10
Pergamon CQ 1994’&.vicr Science Ltd Printed in Great Britain. IMPACT IN A RAILWAY TRUSS BRIDGE T. L. WANG Department of Civil and Environmental Engineering, Florida International University, Miami, FL 33199, U.S.A. (Received 30 July 1992) Abstract-The aim was to investigate the dynamic interactions between an open deck steel truss bridge and a moving freight train. A non-linear, lOO-ton (gross weight 131.5 tons), freight car vehicle model and the 200-ft, open deck, Warren-type steel bridge model were used in this study. Equations of motion for the vehicle, bridge, and bridge/vehicle interactions were also presented. The track irregularities on the approach and the bridge were generated from power spectral density functions for Federal Railroad Ad~~stration (FRA) class 4 track (maxims speed 60 mph). Both zero and two percent of the critical damping were assumed for the bridge. Impact percentages in the bridge due to a three, loll-ton freight car train operating at 20, 40, and 60 mph were calculated. These were compared with the data obtained from an earlier field investigation and those specified by the American Railway Engineering Association (AREA) specifications. 1. INTRODU~ON Dynamic loads in railway bridge members are devel- oped due to the interaction between the moving vehicle(s) and bridge structure. The problem of rail- way bridge vibrations caused by moving vehicle(s) has been studied for a long time. A brief review of the literature can be found in [l&6]. The field investigation of a 200.ft long, open deck steel truss bridge was studied by the Association of American Railroads (AAR) [7]. It consists of stress and impact analyses of various critical members of the bridge. The impact values that were obtained were signi~~ntly lower than AREA-specified values. The objective of this study is to investigate the dynamic interaction between an open deck steel truss bridge and a moving freight train by using the computer modeling technique [5]. The impact values are proved by comparison with the experimental data 171. A brief description of the vehicle and bridge models is given first. Then, bridge/vehicle interactive equations and rail irregularities are introduced. The results of a case study are given, which show the impact percentages and stress time-histories of some critical members of the bridge. They are also com- pared with the test results studied by AAR [7]. 2. VEHICLE AND BRIDGE MODELS 2.1. Vehicle model A vehicle model (refer to Figs l-3) was developed to represent a conventional freight car, consisting of a carbody, two bolsters and two truck assemblies, each of which comprised of two-wheel-axle sets. In the model, the carbody was assumed to be rigid and assigned five degrees of freedom, corresponding to the vertical (y) and lateral (z) displacements, rotation about the verticle axis (yaw or $), rotation about the transverse axis (pitch or e), and rotation about the longitudinal axis (roll or 4). Each bolster was as- signed three degrees of freedom, corresponding to vertical and lateral displacements and roll motion. Each truck frame/axle set was treated as single rigid mass in the lateral and yaw directions, whereas in the verticle direction, one-fourth of the mass of the truck frame/axle set was considered to be concentrated at each wheel and treated as sprung mass. Each sprung mass was assigned one degree of freedom, corre- sponding to vertical displacement. The total degrees of freedom in the model was 23. However, only 21 independent equations of motion were used because two equations were obtained by constraining the lateral motion between the carbody and each bolster. All of the geometric and suspension nonlinearities of the freight car were included in the vehicle model. The total potential energy, V = Xvi, of the system is then computed from the spring stiffness and rela- tive displacements, whereas the dissipation energy, D = ZDi, of the system is obtained from the damping forces. The total kinetic energy, T = CT, of the system is calculated using the mass, mass moments of inertia, and rotational as well as translational vel- ocities, of the system components. The moment of inertia of all components is assumed to be constant and the weight of each component is considered as the external force on that component. The equations of motion of the system are derived, using Lagrange’s formulation, as follows: _T+V+D=*, 34, aqi a4i (1) 1045

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Page 1: Impact in a railway truss bridge

Pergamon CQ 1994’&.vicr Science Ltd

Printed in Great Britain.

IMPACT IN A RAILWAY TRUSS BRIDGE

T. L. WANG

Department of Civil and Environmental Engineering, Florida International University, Miami, FL 33199, U.S.A.

(Received 30 July 1992)

Abstract-The aim was to investigate the dynamic interactions between an open deck steel truss bridge and a moving freight train. A non-linear, lOO-ton (gross weight 131.5 tons), freight car vehicle model and the 200-ft, open deck, Warren-type steel bridge model were used in this study. Equations of motion for the vehicle, bridge, and bridge/vehicle interactions were also presented. The track irregularities on the approach and the bridge were generated from power spectral density functions for Federal Railroad Ad~~stration (FRA) class 4 track (maxims speed 60 mph). Both zero and two percent of the critical damping were assumed for the bridge. Impact percentages in the bridge due to a three, loll-ton freight car train operating at 20, 40, and 60 mph were calculated. These were compared with the data obtained from an earlier field investigation and those specified by the American Railway Engineering Association (AREA) specifications.

