safety and riding comfort.
The corresponding ratio of deflection/span under ZK live load is 1/5000. With regard to current solution of 3.0m depth,
the flexural moment of inertia in cross-section of mid-span is 120% of the value in the solution of 2.8m depth. The
solution of 3.0m depth for the bridge is reasonable and with proper margin.
REFERENCES
1. Wu Wangqing. A Study on suggested values for track irregularity management criteria on 300 km/h
comprehensive experimental section of Qinshen Passenger Dedicated Railway. Railway Standard Design, 2003;
(4): 1-3 (in Chinese).
2. Gao Mangmang. Studies on Train-Track-Bridge Coupling Vibratiion And Runnablility of Train on High-Speed
Railway Bridges. Doctor Thesis, China Academy of Railway Science, Beijing, China, 2001 (in Chinese).
3. State Standard of PR China. Railway Vehicles-Specification for Evaluation the Dynamic Performance and
Accreditation Test. GB 5599-85, Railway Press, Beijing, China, 1985.
4. Shinozuka M, Jan CM. Digital simulation of random processes and its applications. Journal of Sound and
Vibration, 1972; 25(1): 111-128.
5. Ministry-Issued Standard of PR China. Temporary Provisional Regulations on Design of Newly Built Railway
Lines for Passenger Traffic of 300-350 km/h. Railway Press, Beijing, China, 2004.
6. Ministry-Issued Standard of PR China. Code for Rating Existing Railway Bridges. Railway Press, Beijing, China,
2004.
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