Key Words Longitudinal sleeper, Ladder sleeper, Ladder track, Ballasted track, Continuously welded rail, Track Buckling, Lateral stability

Transcription

1 Lateral Stability Analysis of Ladder Sleeper Track Systems By Ernest J. Barenberg Professor Emeritus-Civil Engineering University of Illinois 205 N. Mathews; Urbana, Illinois 61801, U.S.A. Tel ; Fax Dr. Hajime Wakui and Nobuyuki Matsumoto Railway Technical Research Institute of Japan Dr. Yulin Bao Former Graduate Student at University of Illinois Summary The Railway Technical Research Institute of Japan is evaluating a unique ballasted rail-track system. The system consists of pre-stressed concrete sleepers placed parallel to the rails with one sleeper under each rail. Transverse sleeper spacing is maintained by heavy metal pipes cast between the sleepers at approximately three meter intervals. In plan view, the transverse pipe-spacers gives the track system the appearance of a large ladder. Hence the track system is referred to as a ladder sleeper track system. A 3-D finite element program, called ILLIBUCKLE, was developed by Dr. Bao while at the University of Illinois to analyze the lateral stability of track systems with conventional crossties. The program was modified and used to evaluate buckling and lateral stability of the ladder sleeper track system. This paper provides a brief description of the modified finite element program, the type and number of inputs required and typical outputs from the program. Over 200 different ladder sleeper track configurations with continuous welded rail were analyzed for buckling and lateral movements of curved track using the modified ILLIBUCKLE program. Results of these analyses are summarized. Results from the analyzes indicate that for track systems with the same lateral resistance, the ladder sleeper track system has far greater lateral stability and is far less sensitive to initial lateral rail imperfections than rail track systems with conventional cross-ties. Key Words Longitudinal sleeper, Ladder sleeper, Ladder track, Ballasted track, Continuously welded rail, Track Buckling, Lateral stability 1

2 1. Introduction The Railway Technical Research Institute of Japan has developed a unique railway track system called a ladder sleeper track system(1). The system consists of prestressed concrete sleepers placed parallel to the rails, with the horizontal spacing maintained by heavy steel pipe cast into the concrete sleepers at approximately 3 meter intervals. The University of Illinois used a modification of the finite element program ILLIBUCKLE to analyze the lateral stability and buckling resistance of the proposed ladder sleeper track system. Results of these analyses are contained in this paper. 2. Description of Ladder Sleeper System Figures 1 and 2 show the general configuration for the ladder sleeper track system. Since the steel pipe spacers are at 3 meter intervals, the length of each segment will normally be some multiple of 3 meters, i.e. 6m, 9m, or 12 m. The longest segment is usually 12 m to facilitate handling of the segments. Shorter segments are used in the curves to facilitate placing the rail in a smooth continuous arc. The segments are fastened end to end with one of several types of fasteners designed for this purpose. The rail is continuous welded rail and is fastened to the longitudinal sleeper with normal rail-to-tie clips (i.e. Pandrol- e clips). Figure 1 Ladder sleeper track Figure 2 Typical cross section of longitudinal sleeper 3. Analysis Model and Inputs The basic model to analyze the lateral stability of the ladder sleeper is the same model developed by Bao(2) for conventional track systems with cross-ties. The model was revised to compensate for the different geometry of the ladder sleeper system. The revisions required were: 1) redefining the track system geometry, 2) developing the appropriate finite element meshes to describe the ladder sleeper system, and 3) defining the necessary inputs for the model. Criteria for buckling and lateral track movement in curved track segments were the same as used by Bao in his model for track with conventional cross-ties. The model was developed using a commercially available finite element program called ABAQUS (3) (A license is required for this program). The rail in both the ladder sleeper and the conventional cross-tie track systems was simulated using the B3IOS element in ABAQUS, which has 7 degrees of freedom, while the longitudinal sleeper in the ladder sleeper system was simulated using the B31 element with 6 degrees of freedom. Rail-padfastener system was simulated using the JOINTC element in ABAQUS. In the JOINTC 2

