# Jointed track response to rail temperature variations

The thermal behaviour of the jointed track can be analysed throughout a full annual temperature variation and used to define the joint expansion gap variations. The joint gap varies between the maximum value and zero and this is related to the way the rails are allowed by the track resistance forces to contract and expand in the joint. The theoretical envelope of the possible joint gap variations can be defined based on track thermal response calculations (Radu – 1975), considering this temperature variation cycle: installation temperature → continuous increase to maximum rail temperature (53 °C) → continuous decrease to minimum rail temperature (-14°C) → continuous increase to maximum rail temperature (53 °C).. This will define for an ideal track the envelope of the joint gap temperature variation. All the daily gap variations will be theoretically contained within this envelope.

For a restrained thermal expansion jointed track formed by long rails the full temperature cycle will be defined by 14 main stages presented below.

The main assumptions of this calculation are:

• The track is considered ideal, with all the track components working perfectly.
• All the rails are installed at the same temperature and using identical ideal track components.
• The cross-section symmetry of the track allows the calculation for one rail instead of the full track. The track longitudinal resistance is considered for one rail, half of the value of the force for full two rail track, but we will not refer to it as “rail longitudinal resistance” to highlight its nature – it is the resistance to rail movements provided by different track components. This resistance force is acting at different levels of the track however for this calculation is considered in the rail axis, not causing any bending on the rail.
The force distribution along each rail is presumed to be symmetrical longitudinally, with the rail mid-point being the centre of symmetry. Consequently, this is presumed immobile throughout the entire process. The joint gap calculations consider only the variation of the two abutting half-length rails.
• The track is presumed free of any traffic loads throughout the entire temperature cycle.
• The joint resistance force for all the joints is constant for all joints.
• The track longitudinal resistance is constant along the rail and throughout the entire complete temperature cycle.

The stage drawings show the thermal force diagrams, the force distributions on the rail, the formulas and the calculation results for all the stages discussed in this example.  The compression forces are represented conventionally above the reference line while the tension forces are shown below. The joint gap variation is calculated considering the expansion or contraction for both rails of the joint. This is based on the thermal linear expansion equation, taking into account the evolution of the resistance and thermal forces from one stage to the next.

During the full temperature cycle, the track will pass through the following main stages (Radu – 1975):

Stage description Force diagram
Stage 0. Installation

At installation the rail is stress free. The joints are tightened with rail fastenings installed but no internal thermal stress or external resistance forces are active.

From this point the temperature is increasing and the next main stage is reached when the joint resistance force is fully activated.

Stage 1. Joint resistance force activation

The rail expansion and joint gap variation only occur when the rail thermal force increases sufficiently to overcome the joint resistance force (N1 = R). Over the entire rail length there is a constant compression force.

If the rail temperature increases further, the joint friction forces are overtaken by the thermal force and the joint gap variation begin to develop.

Stage 2. Activation of the track longitudinal resistance on the entire rail length

The activation of the longitudinal resistance, p, occurs gradually, sleeper by sleeper, from the rail ends towards the middle.

During the activation of the track longitudinal resistance the rail expansion is restrained at each sleeper. Once the resistance of each sleeper is overcome, the rail expands and moves, meeting the resistance of the following sleeper. The process takes place from both rail ends towards the middle.

At this stage the two resistance forces (R and p) are fully active and there is no other resistance to oppose the rail expansion. The next main stage of this process is closure of the joint gap.

Stage 3. Joint gap closure

The temperature variation required to reach this stage needs to expand both rails to close the joint gap and consume the gap left from stage 2. From stage 2 to 3 the rail thermal stress does not increase and the force diagram is identical with the result at stage 2. The track resistance forces are active and rails are expanding freely until the joint gap is closed.

Stage 4. Maximum rail temperature

Once the joints are closed no further rail thermal expansion is possible and any further temperature increase will result in higher rail thermal forces. Rail movement in the joint is no longer possible and the thermal forces are transferred from one rail directly to the other, without the mediation of the joint fishplates.

This compression force increases and reaches a maximum value when the rail reaches the maximum rail temperature.

From this stage the rail temperature decreases continuously to the minimum rail temperature.

Stage 5. Zero compression force at joint

As the temperature decreases, at some point the rail will tend to contract but for this to begin the compression force at the joint must first decreased to zero. When this point is reached no compression force is transferred through the rails at the joint although both rails are still in contact.  The compression force at the joints is null but the longitudinal resistance p maintains a variable compression force along the rail with a maximum value at the rail mid-point.

