TECHNICAL ARTICLE

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The Journal January 2017 Volume 135 Part 1

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An introduction to rail thermal force calculations

Article 1

AUTHORS:

Constantin Ciobanu

CEng, FPWI

Principal Track EngineerWSP | Parsons Brinckerhoff

Levente Nogy

CEng, MICE, FPWI

Senior Design EngineerNetwork Rail

This article is the first in a series of articles intended to present complementary information to what is currently available in the formal continuous welded rail (CWR) training courses and Railway Group or Network Rail Company Standards. This first article presents the general principles and theoretical considerations on the rails affected by thermal variation.

INTRODUCTION

This series of articles will discuss the general principles and theoretical considerations on railway track which is subjected to thermal effects. These can be used by Track Engineers for a better understanding of the physics behind the rails’ behaviour when subjected to temperature variations and can also help to understand the specifications and requirements mandated in the current standards covering the management of jointed respectively CWR track. Multiple thermal force diagrams will be presented for both jointed track and CWR applications.

THE PHYSICS OF RAIL THERMAL EXPANSION

When exposed to temperature variations, the steel rail tends to vary in length and, if nothing is constraining this length variation, the rail’s mid-point will stay fixed and each abutting half-rail length will expand equally by ΔL/2 (figure 1).

The total length variation can be computed as:ΔL = α•L•ΔT

Where:

- ΔL is rail extension- α is the expansion coefficient of the rail steel = 1.15•10-5 per °C for normal grade rail (NR/L2/TRK/3011).- L is the rail length - ΔT is the rail temperature variation

If the temperature is increasing, the rail will expand. If the temperature is decreasing, the rail will be subjected to a negative expansion – a contraction. The standard nominal rail length is defined for +15°C and length measured at other temperatures is corrected to take into account the expansion or contraction of the rail (BS EN 13674-1:2011).

On the other hand, if the expansion is not permitted and both ends of the rail are fixed, the thermal variation ΔT will generate a constant compression internal force, N (figure 2).

This internal force is defined by:N = (α•E •ΔT)•A = σ•A

Where:

- E is the steel’s elasticity modulus (2.1•108 kN/m²) - A is the area of the rail section- σ is the normal stress generated by the temperature variation, ΔT.

For a positive thermal variation of ΔT= 27°C the compression force would be 500 kN. This is the equivalent of an average freight car weighing 50 tonnes, pressing axially on a single CEN60 rail (figure 3).

The rail alone is unable to withstand this high compression force without deflecting at some point along its length and thus the entire track superstructure must be designed to provide the required stability and to avoid buckling in hot weather and rail breaking in cold weather.

TRACK PARAMETERS INFLUENCING RAIL THERMAL BEHAVIOUR

The behaviour of the track due to temperature variations is influenced by numerous and complex parameters (UIC Code 720). This article focuses on a simplified approach and considers only the main parameters in evaluating the rail thermal behaviour:

• Installation parameters• Rail type• Rail temperature• Track longitudinal resistance• Joint minimum and maximum gap - for

jointed track• Joint resistance - for jointed track• Expansion joint gap or adjustment switch

gap - the range of thermal “breathing” of the devices used on the transitions between sections of CWR track or between CWR track and jointed track.

The entire process presented here presumes an ideal and homogenous track structure.

Table 1: Cross sectional area (A) for different rail types used in the UK

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INSTALLATION PARAMETERS

Track installation defines the starting point of the track thermal behaviour. From this perspective, the main track installation parameters are the following:

• joint installation gap and installation temperature for jointed track

• stress free temperature (SFT) for CWR track

RAIL TYPE

The rail type influences the thermal behaviour through the cross sectional area (A) which defines the amount of thermal force developed in the rail due to the temperature variation. (See table 1)

RAIL TEMPERATURE

The rail temperature has a direct influence on the rail thermal stress and force. The rail temperature is in relation to the air temperature but is also influenced by other factors. The location of the track has a significant impact - if the rail is exposed to direct sunlight it will absorb more caloric energy and its temperature will be higher compared to a rail located in shadow, in a cutting or a tunnel, or embedded in a material with thermal insulation properties. Exposure to wind, humidity or condensation can also cause a difference between the rail temperature and the conventional air temperature.

