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from the paper:
Viggo Tvergaard and Alan Needleman, "On localized thermal track buckling", Int. J. Mech. Sci, Vol. 23, No. 10, pp. 577-587, 1981
Fig. 1 Sketch illustrating the localized track buckling mode typically observed (top view)
Summary--The thermal buckling of a railroad track in the lateral plane is analyzed with the track modelled as an elastic beam resting on an elastic-plastic foundation representing the ballast. The nonlinearity of the resistance forces exerted by the ballast on the track is accounted for, both in the lateral and axial directions. For a perfectly straight track the critical bifurcation mode is a periodic one and the effect of periodic imperfections on the instability temperature is analyzed numerically. The transition to the localized buckling pattern observed in practice takes place by a bifurcation from the periodic deflection pattern. The transition to this localized mode can occur with only little growth of the periodic deflections. The instability temperatures for some tracks with various nonperiodic initial imperfections are also determined. It is shown that the instability temperature depends on both the magnitude and the form of the initial imperfections.
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