**ΔG = αLΔT°. Free expansion**

For a *free thermal expansion jointed track* the rails expand and contract freely and the track components do not provide any resistance to oppose this rail length variation. The joint gap varies linearly relative to the rail temperature.

The figure below presents the joint gap variation for a jointed track formed by 18.288 m rails, installed at 15°C and joint gaps of 8 mm.

In the figure is highlighted a theoretical rail temperature variation for one day after installation.

It starts at the installation: 15°C, with joint gaps of 8 mm.

The rail temperature is assumed to increase to 40°C that day. The rails freely expand in the joint. The expansion can be calculated using the free thermal expansion formula ΔG = αLΔT°. For the 25°C temperature increase from 15°C to 40°C the joint gap will very by:

ΔG = αLΔT° = 1.15 10^{-5} x 2 x 18.288/2 m x (1000 mm/m) x 25 = 5.3 mm

The rails are assumed to expand freely and equally relative to their mid-point. This is why we have 2 (two rails) x 18.288/2 (half rail length) in the equation.The joint gaps are reduced to 8 – 5.3 = 2.7 mm.

During the night the rail temperature decreases to 10°C. Using the same calculation we can evaluate the joint gap variation to be 6.3 mm. The joint gaps to 9 mm. The next morning the rail temperature reaches again 15°C. The rails expand and consume 1 mm from the joint gaps, returning to the 8 mm joint gaps installed the previous day.

For the simplified theoretical model of the *free thermal expansion track superstructure* this theoretical rail temperature cycle can be run with any other temperature variations. Every time the rail temperature will reach again 15°C the joint gaps will be 8 mm. The linear relation between rail temperature and joint gap will always be maintained.

This model presumes that the joint gap will always be the same if measured at the same temperature. In the day of installation, next day, months after, the well-behaved rails will expand and contract freely and the gap variation will follow the dark green line shown in the graph.

As you can imagine, dear reader, this is too good to be true … because the rails are subjected (at least) to …

**Restrained expansion**

The *free thermal expansion* calculation model assumes that no resistance is opposed by any of the track components to the rail thermal length variation. This over-simplifies the reality, even for old types of track components. The fishplated expansion joint provides a resistance force to rail expansion and even old fastenings are developing some rail clamping forces, preventing the rail from creeping. The modern track components are designed to provide well defined resistance forces to rail expansion and contraction; therefore the calculation presented above does not accurately describe the behaviour of modern track components. The other calculation model – *restrained thermal expansion track superstructure* – needs to be considered. The calculation process was described in a previous blog article – Jointed track response to temperature variation – and, in a greater detail, in an article presented in the next issue of the Journal of the Permanent Way Institution.

The same day, in the life of a *restrained thermal expansion track*, will evolve differently.

Lets consider the same installation parameters (temperature of 15°C and joint gaps of 8 mm) and a track formed by 18.288 m CEN 56 rails. The restrained thermal expansion is ensured at the joints by the joint resistance force. For a joint bolt installation torque of 275 Nm the (installation) joint resistance force is around 100 kN. The track is assumed freshly tamped which will provide a low longitudinal resistance force. We assume that to be 5 kN/m of rail.

If, after installation the rail temperature evolves in the same cycle as in the above example (15°C → 40°C → 10°C → 15°C), the track will pass through the joint gap cycle shown below:

**0**, installation 15°C, gap 8 mm**1**, activating the joint resistance force, R, opposing the rail expansion**2**, activating the track longitudinal resistance, p**3**, maximum temperature of the day, 40°C, joint gap is 4.6 mm**4**, the temperature is decreasing and the compression force at the joint is null**5**, activating the joint resistance force,R, opposing the rail contraction**6**, activating on the entire rail length the track longitudinal resistance, p, opposing the rail contraction**7**, minimum temperature of the night, 10°C, joint gap is now 7.7 mm**8**, temperature increase to 15°C, 24 hours after installation. The joint gap remains 7.7 mm.

The joint resistance force is bringing the track closer to the behaviour of the *free thermal expansion track* presented at the beginning of this article. At 15°C and 24 hours after installation, the joint gap is reduced only by 0.3 mm and in the rail there is a tensile force with a maximum of -59 kN. This status is dependant on the history of temperature variation. We see that, at 40°C the joint gap was reduced to 4.6 mm compared to the 2.7 mm calculated above for a *free thermal expansion track* structure.

Lets consider the same track with the same parameters except one – we assume now a joint bolt torque of 475 Nm which will produce a joint resistance force of around 180 kN, at installation.

For the same temperature cycle, the stages, in this case, are as follows:

**0**, installation 15°C, gap 8 mm**1**, activating the joint resistance force, R, opposing the rail expansion**2**, activating the track longitudinal resistance, p**3**, maximum temperature of the day, 40°C, joint gap is 5.8 mm**4**, the temperature is decreasing and the compression force at the joint is null**5**, activating the joint resistance force,R, opposing the rail contraction**6**, minimum temperature of the night, 10°C, joint gap is now 6.1 mm**7**, temperature increase to 15°C, 24 hours after installation. The joint gap remains 6.1 mm.

At this stage, 24 hours after installation the rails have expanded and reduced the joint gaps by 1.9 mm. However, in the rails there is now a tensile force with a maximum of -135 kN.

It took only 24 hours for this track to produce this apparent anomaly. The joint gap at the same temperature as the installation, 24 hours after, is in this case measurably different from the installed one. This difference is dependant on the evolution of the rail temperature. If the maximum rail temperature that day would have been only 30°C, the joint gap next day, at 15°C, would be almost the same as the installed one, around 8 mm.

A real track, which has variable and non-homogeneous resistance forces, influenced by traffic and other factors, will have an even more complex behaviour than the one described here. But, as this theoretical calculation proves, for a track installed with components able to provide resistance to thermal expansion, the behaviour of the track, the joint gap variation and the status of the rail thermal forces is dependant not only on the current temperature conditions but also on the history of rail temperature variation. Other factors bring their contribution also.The resistance forces enable the track to retain thermal forces. The next day after installation these forces are already significant, even though the rail temperature has not reached extreme values.

*At installation the thermal forces through the length of the rail are consistently null. This state will never return naturally throughout the service life of the track. The thermal forces will never be consistently equal through the entire length of the rail, unless joint gap resetting or any similar maintenance works are undertaken.*

that is not a bad idea, Robert!

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I like your titles.

Wondering what may be next. A Welder’s Hard Day’s Night? 😉

Very interesting article!

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