In the summer, the rail temperature can be up to 15-20°C higher and in the winter can go 5°C below the conventional air temperature. Based on this and on the annual weather temperature variations, various railway administrations have adopted their own specific temperature ranges.

On the Network Rail railway infrastructure this range is considered to be [-14°C, 53°C] (NR/L2/TRK/3011).This gives a possible maximum annual rail temperature variation of 67°C. In continental Europe, for the countries of temperate-continental climate, the rail temperature range is [-30°C, 60°C]. This gives a potential maximum annual rail temperature variation of 90°C.

The rail coating and colour also influences its caloric absorption properties. A rail painted in white reflective coating, exposed to direct sunlight can be up to 6-10°C cooler than an uncoated rail (Ritter, Al-Nazeer - 2014). Painting the rails white to reduce the rail temperature is a method used efficiently in the UK and around the world (image 1).

TRACK LONGITUDINAL RESISTANCE

When the rail tends to move longitudinally, either due to temperature variations or due to other factors, the track will oppose this movement through a force called track longitudinal resistance (or track creep resistance).

This track longitudinal resistance has three levels of action (figure 4).

1. LONGITUDINAL RESISTANCE P1 BETWEEN THE RAIL AND THE FASTENING

At this level, the longitudinal resistance is generated by the friction between the rail and the fastening components. The European Norm BS EN 13841 mandates that the fastening longitudinal resistance on ballasted track should be at least 7 kN per fastening. For a normal sleeper interval of 600 mm, this is equivalent to a distributed longitudinal resistance of around 12 kN/m of rail. The fastenings used for high speed railway track (V>250 km/h) are required by the same norm to provide a minimum longitudinal resistance of 9 kN (around 15 kN/m of rail). (See image 2)

At installation and in ideal conditions, all fastenings can provide a certain toe load and consequently a longitudinal resistance. But often, for old types of fastenings (images 3, 4) this toe load cannot be maintained at a constant reliable level throughout the service life of the fastening. Consequently, for such fastenings the longitudinal resistance is usually ignored in the evaluation of the track thermal response.

Special fastening systems are sometimes installed on very long bridges with direct fixing, or in other specific cases, to provide a reduced longitudinal resistance and separate the thermal expansion forces of the track from the ones of the supporting structure. An example is the Pandrol Zero Longitudinal Restraint (ZLR – image 5) that is designed to provide practically no longitudinal fastening resistance.

2. LONGITUDINAL RESISTANCE P2 BETWEEN THE FASTENING AND THE SLEEPER

The highest longitudinal resistance is encountered between the fastening and the sleeper. For Pandrol fastenings, the clip’s shoulder is embedded in the sleeper’s concrete so no relative movement will happen at this level. The alternative screwed fastening is similarly good and generally all modern fastenings are designed to have a very high longitudinal resistance at this level. Since the resistance at this level is significantly higher than the other two, in the calculations that model the railway track thermal behaviour, no movement is considered to happen between

Figure 1: Free rail thermal expansion

Figure 3: Rail thermal force

Figure 2: Restrained rail thermal expansion

Figure 4: The three levels of action of the track longitudinal resistance

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TECHNICAL

Image 1: White coated rails on Italian main lines (Milano Afori railway station)

Image 2: Pandrol Fastclip fastening

Image 3: BR2 baseplate with Macbeth spring spike anchors

the fastening and the sleeper in which it is practically embedded.

3. LONGITUDINAL RESISTANCE P3 BETWEEN SLEEPER AND BALLAST

This resistance is dependant on a complex set of factors e.g. the type, shape, dimensions and weight of the sleeper, the ballast volume, compaction and content of fines. The friction forces at this level are not homogeneous along the track – there can be spots with high or low resistance, even for apparent similar track conditions. Consequently, the longitudinal resistance at this level is usually the lowest and the most difficult to control. This resistance is typically considered in the range of 6 (tamped) to 10 (consolidated) kN/sleeper (Van – 1996), the equivalent of 5 to 8 kN/m of rail.

The track longitudinal resistance is the minimum of these three resistances (P1, P2, P3) and the track movement due to temperature variation will take place at that lowest resistance level.

For modern superstructure and ballasted track, usually, the lowest longitudinal resistance is between the sleeper and the ballast.