1. INTRODU~ON

Dynamic loads in railway bridge members are devel- oped due to the interaction between the moving vehicle(s) and bridge structure. The problem of rail- way bridge vibrations caused by moving vehicle(s) has been studied for a long time. A brief review of the literature can be found in [l&6].

The field investigation of a 200.ft long, open deck steel truss bridge was studied by the Association of American Railroads (AAR) [7]. It consists of stress and impact analyses of various critical members of the bridge. The impact values that were obtained were signi~~ntly lower than AREA-specified values.

The objective of this study is to investigate the dynamic interaction between an open deck steel truss bridge and a moving freight train by using the computer modeling technique [5]. The impact values are proved by comparison with the experimental data 171.

A brief description of the vehicle and bridge models is given first. Then, bridge/vehicle interactive equations and rail irregularities are introduced. The results of a case study are given, which show the impact percentages and stress time-histories of some critical members of the bridge. They are also com- pared with the test results studied by AAR [7].

2. VEHICLE AND BRIDGE MODELS

2.1. Vehicle model

A vehicle model (refer to Figs l-3) was developed to represent a conventional freight car, consisting of a carbody, two bolsters and two truck assemblies, each of which comprised of two-wheel-axle sets. In the model, the carbody was assumed to be rigid and

assigned five degrees of freedom, corresponding to the vertical (y) and lateral (z) displacements, rotation about the verticle axis (yaw or $), rotation about the transverse axis (pitch or e), and rotation about the longitudinal axis (roll or 4). Each bolster was as- signed three degrees of freedom, corresponding to vertical and lateral displacements and roll motion. Each truck frame/axle set was treated as single rigid mass in the lateral and yaw directions, whereas in the verticle direction, one-fourth of the mass of the truck frame/axle set was considered to be concentrated at each wheel and treated as sprung mass. Each sprung mass was assigned one degree of freedom, corre- sponding to vertical displacement. The total degrees of freedom in the model was 23. However, only 21 independent equations of motion were used because two equations were obtained by constraining the lateral motion between the carbody and each bolster. All of the geometric and suspension nonlinearities of the freight car were included in the vehicle model.

The total potential energy, V = Xvi, of the system is then computed from the spring stiffness and rela- tive displacements, whereas the dissipation energy, D = ZDi, of the system is obtained from the damping forces. The total kinetic energy, T = CT, of the system is calculated using the mass, mass moments of inertia, and rotational as well as translational vel- ocities, of the system components. The moment of inertia of all components is assumed to be constant and the weight of each component is considered as the external force on that component.

The equations of motion of the system are derived, using Lagrange’s formulation, as follows:

_T+V+D=*, 34, aqi a4i

(1)

1045

Page 2: Impact in a railway truss bridge

1046 T. L. WANG

Fig. 1. Side view of the vehicle model.

where qi and gi are the generalized displacements and velocities. Details of derivation are presented in [5].

2.2. Bridge model

A 200-ft long, open deck, single track, riveted steel through truss railroad bridge was adopted in this study. The bridge has been used in the field investi- gation study [7J. The details about the bridge were shown in Fig. 4.

The bridge was modeled as a simply supported space truss with pin-connected joints. All the bridge members, including the floor beams, were treated as axial force members. Each joint in the bridge was assigned three translational degrees of freedom (DOF). The structural stiffness matrix of the bridge was generated using the direct stiffness approach [8]. For setting up the mass matrix of the bridge, a lumped mass model was used. Since the interaction between the bridge and vehicle(s) occurs at the bridge deck level, the verticle and lateral DOF correspond- ing to the lower chord joints of the space truss were retained in the analysis. The remaining DOF were

condensed from the mass and stiffness matrices of the bridge, using the Guyan condensation scheme [9].

The equation of motion of the bridge can be written as

wfB1I4s~ + v?91bhJ + Mh#I = VW1 (2)

in which [M,] is the condensed mass matrix, [KB] is the condensed stiffness matrix, {FBy} is the vector of vertical and lateral interactive forces between the bridge and vehicle(s), {q8}, {gB}, {ds} are displace- ment, velocity, and acceleration vectors, correspond- ing to the primary DOF at the lower chord joints, and [DB] = 20,c,[M,] is the bridge damping matrix. o, is the fundamental natural frequency of the bridge and c, is the percentage of critical damping for the bridge.

