It is well known that metals expand and contract with increase or decrease in temperature, i.e., undergo thermal expansion. Thus, a rail expands and contracts depending upon the variations in temperature. The expansion of a rail is a function of the coefficient of the linear expansion of the rail material, the length of the rail, and the variations in the rail temperature. Normally, a free rail would undergo alterations in its length corresponding to the variations in rail temperature, but as rails are fastened to sleepers, which in turn are embedded in the ballast, their expansion and contraction due to temperature changes are restricted. The restraint put on the thermal expansion of rail gives rise to locked up internal stresses in the rail metal. The resulting force, known as the thermal force, is given by the following equation:
P = EAaT (17.1)
where P is the force in tonnes, E is the modulus of elasticity of rail steel = 2.15 x 106 kg/cm2 or 2150 t/cm2, A is the cross-sectional area of steel in cm2 and depends upon the individual rail section (for a 52-kg rail it is 66.15 cm2), a is the coefficient of linear expansion = 0.00001152 per °C, and T is the temperature variation in °C. Substituting the values of E, A, a, and T, the force for every 1° rise of temperature for a 52-kg rail can be derived as follows:
P = (2.15 x 106) x 66.15 x 0.00 001152 x 1 x 10-3 = 1.638 t per °C
The values of E and a are fixed for each type of rail steel. The value of the cross-sectional area depends upon the sectional weight of the rail. Substituting the value of sectional weight in kg/m Eqn (17.1), the force P can also be given by the formula
P = 31.5AT
where, P is the force in kilograms, A is the sectional weight in kg/m, and T is the temperature variation in °C. For a 52-kg rail P = 31.5 x 52 kg per unit °C = 1638 kg = 1.638 t per °C
17.2.1 Longitudinal Thermal Expansion of LWR and Breathing Length
In the case of LWR, the thermal expansion of the rail takes place at the rail ends because of temperature variations and the inability of the resisting force offered by the rail and the ballast to overcome the same. A long welded rail continues to expand at its ends up to that particular length at which an adequate resisting force is developed towards the centre. A stage is finally reached at a particular length of the rail from its ends when the resistance offered by the track structure becomes equal to the thermal forces created as a result of temperature variations. There is no alternation in the rail length beyond this point. The cumulative value of the expansion or contraction of these end portions of the rail (breathing lengths) is given by the formula
where, 8? is the amount of expansion or contraction of the rail, ? is the breathing length of the rail, a is the coefficient of thermal expansion of the rail, and t is the variation in temperature. This value of expansion or contraction of the rail is half the value that would have been attained if the rail had been free to expand on rollers without any ballast resistance. This alteration in length is confined to only a certain portion at the ends of the LWR. The central portion of the LWR, where the force is constant, is immobile and does not undergo any change in its length.
The portion at the end of the long welded rail, which undergoes thermal expansion, is called the breathing length. On Indian Railways this length is equal about 100 m at either end of the rail in the case of BG tracks.
Example 17.1 Calculate the minimum theoretical length of LWR beyond which the central portion of rail would not be subjected to any thermal expansion, given the following data: cross-sectional area of a 52-kg rail section = 66.15 cm2, coefficient of thermal expansion of rail steel = 11.5 x 10-6 per °C, temperature variation = 30° C, modulus of elasticity of rail steel = 2 x 106 kg/cm2, sleeper spacing = 65 cm, and average restraining force per sleeper per rail = 300 kg.
(a) Using Eqn (17.1) and the given values, the thermal force P is given by
P = (2 x 106) x 66.15 x 11.5 x 10~6 x 30 = 45.6t
(b) Resistance offered per sleeper = 300 kg = 0.3 t
(c) Number of sleepers required to restrain a force of 45.6 t
=-= 152 sleepers at each end
(d) Breathing length at each end when sleeper spacing is 0.65 m
= 152 x 0.65 = 98.8 m = 100 m approx at either end
(e) Total breathing length on both sides presuming the zero portion in centre
= 100 x 2 m = 200 m
Therefore, the minimum theoretical length of the LWR is 200 m.