3 element, 6 springs are used to simulate the stiffness of the rail-pad-fastener system. Spring elements were also used to simulate the lateral, vertical and longitudinal resistance of the ballast and subgrade. The spring elements can simulate resistance with as either linear, bilinear or nonlinear properties. Stiffness of connectors between ends of the sleepers was simulated using linear spring elements. For each track segment modeled there are: 4 rail elements, 4 ladder sleeper elements, 2 JOINTC elements, 4 ballast lateral resistance elements, 4 ballast vertical resistance elements, 4 ballast longitudinal resistance elements. The large number of inputs required for the program (up to six thousands lines) necessitated that a program be written to facilitate the inputs. For the original ILLIBUCKLE program, the input program was written in fortran, but for the ladder sleeper track system the input program was was written in C ++. Due to space limitations this program is not shown here. 4. Typical Results 4.1 Introduction A number of cases of ladder sleeper track structures were analyzed using the ABAQUS and RIKS programs in ABAQUS. The cases included both straight and curved track, cases with no lateral rail imperfections and both straight and curved track with lateral rail imperfections. In all a total of 204 individual cases were analyzed. Too many cases were analyzed to discuss the results individually. Instead, the results have been grouped into broad categories and results within the categories discussed with emphasis on those factors that affect the results. Where possible, results from the analyses on the ladder sleeper track systems were compared with results obtained by Bao for track systems with conventional cross-ties. 4.2 Sleeper End Restraints-Straight Track End restraints between the sleepers were modeled as linear springs. End spring restraint was varied from 1000 pounds per inch to 1,500,000,000 pounds per inch. The highest value essentially represents the condition of direct contact between sleepers, that is, the condition where there was no gap between the sleeper ends. Results shown are from analysis made on straight track with a uniform lateral resistance of 500 pounds per inch and no lateral track imperfections. The effect of end sleeper restraint on curved track is presented later. Figure 3 shows a typical buckled shape of a straight ladder sleeper track system after reaching the buckling temperature. The shape of the buckled track was similar for all end restraint levels for straight track. The important aspect of these results is that the sleeper end connection stiffness had no apparent effect on the buckling temperature of the ladder track system. Conventional track structures do not have comparable end restraint factors for comparison. 3

4 4.3 Lateral Restraint-Straight Track Lateral resistance for the ladder sleeper track system was shown to have a significant effect on the track buckling temperature. For these analyses the lateral resistance was allowed to vary from 50 to 500 pounds per inch. The ladder sleepers were analyzed with a gap and no connection between sleepers. A summary of results from analyses of ladder sleeper straight track systems with no lateral imperfections is shown in Figure 4. As seen in the figure, for a constant neutral temperature, there is an almost linear increase in the buckling temperature with increasing lateral resistance. This result emphasizes the need to maintain a high lateral resistance for the ladder sleeper track system. Figure 3 Typical buckled shape-straight track Figure 4 Effect of lateral restraint-straight track Buckling temperature for the ladder sleeper track system is significantly higher than the buckling temperature for track systems with conventional cross-tie system with the same lateral resistance. For example, with a lateral resistance of 500 lb/in, the buckling temperature above the neutral temperature for straight track with the ladder sleeper system with no lateral imperfections is shown to be approximately 275 o F. According to Bao, for the conventional track structure under comparable conditions, the buckling temperature above the neutral temperature is approximately 92 o F. This is a difference of approximately three to one in terms of the buckling temperature above neutral. 4.4 Lateral Resistance-Curved Track Lateral resistance for curved ladder sleeper track was shown to have a significant effect on lateral track stability. For these analyses the lateral resistance was varied from 50 to 500 pounds per inch. These systems were analyzed with a gap between ends of the sleepers and with no connections between the sleepers. Lateral resistance was assumed to be uniform and there were no lateral imperfections in the track structure. Both 5 degree and 10 degree curves were analyzed. A summary of the effect of lateral resistance on track buckling temperature for a ladder sleeper track structure with a 5 degree curve is shown in Figure 6. For the track structure with 5 degree curve there was a nearly linear increase in the buckling temperature with an 4

5 increase in lateral resistance above approximately 100 pounds per inch. Below approximately 100 pounds per inch lateral resistance there was almost no effect of lateral resistance on the buckling temperature. Figure 5 Effect of lateral resistance-5 deg. curve Figure 6 Effect of lateral resistance-10 deg. curve The shape of the buckled ladder sleeper track system with a 10 degree curve is identical with the shape for the straight track and track with 5 degree curve. For ladder sleeper track systems with both 5 and 10 degree curves, the location of the apparent buckle is at the ends of the sleeper. This suggests that the shape of the buckled track system will be a series of essentially straight track segments with sharp angles at the ladder sleeper ends. A summary of the effect of lateral resistance on track buckling temperature for a track structure with a 10 degree curve is shown in Figure 6. With track structures having a lateral resistance above approximately 200 pounds per inch, there was a nearly linear increase in the buckling temperature with increasing lateral resistance. However, a lateral resistance below approximately 200 pounds per inch there was an inverse relationship between the buckling temperature and lateral resistance. The difference in buckling temperature/lateral resistance relationship is probably due the greater ability of the track structure to move in and out with the buildup in forces when the lateral resistance is low. For track structures with both 5 and 10 degree curves, the buckling temperature for the ladder sleeper track was quite high. Direct comparisons with conventional track structures are very difficult, as for most conventional track structures the analyses by Bao were made with 4 or more ties missing. There is no comparable situation for analysis of the ladder sleeper system. Also, the buckling/lateral displacement responses for the two systems are not identical. However, making the best comparisons possible, it appears that the ladder sleeper track system will sustain a significantly higher temperature increase above neutral temperature in the rails without experiencing buckling than the conventional track system. As with straight track system comparisons, it appears that the ladder sleeper track system will sustain between 2 and 3 times the increase in temperature above the neutral temperature without buckling over that obtained from the analysis of conventional systems. 4.5 Sleeper End Restraint-Curved Track A series of analyses were also made on track structures with 10 degree curves in which an end restraint was placed between the ends of the ladder sleepers. For these analyses the 5