On slab track, the third level of resistance, P3, is very high as the sleeper or the rail fastenings are embedded in the concrete slab and the only longitudinal resistance to be considered is the one developed at the first level – the fastening’s longitudinal resistance.

The longitudinal resistance for each sleeper will start to act as soon as the rail will tend to move due to temperature variation (figure 5). The nearest sleeper to the joint will be the first one to oppose the rail expansion when the temperature is increasing. Once the thermal force has increased above the longitudinal resistance P of the first sleeper, the rail movement will be allowed at this sleeper and the further expansion will be opposed by the resistance at the next sleeper. The process continues in relation with the rail temperature increase, until the longitudinal resistance is activated on the entire rail length.

For calculation purposes, the individual longitudinal resistance per sleeper is converted into a distributed longitudinal force, p, defined per 1m of rail.

JOINT RESISTANCE FORCE

For jointed track, the fishplated joint provides two other parameters which are considered in the rail thermal force calculations. The first one is the joint resistance force.

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The bolt tightening torque develops a tensile force, T, in the bolt. This tensile force induces a set of normal forces, N, at the contact areas between the fishplates and the rail head and foot. These normal forces can be found by decomposing the force vector T onto the two contact areas between the fishplate and the rail.

If the rail will tend to move in the joint, the normal forces will cause a set of friction forces between the fishplate and the rail (figure 6). Any such force will be defined as N•f – where N is the value of the normal force and f is the coefficient of friction between steel and steel.

Theoretically, for each bolt, there are four friction forces that will oppose the rail movement.

The resultant R of these friction forces is called the joint resistance force. This can be defined as:

R = 4•n•N•f

where n is the number of bolts per rail end.

The rails will move within the joint only if this resultant force R is overcome. Only then will the joint gap start closing or opening.

The rail temperature variation required to overcome the joint resistance can be computed from:

For a normal 4 bolt mechanical joint, designed for constrained thermal expansion track superstructure, lubricated and with bolts tightened at a 475 Nm torque (NR/L3/TRK/002), the joint resistance force R is around 175 - 200 kN.

For R = 175 kN and a BS 113A rail jointed track, if the rail temperature variation relative to the installation temperature is less than 10°C no rail movement will appear within the joint.

Some of the old joint types do not have a well-defined installation tightening torque and are also lacking essential components to keep this torque constant throughout the normal usage of the joint. In such cases, the joint resistance force can be ignored in rail thermal expansion calculations even though the joint bolts have a certain tensile force and, consequently, will develop a joint resistance force. Such a joint resistance is not continuous, constant and above a well-defined limit value and it cannot be used to define the thermal behaviour of the track in a safe manner.

Image 4: Bullhead rail Panlock chair fastening

Image 5: Pandrol ZLR fastening (source of the image – Pandrol ZLR brochure)

Figure 5: Track longitudinal resistance

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MAXIMUM JOINT EXPANSION GAP

The fishplated rail joint is designed to allow a gap variation and, as such, has two well defined limits of this gap. The minimum gap of a mechanical rail joint is null and that is usually achieved without exerting shear force on any of the joint bolts.

The maximum expansion gap of a mechanical rail joint is dependent on the position of the joint holes and on the diameter of the holes and bolts of the joint (Radu - 1999). This maximum gap is reached when the rails contract due to temperature decrease and are starting to exert shear forces on the joint bolts (figure 7).

The maximum expansion gap Gmax can be computed as:

Gmax = Bf + Df + Dr – 2Db – 2Br

Where:

Bf – the distance between the centres of the middle holes of the fishplate;Df – the fishplate hole diameter;Br – the distance from the end of rail to the first rail hole centre;Dr – the rail joint hole diameter;Db – the joint bolt diameter.

The maximum gap is designed for the longest rail used on jointed track to avoid the shearing

of the joint bolts when the rail contracts and it is an important parameter in the analysis of the jointed track’s behaviour due to rail temperature variations.

The track parameters described above define two distinct types of railway track superstructure:

• Free thermal expansion track superstructure

• Restrained thermal expansion track superstructure

FREE THERMAL EXPANSION TRACK SUPERSTRUCTURE

The free thermal expansion track superstructure allows the rail to vary its length freely due to temperature variation. This is a case in which the joints and fastenings do not provide reliable and continuous resistance forces to the rail thermal length variation and for calculation purposes all potential resistance forces are ignored. Most of the old types of fastenings and joints can be placed in this category.

