Thursday, May 1, 2014

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How to calculate Minimum number of Sleepers, Breathing and Theoretical Length in Long Welded Rails

Continuous or Long Welded Rails
Continuous or Long Welded Rails
Long Welded Rails is welded rail in which central part does not undergo any longitudinal movement due to temperature variation. A length of greater than 250 metre on Broad Gauge (BG) and 500 metre on Metre Gauge (MG) will normally function as LWR (Fig). The maximum length of LWR under Indian conditions shall normally be restricted to one block section.
The portion that undergoes expansion and contraction is known as breathing length. Steel rails experience longitudinal movement due to varied temperature conditions. To compensate this movement in long welded rails, expansion joints are placed at certain distances. These joints have pre-calculated expansion gap to allow longitudinal movement of rails, thus preventing thermal stresses.
If length of rail is L, due to temperature increase,
Increase in Length ẟL = L x ἀ x T
Thus, Strain prevented if this increase or decrease in length in prevented
Strain = ẟL/L
= L x ἀ x T/L
= ἀ x T
Stress developed, fs = E x Strain = E x ἀ x T

If area of cross- section of one rail = As, Force developed in rail
Ps = As x fs
= As x E x ἀ x T
If one sleeper provide R Resistance, Minimum number of sleepers required in any direction so that rail does not move
N = Ps / R, where “R” is resistance per sleeper.

If “S” is the spacing of sleepers, one Breathing Length required, l = (N-1) S
Minimum length of LWR so that central portion does not move in any direction = 2l
Breathing Length
Breathing Length


Determine the minimum theoretical length of LWR beyond the central portion of a 52kg rail would not be subjected to any longitudinal movement due to 30 degree Celsius temperature variation.
c/s area = 66.15 sq. cm
Es = 2.10 x10^6 kg/ sq. cm
ἀs = 11.5 x 10^-6 / deg. C
Sleeper Spacing = 60cm
Avg. resistance per sleeper per rail = 300kg

The forced developed in rail if no movement is allowed,
Ps = As x E x ἀ x T
= 2.10 x10^6 x 11.5 x10^-6×30 x 66.15
= 47926 kg
Resistance per sleeper, R= 300kg
Number of Sleepers required in one direction N = Ps/R = 47926/300 = 160
Minimum breathing length in one directio,
l = (N-1) x Spacing
= (160-1) x 0.60
= 95.4 m
Minimum Theoretical Length of long welded rails = 2l = 2x 95.4 = 190.8m = 191 m


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