3. BRIDGE/VEHICLE INTERACTION

The vertical interactive force is given as

F’,,=(1/4).m,,. (g - tiByi) + F$t * F&r (3)

Side frame /

T- 4 4

Left rail

Sl b

I , 1L / I/ 3

Q CX

Axle #2 Jltl Axle Ul

Sl b Bolster

wheel #3 wheel#l _ _ Right rail

I L’ tl

I- LZ --I, Lz

-I

Fig. 2. Top view of the front truck in the vehicle model.

Page 3: Impact in a railway truss bridge

T

Impact in a railway truss bridge

i’ %-&WC

Car body

1047

Flange clearance

l- b b

Sl a1 -I

M ,_ Cl

Fig. 3. End view of the front truck in the vehicle model.

where tn,, is the mass of the Jth truck (J = 1 indicates the front and J = 2 the rear truck); g is the gravita- tional acceleration; tiB,,i is the bridge acceleration under the ith wheel; and F& is the vertical force due to the suspension springs as shown in Fig. 3. Details of the equation can be found in [5]. FBY is the additional vertical force due to lateral interactions. From Fig. 5

&, = (F6 + F’B: ‘) (s;lc). (4)

The value of lateral force & is given in the next section.

The deflection wByi was related to deflections at neighboring panel nodes k, k + 1, k’, and k’ + 1 by linear interpolation (refer to Fig. 6).

The lateral interactive force between the ith wheel and the rail is given as

F:, = KBri ’ U,zi i- D,zi . ir,, (5)

in which U,zi is the relative lateral displacement at the ith wheel/rail contact point, KBli is the lateral stiffness of the bridge/track system under the ith wheel, and D,2i is the lateral track damping under the ith wheel.

The bridge lateral displacement and force under the ith wheel were related to the lateral nodal dis- placements and forces nodes k, k + 1, k’, and k’ + 1 by linear interpolation.

4. RAIL IRREGULARITIES

The geometric parameters used to quantitatively describe the rail irregularities are: vertical profile (&,), cross level (2&), alignment (&), and gage (2&J, as shown in Fig. 7. If R and L refer to the right and left rails, respectively, then the vertical (u,) and lateral (u,) irregularities are expressed as

The power spectral density (PSD) functions devel- oped by Hamid et al. [IO] for wave lengths from 10 to 1000 ft were used to generate the rail irregularities. These PSD functions are as follows.

For the vertical profile and alignment irregularities

S(4) = MmJ2 + 43 4”W + 43

(74

Page 4: Impact in a railway truss bridge

1048 T. L. WANG

Ul u7

(a) Rear view

8 @ 25’ - 0” = 200’ - 0”

(b) Front view

4 4

(c) Bottom bracings

(d) Top bracings

Fig. 4. Dimensions, joints, and members for the 200-ft long, single track, riveted, railway truss bridge.

i I i

i i i

I

I

i

I

i

i

i

I

I

I

'i4

f

I

I

I

I i I

I

i

i

I I

I i

I

I

I

%

I- i F DY Fny

Fig. 5. Partial cross-section showing additional vertical nodal forces due to lateral interaction.

Page 5: Impact in a railway truss bridge

Impact in a railway truss bridge

k’th node . @

Q’ + 1)tll node ._. ._. ._. ._. _,_._.-.-._._._._. .-.-.-.-.-.-.-

FnY (do-) up Eof re1r truss

-

1049

di Left rail

C

I, ith wheel Right rail 1, di

di

Fig. 6. Plan view of the wheel load position in a bridge panel and additional vertical nodal forces due to lateral interaction.

For the cross-level and gage irregularities 5. TRACK SUPPORT STIFFNESS

where S(4) is the PSD (in2/cycle/ft); 4 is the wave length (cycle/ft); c$,, c#+, are the break wave lengths (cycle/ft); and A is the roughness parameter (in2- cycle/ft).

The values of A, 4,) C#J~ and & for Federal Railroad Administration (FRA) class 4 track are listed in Table 1. The rail irregularities from these PSDs were generated by using the method similar to that dis- cussed by Wiriyachai [6]. In this study, the sample length was taken as 512 ft and 1024 (2”) data points were generated for this distance. The random num- bers for vertical profile, alignment, cross level, and gage irregularities were then generated. The average vertical and lateral irregularities for each rail com- puted from 25 simulations are shown in Fig. 8 for class 4 track.