6 lateral track resistance was varied from 50 through 500 pounds per inch with no initial lateral imperfections. A typical result from this analysis is shown in Figures 7 for a ladder sleeper tracks systems with 10 degree curve and end sleeper restraints of 100,000 pounds per inch. The shape of the buckled track structure for mode 1 buckling is identical with the shapes shown earlier for both straight and curved track (Figure 3). As before, the buckled shape appears to be a series of straight track segments with sharp angle changes in the track system near the ends of the sleepers. The effect of lateral resistance on buckling temperature of ladder sleeper track systems with 10 degree curves and end restraints of approximately 2,00,000 lb per in and 1,500,000,000 are shown in Figures 8 and 9 respectively. Figure 7 Effect of lateral resistance-10 deg. Curve; End restraint = 100,000 lb/in Figure 8 Effect of lateral resistance-10 deg. Curve; End restraint = 2,000,000 lb/in As seen in Figures 7, 8, and 9, when the end restraint is placed between the sleepers the change in buckling temperature is no longer a linear function of the lateral resistance but changes with the level of end restraint. The probable reason for the buckling temperature to decreases with increasing lateral resistance at low levels of lateral resistance is the ability of the track structure to move in and out in the curves with the change in longitudinal forces. Clearly there is a significant impact on the behavior of the ladder sleeper track structure when high end-restraints are placed between the sleepers. However, the buckling temperature remains high so there is very little danger of buckling for the ladder sleeper track system even when the sleeper ends are highly restrained. As with the straight track analysis, there is no comparable situation with the sleeper end restraints with the conventional track systems. Therefore no direct comparisons can be made. 6

7 Figure 9 Effect of lateral resistance-10 deg. Curve; Figure 10 Effect of lateral resistance on End restraint =1,500,00,000 lb/in lateral displacement 4.6 Curved Track Systems with non-linear Lateral Resistance. A series of analyses were made of ladder sleeper track systems with both 5 and 10 degree curves, using a non-linear buckling analysis. Purpose of this analysis was to evaluate the lateral movement of the track system in the curves prior to the track system reaching the buckling temperature. For this analysis, the RIKS analysis in ABAQUS was used. All systems had a uniform lateral resistance and no initial imperfections. All analyses started at the neutral temperature and the temperature allowed to increase in 5 degree increments. A summary of the results from these analyses on track movement is shown in Figure 10. Note that the maximum deflection of the curved track system is highly dependent on the lateral resistance and the temperature above the neutral temperature. A maximum deflection of two inches occurred at a temperature of 20 degrees F above the neutral temperature for a track system with only 50 pounds per inch, whereas, it required a temperature increase of approximately 100 degrees F to get this same deflection when the lateral resistance was 500 pounds per inch. These results clearly indicate the need to maintain a high lateral resistance to maintain good track alignment and appropriate curve geometry. Comparable analyses were not run to determine the effect of decreasing temperatures below the neutral temperature, but logic indicates the movements should be essentially a mirror image of the deflections due to increasing temperature. The only difference is that with decreasing temperatures there is no tendency for the track structure to buckle. A summary of the deflection versus temperature increase and lateral resistance for ladder sleeper track systems with a 10 degree curve is shown in Figure 11. For this system, a temperature increase of nearly 30 degrees F is required to produce a lateral movement of 2 inches for the track system with 50 pounds of lateral resistance, and a temperature increase of approximately 70 degrees F required to produce the same two inch movement with a track system having a lateral resistance of 500 pounds per inch. The results show there is a clear difference in the track system response to temperature for ladder track systems with 5 and 10 degrees of curvature. Further analyses would be required to fully understand the implications of the degree of curvature on track lateral movement. 7

