# Buckling Analysis of Interspersed Railway Tracks
## 2 Methodology
### Material Properties and Boundary Conditions  
It should be noted that the ballast lateral resistance, which is a major factor to resist track buckling, was based on the initial stiffness within the linear elastic zone obtained from single sleeper (tie) pull/push tests (STPT) that have been carried out by many researchers [45; 46; 47; 48]. The effects of ballast, such as shoulder height, shoulder width, crib height, ballast depth, ballast compaction, etc., play a significant role in the resistance in the lateral plane. Assuming the same ballast geometry, the differences between the lateral resistances of concrete and timber sleepers depend on the sleeper-ballast contact and friction, which are mainly affected by the sleeper shape, weight, and geometry. Hence, track lateral resistance mainly comes from ballast-sleeper interaction. It is noted that the three main contributions are sleeper side friction (crib), sleeper bottom friction, and end or ballast shoulder restraint. For these reasons, inconsistency in the lateral resistance of interspersed tracks needs to be considered for buckling analysis. Some researchers have studied the difference in lateral resistance between timber sleepers and concrete sleepers. The typical values of timber sleeper peak resistance are lower than those of concrete by about 500 lbs/sleeper [44]. Others have found that the lateral resistance ratio between timber track and concrete track is about 0.6-0.8 [41]. In this study, it was assumed that the ballast lateral resistance of timber sleepers is about 60% that of concrete sleepers as this represents the worst-case scenario for the lateral resistance of sleepers. The parameters used for lateral resistance were calculated from the slope of the load-deflection relationship obtained in the linear elastic zone. Apart from lateral resistance, other factors influence the buckling strength of railway tracks, such as torsional resistance, track curvature, track misalignment, etc. The torsional resistance is provided by the sleeper and rail connections via fastening systems. The main function of fastening systems is to resist rail creep and provide torsional and translational resistances to maintain rail gauge and track stability [42]. According to previous experiments, the resistance values for rail fastenings to timber sleepers are higher than those for concrete sleepers [27]. The different fastening systems and types of sleepers were tested [39]. It was found that the sleeper types do not influence torsional resistance, while the fastening systems for timber sleepers are much stronger than those for concrete, resulting in higher torsional resistance. Nominal values of torsional resistance per fastener were proposed [27]. It should be noted that the torsional stiffness of the fastening systems for concrete sleepers is about 40-70% that for timber sleepers [26; 27]. In this study, assumptions on the values of lateral and torsional resistance were drawn based on previous testing. The material properties are presented in Table 1. For the worst-case scenario, it was assumed that the lateral resistance, represented by the spring at one sleeper end, of the span of timber sleepers is 60% that of concrete sleepers, while the torsional resistance of the fastening on timber sleepers is 2.5 times that of concrete sleepers. In reality, track buckling can be often found in symmetrical (Figure 3a) or antisymmetrical (Figure 3b) shapes. Generally, there are two regions, consisting of buckled regions and adjoining regions. The buckled zone is the zone of change in shape of the track geometry in the lateral plane, while the rails are deformed longitudinally in the adjoining zone. In the analytical solutions for track buckling [37], the track consists of a buckled region (\(0\leq x\leq l\)) and adjoining region (\(l\leq x\leq l_{0}\)) which has a length of a. Subscripts 1, 2, and 0 represent the buckled regions 1 and 2 and the area beyond the adjoining regions, respectively. The differential equations are solved to get the resulting equations.