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In Sweden, were cold winters can be followed by rather hot summers, the temperature in the rail can vary from -40 to 55 degrees (C). Such large differences in temperature puts high demands on the rail. In such conditions, buckling of the rail can occur, a phenomenon known as a lateral buckling of the track (in Swedish "solkurva"). Here, a brief description of this phenomenon is given. Hopefully, it can be understood without any prior knowledge of mechanics.
When the temperature is rising, the rail is expanding. As an example, a rise in the temperature with 50 degrees (C) on a 50 meter long rail results in an expansion of 29 mm. If this thermal expansion is prevented (intentionally or unintentionally) the thermal heating will give rise to normal forces acting in the long direction of the rail. The magnitude of these normal forces is equal to the magnitude of forces necessary to compress the rail the length it would have expanded if this expansion was not prevented. In the example above, the forces necessary to prevent expansion would be equal to the forces needed to compress the rail 29 mm. The relation between the forces acting on the rail (F) and the expansion of the rail (d) can be written as
F/A = E*d/L
where A is the cross sectional area of the rail, E is Young's Modulus (different for different materials) and L is the length of the rail. The expansion of the rail (d) is linked to the rise in temperature by the relation
d = c*T*L
where c is the coefficient of thermal expansion (different for different materials), T is the rise in temperature and L is still the length of the rail. Thus, the forces in the rail can be expressed as
F = E*c*T*A
As an example, in a UIC60-rail (an international standard rail with a weight of 60 kg per meter) a force of 19 kN (that's the force you have to use to lift a weight of 1900 kg) is introduced for every degree the temperature is rising!
When the normal forces exceed a critical value, buckling of the rail will occur. You can try this yourself, in a slightly smaller scale, by pressing on the short sides of a ruler (watch out, the ruler might brake). You can say that the buckling of the rail is its way of "relieving the pressure". Also, the buckling of a rail is, just as buckling of a ruler, a very fast process.
Two basically different techniques in avoiding lateral buckling of the track are used in the jointed and all welded tracks.
In the jointed track, the rails are allowed to expand freely. At each end of the rail segment, there are open joints. The problem comes when the rail is expanding (or has moved) so much that these joints are closed. In this case, further heating will introduce normal forces, which the track is not constructed to resist. The result will, in many cases, be a lateral buckling of the track.
In the all welded track, no movement of rails is allowed. Thus, large normal forces are introduced and the track has to be constructed to resist these forces. The strategy in doing so is to use a lot of ballast with high inner friction, to use concrete sleepers and to use good fastenings.
So, what is the critical value of buckling? Well, that is not so easy to say. The basic rule is that the destabilising forces (i.e. the normal forces in the rail) should be equal to the stabilising forces, such as the pressure from the ballast on the track, the stiffness of the rail etc. However, several factors are influencing both types of forces. Here are some aspects:
The "neutral temperature" is the temperature at which there are no normal forces in the rail. This temperature is of great importance. You could choose to have this temperature very high and thus prevent the lateral buckling of the track (because the temperature will never rise much above this value). The problem is that temperatures below this value will introduce tensile stresses in the rail. This increases the risk of rail breakage due to cracks since it is the same effect as if something is pulling the rail in both ends. As a compromise, you try to find a temperature which will not give to large compressive forces at the highest temperatures and not to large tensile stresses at the lowest temperature. In Sweden, this temperature is chosen at about 15-20 degrees (C) (local variations).
In curves, part of the normal forces will try to "push" the rail out of the curve (i.e. towards the convex side). This will give an increase in the destabilising forces acting on the track and thus increase the risk of lateral buckling of tracks.
Some 60% - 70% of the stabilising forces are due to the influence of the ballast and sleepers. The influence of the ballast and sleepers is depending on the weight and the friction they can mobilise. Thus, on all welded tracks, concrete sleepers and rugged stones are used. Also, a ballast shoulder is placed outside the track, especially in curves. A good horisontal stabilisation (through ballast and sleepers) will also cause a possible lateral buckling to be a slower and less dramatic process. This increases the chance that the "suncurve" is spotted before it becomes so big that it may cause derailments.
A rail with larger area will be more stable. However, an increased area will give rise to increased forces acting on the rail. The solution to this dilemma is to shape the area in an "intelligent way" so that you will get out a maximum of stability from the area used. This is one of the reasons why rails have the shape that they have. Also, good fastenings are increasing the stiffness of the track enabling the rail and the sleepers to form a stiff "frame".
During track reconstruction, the stability of the track can be reduced, for instance due to removal of ballast. Also lateral movements of the track, in order to alter the track geometry, has a great influence of the acting forces.
So far, the consequences of lateral buckling of the track are not very severe even though they are occurring rather frequently. This is due to different factors. First, a lateral buckling of the track is not directly leading to a derailment. Modern trains can travel on a track with minor defects due to lateral buckling without derailing. Secondly, modern passenger trains are constructed to withstand a derailment without too severe damages. So far in Sweden, only 4 persons have died in derailments due to lateral buckling of the track and only 1 of them in "modern time". However, material damages of large values have been caused by lateral buckling of tracks.
The text above is mainly compiled from
[1] Fältman C., Solkurvor, Järnvägsinspektionen
The case of lateral buckling of tracks is closely related to the case of thermal buckling of pipelines (thanks to professor Hobbs for pointing this out), which you can read about in
[2] Hobbs R.E., In-service buckling of heated pipelines,
ASCE, Journal of transportation engineering, 110(2), 1984
[3] Hobbs R.E., Liang F., Thermal buckling of pipelines close to restraints,
In 8:th Int. conf. on offshore mechanics and arctic engineering, Vol.5,
pp. 121-127, Hague, 1989.
As for general buckling in general, and buckling of beams in particular, the most widespread textbook is probably
[4] Timoshenko S.P., Gere J.M., Theory of elastic stability,
2:nd Ed., McGraw-Hill, 1961.