5.1. Vertical stiffness of track on the approach and

the bridge

The vertical track stiffness per rail on the approach was assumed to be 125 kips/in [1 11. The vertical stiff- ness of the bridge/track system was taken as that of the bridge itself. It was found that the stiffness of the track structure, consisting of rails, wooden ties, stringers, and floor beams, is much higher that that of the bridge. When this stiffness was combined with the bridge stiffness, the bridge stiffness dominated [l]. The bridge stiffness is given by the term [KB]. {qs} in

eqn (2).

5.2. Lateral st@ness of the track and the track bridge system

Lateral stiffness of the track &) was taken as 83.33 kips/in on both the approach and the bridge [l I]. The lateral stiffness of the bridge (KBzi) was determined as follows. A lateral unit load was

Crosr level

*&.&

Horizontal “*

Alignment

Gage

Fig. 7. Definition of track alignment, gage, cross-level, and vertical profile.

Page 6: Impact in a railway truss bridge

1050 T. L. WANG

Table 1. Spectrum model constants for FRA class 4 track Kzi = l/Azi. The combined stiffness

Profile Cross-level Alignment Gage

A x IO-' c#. x IO-’ lpb x 10-Z

1.40 7.10 4.00 0.74 7.10 4.00 0.89 10.00 5.60 0.89 8.90 7.10

obtained as I& was then

(8)

Note: units A: in2 cycle/ft, rj~,,, &: cycle/ft.

applied at the ith point between ith and (i + ljth

6. SOLUTION OF EQUATIONS OF MOTION AND IMPACT FACTOR

.I I

nodes. The simple beam reactions on these nodes The equations of motion of the vehicle(s), bridge, cause lateral deflection. The lateral displacement and bridge/vehicle interaction were solved by using a Azi under the ith wheel was determined from fourth-order Runge-Kutta scheme, with an inte- the linear interpolation of the calculated deflections. gration time step of 0.00024 sec. Such a small time The lateral stiffness of the bridge was taken as step was necessary to avoid numerical instability.

200 300 400 500 c Distance along track (FT)

(a) Vertical irregularity, right rail

200 300 400

Distance along track (FT)

(b) Vertical irregularity, left rail

1 500 600

0 100 200 300 400

Distance along track (FT)

(c) Lateral irregularity, right rail

500 60(

0 100 200 3io 400 Distance along track (FT)

(d) Lateral irregularity, left rail

500 600

Fig. 8. Vertical and lateral rail profiles for class 4 track based on PSD spectra.

Page 7: Impact in a railway truss bridge

Impact in a railway truss bridge 1051

t 7

2 3 4

Time (set)

Fig. 9. Amplification factor for bottom chord member L,L, with 0% of the critical bridge damping and operating speed of 60 mph.

2 3

Time (set)

Fig. 10. Amplification factor for top chord member U, U., with 0% of the critical bridge damping and operating speed of 60 mph.

E 0.8

&! 0.7

.g 0.6

.Fj 0.5 % -& 0.4

$ 0.3

0.2

0.1

2 3

Time (set)

Fig. Il. Amplification factor for end post member b U, with 0% of the critical bridge damping and operating speed of 60 mph.

Page 8: Impact in a railway truss bridge

1052 T. L. WANG

0.8

"0 ;j 0.7

' 0.6 s f= 0.5 0" 5 0.4

z $ 0.3

0.2

-0.1 : 0 I 2 3

Time (set)

Fig. 12. Amplification factor for diagonal member U,L, with 0% of the critical bridge damping and operating speed of 60mph.

The real percentage of impact acquired from the The impact values were also computed by using the study is defined as formulas given in the American Railway Engineering

Association (AREA) specifications [12]. The AREA

Imp(%) = [(R&J&J - I] x 100 (9) empirical formulas are as follows:

in which R,,,,, and R, are the absolute maximum responses for dynamic and static studies, re-

Imp(%)=~+40-g, for L <8Oft (11)

spectively. The amplification factor A, and A, for dynamic

and

(R,,) and static (R5) responses are, respectively

4 = &IR,, A, = W&m. (10)

Imp(%) = 00 -t- 16 +$, for L > 80ft (12) s

where s is the center-to-center distance between the The typical amplification factors for the selected trusses, in ft, and L is the span length of the stringer, members are shown in Figs 9-12. when calculating impact for hanger, ffoor beams, and

Table 2. Comparison of the impact in selected members at various speeds for 0% and 2% of the critical bridge damping, with class 4 track irregularities on both the approach and the bridge

Maximum Vehicle static live load Swed

0% of the critical 2% of the critical bridge damping bridge damping

Maximum Maximum dynamic live dynamic live

load force Imnact load force Imoact Member (kiP) (mph) (kip) (%) (kip) &,

L2L 20 - 346.41 1.18 - 346.03 1.07 bottom - 342.36 40 - 357.43 4.40 - 355.65 3.88 chord 60 - 359.20 4.91 - 358.81 4.80