Rail anchors (image 6) are sometimes used on this type of old superstructure to provide a certain track longitudinal resistance and reduce the rail creep on high gradients or on sections with frequent braking or acceleration.

JOINTED TRACK - SHORT AND LONG RAILS

For the free thermal expansion track superstructure, the joint and track resistances are ignored, hence the gap variation is linearly dependant on the temperature variation. The linear joint gap variation due to the rail temperature is displayed in the following graph (figure 8).

This graph defines a reference rail length Lref which will have the gap varying between the points A and B – closing the gap, when the temperature reaches its maximum value (B) and opening the gap to Gmax when the rail temperature reaches its minimum value (A).

In both cases no rail force is presumed to be transferred through the joint and the axial rail force will be null.

The reference rail length Lref can be computed from:

This reference length, Lref, defines a limit in the way the rail behaves.

For any rail shorter than the reference rail length, if installed and maintained correctly, the joint should never close to 0 or open to Gmax.

Figure 6: Friction forces generated by the joint bolt

Figure 7: Mechanical joint maximum gap

Image 6: A1 spike fastening with rail anchors

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Also, theoretically, for such a rail length, there will never be any thermal stress in the rail. This type of rail length is called short rail.

For any rail longer than the reference rail length, if installed correctly, the joint will close before Tmax, in a point F. If the temperature will increase to Tmax, the rail cannot expand any longer and a thermal compression stress will appear in the rail. In addition, as the temperature decreases, the rail will open to Gmax before reaching Tmin, in a point E. As the temperature goes further down to Tmin, a thermal tensile stress will appear in the rail and joint.

This type of rail length, for which the thermal stress will naturally appear during the annual temperature variation, is called long rail. But even a short rail may develop thermal forces if installed incorrectly and the joint gap closes before the maximum rail temperature or opens to the maximum gap before reaching the minimum rail temperature.

Table 2 presents the maximum gaps for several types of mechanical joints, applying the maximum gap formula and the standards defining the parameters of the rail joint components used in the UK.

For the majority of flat-bottom rail types quoted in table 2, the reference rail length, Lref, is approximately 18.5 m.

Hence, for these rail types and joints, the British standard rail length of 18.288 m (60 feet) is practically the reference length (figure 8), defining the limit between short and long rails.

JOINT CLOSURE TEMPERATURE (JCT)

Since the thermal expansion and contraction is freely allowed, the Joint Closure Temperature (JCT) can be computed using this formula:

Where:

- JCT is the estimated joint closure temperature- Tr is the measured rail temperature - Gr is the measured joint gap- α is the steel’s expansion coefficient - L is the rail length

The Joint Closure Temperature Table (NR/L2/TRK/001/Mod14) can be replicated using this formula. Table 3 shows the joint closure temperature (JCT) for 60 feet rails on free thermal expansion track superstructure.

The JCT values computed this way are rather theoretical and presume the free linear expansion of the rail, without the presence of any resistance force, either at the joints or of the track longitudinal resistance.

Figure 8: Joint gap variation. Short and long rail definition

Table 2: Maximum joint gap computation. Long rail definition

Table 3: Joint closure temperature for a 60 feet rail

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Depending on the real life track’s behaviour and on how these resistance forces are actually mobilised along the track, the real joint closure temperature (JCT) may be a few degrees lower (or sometimes higher) than the calculated one.

EXAMPLE OF LONG RAIL THERMAL BEHAVIOUR

The thermal behaviour of the rail and track can be analysed throughout the full annual temperature variation. This can be idealised as a continuous temperature variation, increasing from the installation temperature up to the maximum rail temperature and decreasing to the minimum rail temperature. From here this ideal cycle can be continuously repeated between the minimum and maximum rail temperature.

A free thermal expansion track superstructure with 22.710 m rails is used for this example. The rails are installed at 20°C and 6 mm joint gaps – according to NR/L2/TRK/001/mod04, table 2. The track will pass through the following main stages:

STAGE 0 - INSTALLATION(Figure 9)

T = 20°C and 6 mm joint gaps. At this stage the rail is stress free.