The rail will tend to contract due to the further temperature decrease and this activates the static friction force at the joint which opposes this tendency.

Stage 6. Activating the joint resistance force

As the temperature decreases the joint resistance force opposing the rail contraction is activated. The joint is still closed but, along the rail there are sections of tension stress – close to the joint – and residual compression stress in the mid-section of the rail.

Beyond this point, with the joint resistance activated, any decrease in temperature will allow the rail contraction and the joint will start to open. The next resistance to oppose the rail contraction is the track longitudinal resistance.

Stage 7. Reversing the track longitudinal resistance on the entire rail length

As the rail temperature decreases further, the track longitudinal resistance at each sleeper reverses its direction, opposing rail contraction.

The track longitudinal resistance, p, is reversed on the entire rail length to oppose the rail contraction. During this stage the rails contract, opening the joint gap. As the temperature continues to decrease the rail continue to contract freely until the joint gap is fully open.

Stage 8. Joint open to maximum gap

The free contraction of the rails opens the joint to the maximum gap. The decrease in rail temperature required to achieve this can be calculated based on the required gap variation.

The maximum gap of the joint is achieved when the bolts come in contact with the joint holes of the rails and of the fishplates. When this contact is made, further joint gap variation ceases and any increase of the tension thermal forces will affect the bolts through shear forces.

Stage 9. Minimum rail temperature, -14°C

When the joint is fully open, further contraction of the rail is prevented by the joint bolts and the further decrease in temperature increases the rail tension force. This next stage of the process is reached at the minimum rail temperature.

From this stage the rail temperature increases continuously towards the higher end of the temperature range.

Stage 10. Zero tension force at joint

As the temperature increases, at some point the rail will tend to expand and for this to begin, the rail tension force at the joint must decrease to zero and only then the joint resistance force opposing the rail expansion can be activated.

At this stage the joint bolts are free of shear force but still in contact to the fishplate and rail holes.  The longitudinal resistance, p, maintains a variable tension force along the rail with a maximum value at the rail mid-point.

Following further temperature increase, the rail will tend to expand and this activates the resistance force at the joint opposing this tendency.

Stage 11. Activating the joint resistance force

As the temperature increases further, the joint resistance force opposing rail expansion is activated. The joint is still open at the maximum gap and on the mid-section of the rail there is residual tension stress and compression stress close to the joint.

Beyond this point, with the joint resistance activated, the joint gap begins to close following further temperature increase. The next force to oppose the rail contraction is the track longitudinal resistance.

Stage 12. Reversing the track longitudinal resistance on the entire rail length

As the temperature increases, the track longitudinal resistance at each sleeper reverses its direction, opposing rail expansion. The track longitudinal resistance, p, is reversed on the entire rail length. During this stage the rails expand at the joint reducing the gap.

Once this stage is reached, there is no resistance to rail expansion and as the temperature continues to increase the rail will expand freely until the joint gap is fully closed.

Stage 13. Joint gap closure

The temperature variation required to reach this stage closes the joint gap left from stage 12. The increase in rail temperature can be calculated based on this joint gap variation. The joint closure temperature (JCT) is identical to the one which can be calculated for stage 3.

From stage 12 to 13 the rail thermal stress does not increase and the force diagram is identical with the one resulted at stage 12. However, even though the rail is expanding freely until the joint gap is closed, the track resistance forces are active.

The resulting rail temperature and thermal force diagram at stage 13 are identical to stage 3. If the temperature increases further to the maximum temeprature the result will be identical to stage 4.

Stage 13 closes the ideal cycle.

The cycle of rail temperature variation can be presented in a temperature/joint gap graph (Radu – 1975, Alias – 1984, Giunta – 2013):

The graph shows the delayed rail response to temperature variation, caused by the presence of the resistance forces. The process analysed here is based on the continuous variation between the extreme rail temperatures and defines a theoretical envelope loop of the joint gap variation. For any rail temperature, if the joint gap is not closed or fully open, there will be a range of values defined inside this envelope loop.

The theoretical ideal evolution of the rail thermal forces throughout the temperature cycle is also shown in the figure second graph. Compared to a free thermal expansion track, the thermal forces of a restrained track are significantly different and also define a range within which the rail forces can vary.

For a free thermal expansion track the thermal force will appear only after the joint gap is closed or fully open. The restrained thermal expansion track, by comparison, retains thermal forces throughout the entire temperature cycle. The magnitude of these retained forces is dependant on the track resistance forces.

A more detailed description of this calculation will be published in the next issue of the Journal of the Permanent Way Institution.

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