GfJW 20 356.42 1.71 355.43 1.43 top 350.43 40 368.76 5.23 366.76 4.66 chord 60 317.20 7.64 375.39 7.12

L,U,, -98.19

20 - 102.37 4.26 - 101.72 3.60 hanger 40 - 102.39 4.28 - 102.16 4.04

60 -111.13 13.18 - t 10.52 12.56

&VI, 20 266.62 0.88 265.90 0.60 end 264.30 40 271.81 2.84 271.01 2.53 post 60 283.95 7.44 283.07 7.10

f.JiL2,

208.85 20 -212.20 1.60 -212.03 1.52

diagonal - 40 -209.57 0.36 -210.09 0.61 60 - 209.62 0.37 -208.08 -0.37

Note: Train of three IOO-ton loaded cars.

Page 9: Impact in a railway truss bridge

Impact in a railway truss bridge

- Area-specified vaIuer

1053

40- LlU2

30- Area design

20-

‘i

[ IO-

.S 0 4 ii- I OQra &_I(, o &

.t

L2Us

3 30- Iz

Area derign

2O-

0 10 -

Ama design

J-l% t

Area design. 44.0

Speed in mph

Fig. 13. Comparison of computed, AREA-specified, and experimental impact results for open deck, steel truss bridge on the Great Northern Railway, with a 200-ft span.

stringers; and span length of the bridge, For all other agreement between the computed and experimental

members. values.

7. SUMMARY AND CONCLUSIONS

The bridge was subjected to dynamic load from a three loo-ton freight car train-consist, operating at 20,40, or 60 mph. The train-consist occupied almost the entire length of the bridges. Both zero and two percent of the critical damping for the bridge were studied in the analysis.

A summary of the impact percentages in selected member is given in Table 2. It may be noticed that the impact percentages increased with speed. It can also be seen that bridge damping slightly decreased the impact.

The computed impact percentages in critical mem- bers, such as the bottom chords, top chords, end posts, and diagonals are plotted in Fig. 13, together with a comparison with AREA-specified values [IZ] and test results 171. The impact values for both zero and two percent of the critical bridge damping are generally below the AREA-specified values which are 44.0% for hangers and 24.7% for the other members. It may also be seen that there is a good

REFERENCES

1. M. H. Bhatti, Vertical and lateral dynamic response of railway bridges due to nonlinear vehicle and track i~egula~ties. Ph.D. thesis, Illinois Institute of Technol- ogy, Chicago, IL (1982).

2. K. H. Chu, V. K. Garg and C. L. Dhar, Railway bridge impact: simplified train and bridge model. J. Srrucr. Div. ASCE 105, 18234844 (1979).

3. K. H. Chu, V. K. Garg and T. L. Wang, Impact in railway prestressed concrete bridges. J. Strucr. Engng, ACE 112, 1036-1051 (1986).

4. V. K. Garg, K. Ii. Chu and T. L. Wang, A study of railway bridge/vehicle interaction and evaluation of fatigue life. J. Earthquake Engng Struct. Dynam. 13, 689-709 (1985).

5. T. L. Wang, Impact and fatigue in open-deck steel truss and bagasted prestressed concrete railway bridges. Ph.D. thesis, Illinois Institute of Technology, Chicago, IL (1984).

6. A. Wiriyachai, Impact and fatigue in open-deck railway truss bridge. Ph.D. thesis, Illinois lnstitute of Technol- ogy, Chicago, IL (1980).

7. Field Investigation of a Truss Span on the Great Northern Railway. Association of American Railroads, Report No. ER-81, Chicago, IL (1968).

Page 10: Impact in a railway truss bridge

1054 T. L. WM4G

8. A. S. Hall and R. W. Woodhead, Frame Analysis. ministration, Report No. DOT-FR-82-03, Washington, Robert E. Kreiger, Huntington, NY (1980). DC (1981).

9. R. J. Guyan, Reduction of stiffness and mass 11. Y. H. Tse and G. C. Martin, Flexible body railroad matrices. Am. Inst. Aeronaut. Astronaut. Jnl 3, No. 2 freight car. Report No. R-199, Association of American (1955). Railroads, Chicago, IL (1976).

10. A. Hamid, K. Rasmussen, M. Baluja and T.-L. Yang, 12. Manual for Railway Engineering. American Railway Anaiyticai deceptions of track geometry variations. En~n~~ng Association, Chapter 15, 15 1 -I to 1 S-9-24 Department of Transportation, Federal Railroad Ad- (1972).