The standard nominal rail length is defined for 15°C (BS EN 13674-1:2011).

The rail length at T = 20°C is L = 22.710 + (20 – 15)•1.15•10-5 = 22.711m

The rail temperature is presumed to increase continuously and the rails will expand freely tending to close the joint gap. The next main stage of this theoretical cycle is the closure of the joint.

STAGE 1 - JOINT CLOSURE(Figure 10)

Due to the temperature increase between stage 0 and stage 1 the rails have expanded freely and the joint is closed.

The temperature variation required to close the rail is:

JCT (Joint Closure Temperature)= 20 + 23 = 43°C

The rail length has increased to 22.717m.

After the joint closes, the temperature will continue to increase until it reaches the maximum rail temperature.

STAGE 2 - MAXIMUM RAIL TEMPERATURE, 53°C(Figure 11)

Following the joint closure, the temperature increase to the maximum rail temperature cannot cause any further rail expansion and thermal stress will develop in the rail generating an internal compression force.

For a BS 113A rail, this thermal compression force is:

The rail length is unchanged, 22.717m.

From this stage of the theoretical cycle the temperature is decreasing continuously until it reaches the minimum rail temperature.

As the temperature decreases, the rail compression force will decrease. The next main stage is reached when the rail compression force is null and the joint can start to open.

STAGE 3 - THE JOINT STARTS TO OPEN

For the rail compression force to turn null, a rail temperature variation ΔT3 is required.

The rail temperature at this stage is the same as in stage 1. From this stage the rail temperature is decreasing further, leading to rail contraction. This contraction continues until the joint opens to its maximum gap, which defines the next main stage.

STAGE 4 - JOINT OPENED TO GMAX(Figure 12)

The temperature variation required to reach this stage can be calculated considering the gap change from 0 (stage 3) to 14.3 mm – the maximum gap:

The temperature at this stage is -11.8°C and the rail length has contracted to 22.703m. The next stage will be reached at the minimum rail temperature.

STAGE 5 - MINIMUM RAIL TEMPERATURE, -14°C(Figure 13)

As the temperature decreases further from the previous stage, the rail contraction is no longer permitted since the joint is already at its maximum gap. An internal tension force will develop in the rail.

For a BS 113A rail, this thermal tension force is:

The rail length is unchanged, 22.703m.

From this stage the temperature is presumed to increase continuously towards the maximum rail temperature.

As the temperature increases, the rail tension force will decrease. The next main stage is reached when the rail tension force is null and the joint gap can start to close.

Figure 9: Stage 0 - installation

Figure 10: Stage 1 - joint closure

Figure 11: Stage 2 - maximum rail temperature

Figure 12: Stage 4 - joint opening to Gmax

Figure 13: Stage 5 - minimum rail temperature

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STAGE 6 – THE JOINT GAP STARTS TO CLOSE

A rail temperature variation ΔT6 is required for the rail tension force to turn null.

The rail temperature at this stage is the same as in stage 4.

From this stage further the cycle repeats thorough these main stages 6-1-2-3-4-5-6. The full cycle is graphically presented in figure 14.

The graph presents the ideal behaviour of the rails on free thermal expansion track superstructure. Nevertheless, in real life, the track components will oppose some resistance to the thermal variation. The wheel loading and rail-wheel contact forces will also have an influence on the joint gap variation, generating longitudinal movements of the rail. Due to all these influencing elements, over time, the joint gap variation could significantly depart from the ideal/theoretical behaviour presented in this example and maintenance works will be required to apply the necessary corrections.The characteristic temperatures and forces for various types of rail can be evaluated using this procedure. An example of these parameters is presented in table 4 for free thermal expansion track superstructure with different rail lengths installed at 20°C and 6 mm joint gaps.

As the actual rail length increases beyond the reference length, the magnitude of compression and tensile forces can become significant, closer to values typical for CWR. In such cases, the jointed track requires maintenance and protection measures similar to the ones required in CWR track.

RESTRAINED THERMAL EXPANSION TRACK SUPERSTRUCTURE

The other type of superstructure is the restrained thermal expansion track superstructure – this superstructure has fastenings and joints designed to provide well-defined and reliable resistance forces to rail thermal variation throughout the service life of the track. Almost all modern fastenings and joints are designed in this way.