8 Figure 11 Effect of temperature and lateral resistance on rail displacement 4.7 Initial Imperfection Analyses A series of analyses were made on straight and curved ladder sleeper track systems having an initial lateral imperfection in the track system. In each case the imperfection was represented by a half-sine imperfection with a lateral displacement of 1 inch. Results of these analyses indicate there was almost no discernable decrease in the buckling temperature when the lateral imperfection was included in either the curved or straight track. These results are somewhat different than results obtained by Bao for track systems with conventional cross-ties. Bao found that the inclusion of lateral imperfections of the type described above in track systems with conventional cross ties had a significant impact on the final results. 4.8 Effect of Missing Rail Pad Fasteners Analyses were made to evaluate the effect of lost rail pad fasteners on the lateral stability of ladder sleeper track systems. Analyses were run with zero, 3 and 5 rail pad fasteners missing. In each case the missing pads were adjacent to each other, and were missing from the center of a ladder sleeper segment. Results from these analyses are shown in Table 1. Note that as the rail pad fasteners were eliminated, the buckling temperature of the ladder sleeper system decreased from approximately 350 o F with no fasteners missing to 280 o F with three fasteners missing to 238 o F with 5 fasteners missing. While these results are still very high, they indicate the need to maintain all fasteners to keep the ladder sleeper track system stable. Figure 12 shows the deflected shape of a ladder sleeper track system after buckling when 5 fasteners are missing. Note that the buckled shape of the track is significantly different from the shapes shown earlier. With the 5 fasteners missing, there is much less symmetry in the buckled shape, and the angle changes in the rail are much sharper, suggesting a greater danger for the operating trains. While not enough runs were made to evaluate all situations, the results shown suggest that if only one or two fasteners are missing, they will have little effect on the stability of the ladder sleeper track system. As the number of 8

9 adjacent fasteners missing increases, however, the missing fasteners could have a profound impact on the lateral stability of the ladder sleeper track system. Figure 12 Deflected buckled shape of ladder sleeper with 5 rail tie fasteners missing. 5 Concluding Remarks A 3-D finite element model developed for analysis of ladder sleeper track systems is presented. The model is capable of analyzing the ladder sleeper track responses due to vertical (train) load, longitudinal track load, transverse load, and load due to change in rail temperature. Track buckling, which is a track response due to increases in rail temperature, is the primary thrust of the analyses presented in this paper. Factors which were evaluated for their effect on buckling of ladder sleeper track systems are: 1. Sleeper end restraint for both straight and curved track; 2. Lateral restraint of both straight and curved track systems with both linear and non-linear lateral restraints; 3. The effect of track lateral imperfections; and 4. The effect of missing rail fasteners. Some of the factors evaluated had a significant effect of track buckling temperature, and other factors had almost no impact on the buckling temperature. Where possible, comparisons were made on the effect the factors on ladder sleeper track systems with the effect of similar factors on conventional track systems with cross ties. In general, it is noted that for the same lateral resistance, the ladder sleeper track systems had a much greater resistance to lateral movement than the track systems with conventional cross-ties. This includes a higher tolerance for temperature increase above neutral temperature without track buckling. For most analyses, the ladder sleeper track system showed an increase in temperature of from two to three times greater than for conventional track systems before buckling. The ladder sleeper track system was also less sensitive to loss of several rail/pad fasteners on buckling temperature. Direct comparison of the loss of rail/pad fasteners with conventional track systems is not valid, but relative comparisons indicate the lower sensitivity of the ladder sleeper track system to loss of three to five consecutive rail pad fasteners. 9

10 The level of sleeper end restraint had a greater effect on curved ladder sleeper track systems than on straight systems. However, the level of sleeper end restraint was not significant, even with end restraint up to 1,500,000,000 lb/in, which is equivalent to having direct contact between the sleeper ends. It is interesting to note that for all cases except when five or more consecutive rail/pad fasteners were missing, the lateral deflections during buckling always occurred at the ends of the ladder sleepers. This suggests that it may be possible to develop a track stiffener system to be installed at the ladder sleeper ends to further increase the buckling stability of the ladder sleeper track system. 10

11 References 1. Wakui, H., Matsumoto, N., Asanuma, K., Reed, D.M., and Laine, K.J., Performance of Ballasted Ladder Track under Heavy Axle Loads at TTCI, Proceedings of 7 th International Heavy Haul Conference, Brisbane, Australia, June 2001, pp Bao, Yulin, Three Dimensional Stability/Lateral Shift Analysis of Continuous Welded Rail (CWR) Track and Innovative Methods to Enhance CWR Track Performance, PhD Thesis, University of Illinois, ABAQUS/STANDARD Version 5.3, Hibbitt, Karlsson & Sorensen, Inc,

12 Table 1 Effect of missing rail tie fasteners on buclking temperature Designation Number of Missing Fasteners Buckling Temperature (See Notes 1,2,3,4,5,6,7) Ls_Imp1 No fasteners missing 354 o F Ls_no3 Three fasteners missing 280 o F Ls_no5 Five fasteners missing 238 o F Notes: 1. Straight track. 2. Gap with no fasteners between sleeper ends. 3. Lateral resistance 500 lb/in. 4. Neutral Temperature 80 o F. 5. No initial imperfections. 6. Fasteners missing from middle of sleeper. 7. Linear Buckling analysis in ABAQUS 12