**Joint resistance**

The normal rail joints are designed to allow the rail length variation due to temperature. To do this the joints have a well-defined maximum gap and a set of installation parameters to provide an optimum behaviour at temperature variation and a good maintenance regime.

Any modern rail joint has a standard bolt tightening torque from which R, the installation joint resistance force, can be computed. This installation value can vary between 100 and 200 kN and often, in the thermal force calculations, is presumed to be reduced by 50% in a long term working regime (Radu – 1975).

The real value of this resistance force varies depending on the lubrication of the joint but is influenced also on the repeated loading of the joint due to the railway traffic and other factors. The passage of the vehicles over the joint is causing significant impact forces which, in long term, tend to reduce the bolts torque, wear the contact areas but also are affecting the contact conditions between the fishplates and the rails. Due to these factors the real value of the joint resistance force can vary from practically null (when the bolt are loose) to almost 3-4 times the theoretical installation value, behaving as a “frozen joint” which is keeping the joint gap locked even for significant temperature variation.

To show this resistance in a common knowledge illustration, we can say that 100 kN is roughly produced by the weight of 4-5 BMWs X5 (ok, in USA 4-5 Dodge Durango). *If a joint resistance force is slightly above this value, the tare weight of 4-5 cars (≈ 10 t) will not vary the joint gap of a rail placed as in the figure below:*

From thermal calculations perspective, this means that the rail thermal force needs to be above R to vary the joint gap and a minimum rail temperature variation ΔT_{R} is required to cause the gap variation:

**Track longitudinal resistance**

Another important track parameter is the longitudinal resistance force produced at each sleeper by the friction forces between rail and fastening (P_{1}) fastening and sleeper (P_{2}) and sleeper and ballast (P_{3}). As explained in another article, this force will develop at the level of minimum resistance, which, for ballasted track, is between the sleeper and the ballast.

This longitudinal resistance force can be null for free thermal expansion track superstructure (where the rail fastening do not provide a reliable clamping force) but it can reach values of 5 to 10 kN/m for modern track components. For a normal track layout, depending on the rail length, to activate the rail longitudinal resistance on the entire rail length, is required a rail temperature variation ΔT_{p} . A sample set of values for ΔT_{p} is presented in the following tables:

**Track thermal force diagrams**

For a temperature increase the rail will tend to expand which will activate both the joint resistance force and the track longitudinal resistance force the force diagram will be as follows:

In the rail will be a compression stress from this early stage.

If the temperature is decreasing the rail will tend to contract and a tension stress will appear in the rail:

*This is the most significant difference between the two theoretical types of railway track superstructure.* For the free thermal expansion superstructure a thermal stress in the rail will only appear if the joint is closed or open at maximum gap. The restrained thermal expansion track superstructure will develop thermal forces for any temperature variation from the installation.

**Joint gap domain**

The joint gap breathing of a free thermal expansion superstructure is defined by a linear variation between the minimum and the maximum joint gap.

For a restrained thermal expansion track superstructure the joint gap variation is more complex and is defined by a domain of values:

Even for an ideal track, for almost any rail temperature Ti, the joint gap is not defined as a single value but by a range, varying between Gti and Gci.

Also, depending on the temperature variation that brought the rail to Ti, the rail stress is either compression or tension.

**Significance of track parameter variation**

The variation of these two main parameters – Joint Resistance Force and Track Longitudinal Resistance – is affecting significantly the rail thermal behaviour and the thermal forces that are developed in the rail within the normal range of rail temperatures.

A temperature variation ΔT_{G} can be defined to measure the distance between the Compression and Tension edges of the joint gap domain:

Also, for each rail temperature Ti within the normal range of temperature variation, the joint gap in ideal condition can be within a range ΔG = Gci – Gti. This ΔG is indicatively:

All these calculations are for CEN56 rail and a normal ballasted track structure.

Practically, in the rail there will be compression stress if the rail temperature is increasing by more than ΔT_{G} – no matter what was the initial temperature (except the extremes). If the temperature is increasing by more than ΔT_{G} the joint variation is placed on (or around) the C1-C2 branch.

Similarly, in the rail there will be tension stress if the rail temperature is decreasing by more than ΔT_{G} . If the rail temperature is increasing by more than ΔT_{G} the joint variation is placed on (or around) the T1-T2 branch.

If this temperature increase (or decrease) is greater than 2(ΔT_{R} + ΔT_{p}) an ideal track will be for sure on the on the compression C1-C2 branch, or, respectively, on the tension T1-T2 branch.

The point C1 in the graph above can be at 0°C or even below and, as strange as might sound, if the temperature is increasing, the track will have compression stress even at this low temperature. If the joint and longitudinal resistances are high, the compression stress will be also significant.

In the ideal track behaviour, if the temperature is increasing further from C1, the rail stress will stay constant and the rail will expand freely until the point C2 (joint closure) has been reached. Only then the rail stress will increase further.

But, the real track is influenced by a wide variety of factors and the rail stress can sometimes grow to the buckling risk level when the rail temperature is starting to evolve toward the hot range, even without reaching the maximum temperature or without closing the joint gap. This is the reason why is very important even for jointed track to have well-defined rail temperature management measures in place and why caution actions are required as soon as the hot season is approaching.

One other important parameter – not yet discussed – is the track lateral resistance. This “gap” will be filled soon!

**References**

- Alias, J. (1984).
*La voie ferrée, techniques de construction et d’entretien*. SNCF – Eyrolles, Paris, France. - Radu, C. (1975).
*Suprastructura căii. Partea I. Probleme.*Institutul de Constructii Bucuresti, Romania. - Giunta, M. (2013).
*Corso di infrastrutture Ferroviarie. Lecture 02.1 – Termica del binario.*Universita “Mediteranea” di Regio Calabria, Italia. - Yakovleva, T. G. (1999).
*Железнодорожный путь (Railway tracks – RU)*. Transport, Moscow, Russia.