For this type of superstructure, closer to the behaviour of the actual track, the thermal variation is more complex (Alias – 1984, Hila et al. – 1975, Radu – 1989). The influence of the two main resistance forces will create a delayed joint gap response to temperature variation – a hysteresis loop (figure 15).

Figure 14: Joint gap variation for a track with 22.710 m rails on free thermal expansion superstructure

Figure 15: Example of gap expansion diagram for restrained thermal expansion superstructure

Figure 16: Theoretical example of ball and claw thermal variation from the median position

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In this case, for almost any rail temperature, Tr, the joint gap will not have a single value but is expected to be within a defined range (between Gr1 and Gr2 – see figure 15). The amplitude of this range is dependant on the track resistance forces and the rail length.

CONCLUSIONS

The principles presented in this article can be used to evaluate the thermal behaviour of the jointed track in a more accurate manner and, in a more complex approach, of the continuous welded rail (CWR) track, especially on the stress transition lengths.

The principles of the restrained thermal expansion superstructure including the delayed response to temperature variation, can also be applied to the devices used to control the interaction between the stock and switch rail (ball and claw or similar – see figure 16). The next articles will present more examples of thermal expansion calculations, for jointed and CWR track and will discuss the associated theoretical and practical implications.

ACKNOWLEDGEMENTS

The authors would like to thank the following for their comments, suggestions and support in writing this article: Constantin Radu, Univ Prof Dr Eng – TUCE Bucharest and Tom Wilson, Technical Discipline Leader (Track) – WSP | Parsons Brinckerhoff.

REFERENCES

Alias, J. (1984). La voie ferre, techniques de construction et d’entretien (The railway track, construction and maintenance techniques). SNCF – Eyrolles, Paris, France.

BSI BS 11:1985. Specifications for railway rails. British Standards Institution.

BSI BS 47-1:1991. Fishplates for Railway Rails – Part 1: Specification for Rolled Steel Fishplates. British Standards Institution.

BSI BS 64:1992. Specification for Normal and High Strength Steel Bolts and Nuts for Railway Rail Fishplates. British Standards Institution.

Code 720, UIC (2005). Laying and Maintenance of CWR Track. International Union of Railways (UIC), 2nd ed.

Cope, G. (1993). British Railway Track – Design, Construction and Maintenance. Permanent Way Institution, Echo Press, Loughborough.

Hila, V. Radu, C. Ungureanu, C. Stoicescu, G. (1975) Cai Ferate. Partea II. Suprastructura caii. (Railway Track. Part II. Track superstructure). Institutul de Constructii Bucuresti.

NR/L2/TRK/001/mod04 (2012). Inspection and maintenance of Permanent Way. Rail Joints. Issue 6. Network Rail.

NR/L2/TRK/001/mod14 (2012). Inspection and maintenance of Permanent Way. Managing track in hot weather. Issue 6. Network Rail.

NR/L2/TRK/2102 (2016). Design and Construction of Track, Issue 7. Network Rail.

NR/L2/TRK/3011 (2012). Continuous Welded Rail (CWR) Track, Issue 7. Network Rail.

NR/L3/TRK/002 (2007). Track – Renew Fishplates, Issue 2. Network Rail.

Radu, C. (1999). Cai Ferate – Suprastructura Caii (Railway – Track Superstructure) – Course notes, Faculty of Railways, Roads and Bridges – Technical University of Civil Engineering Bucharest.

Radu, C. (2001) Realizarea si Intretinerea Caii Fara Joante - curs postuniversitar. Technical University of Civil Engineering Bucharest. (Construction and Maintenance of the Continuous Welded Rail (CWR) Track – post-university course).

Ritter, G. W., Al-Nazeer, L. (2014). Coatings to Control Solar Heat Gain on Rails. AREMA 2014 Conference.

Van, M.A. (1996) Buckling analysis of continuous welded rail track. Delft University of Technology, HERON, 41 (3) 1996.

Table 4: Thermal behaviour characteristic data for free thermal expansion superstructure. * for short rails the joint closure and opening temperatures are only theoretical values, outside of the range of the rail temperature variation.

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