## How to calculate Minimum number of Sleepers, Breathing and Theoretical Length in 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

## Problem

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.

**Given:**

Rails

c/s area = 66.15 sq. cm

Es = 2.10 x10^6 kg/ sq. cm

ἀs = 11.5 x 10^-6 / deg. C

Sleepers

Sleeper Spacing = 60cm

Avg. resistance per sleeper per rail = 300kg

**Solution**

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