Forces Acting on the Track
A rail is subjected to heavy stresses due to the following types of forces.
(a) Vertical loads consisting of dead loads, dynamic augment of loads including the effect of speed, the hammer blow effect, the inertia of reciprocating masses, etc.
(b) Lateral forces due to the movement of live loads, eccentric vertical loading, shunting of locomotives, etc.
(c) Longitudinal forces due to tractive effort and braking forces, thermal forces, etc.
(d) Contact stresses due to wheel and rail contact.
(e) Stresses due to surface defects such as flat spots on wheels, etc.
Vertical Loads
Dead load of vehicles at railwheel contact
The value of dead load is usually taken from the axleload diagram. It is, however, brought out that for various reasons the actual wheel loads, even in the static state on a level and perfect track, may be different from the nominal values. Cases have sometimes come to notice where a steam locomotive had a higher axle load than the nominal load or had different right and left wheel loads.
Dynamic augment of vertical loads
On account of vertical impact due to speed and rail vibrations, etc., the dynamic load is much more than the static load. The dynamic wheel load is obtained by increasing the static wheel load by an incremental amount given by the speed factor. Till 1965 Indian Railways used the 'Indian Formula' for calculating the speed factor. This formula is the following:
V
Speed factor ^{=} ~m (5.1)
where V is the speed in miles per hour and jl is the track modulus in psi.
After 1966, the 'German formula' given by Schram was adopted, which is as follows.
(a) For speeds up to 100 km/h:
V^{2}
Speed factor = (5.2)
30,000 ^{v 2}
(b) For speeds above 100 km/h:
45V^{2} 15V^{3}
Speed factor = (5.3)
10^{5} 10^{7}
where V is the speed in km/h.
At a speed of 60 mph, the Indian formula gives a speed factor of 55%, whereas the German formula, as used by RDSO, gives a speed factor of 30%. Investigations have been carried out by RDSO and different values of speed factors have been recommended for different types of vehicles running at different speeds.
Hammer blow effect
The centrifugal forces due to revolving masses in the driving and coupled wheels of a locomotive, such as crank pins, coupling rods, and parts of the connecting rod, are completely balanced by placing counterweights near the rim of the wheel, diametrically opposite to the revolving masses. The reciprocating masses of the piston, piston rod, cross head, and part of the connecting rod, by virtue of their inertia and oscillatory movement, produce alternating forces in the direction of the stroke and tend to cause the locomotive to oscillate sideways and nose across the track. In order to reduce this nosing tendency, a weight is introduced onto the wheels at the opposite side of the crank. The horizontal component of the centrifugal force of this added weight balances the inertial force in the line of stroke, but the vertical component throws the wheel out of balance in the plane perpendicular to the line of stroke. The vertical component of the centrifugal force of the weight introduced to balance the reciprocating masses causes variation in the wheel pressure on the rail, and is called the hammer blow. The heavier the weight added to balance the reciprocating masses, the greater the hammer blow.
The hammer blow effect occurs only in the case of steam locomotives. The hammer blow can be calculated as follows (Fig. 5.2):
M _{2}
Hammer blow =  r(2pn) sin Q (5 4)
g
Fig. 5.2 Steam effect, hammer blow effect and effect of inertia of reciprocating masses
where M is the net overweight in lbs, r is the crank pin diametre in ft, n is the number of revolutions of the wheel per second, and Q is the crank angle.
Steam effect
A steam locomotive works by converting coal energy into steam energy. Steam pressure acts on the piston and is transmitted to the driving wheels through the
crank pins and connecting rod. The vertical component of the crank pins and connecting rod is at an angle to the piston rod. Its value is given by the formula
where L is the length of the connecting rod in inches, h is the height of the cross head above the centre line of the driving wheel in inches, and Q is the crank angle,
i.e., the angle traversed by the crank since the beginning of the stroke.
The steam effect (Fig. 5.2) does not scynchronize with the hammer blow effect due to overbalance and is additional to the hammer blow only during some part of the revolution of the crank shaft.
Inertia of reciprocating masses
The reciprocating masses, due to their inertia and acceleration, alter the forces on the piston, and hence the force in the connecting rod is also affected during the revolution of the wheel.
where M is the mass of the reciprocating parts, L is the length of the connecting rod, n is the number of revolutions per second, h is the height of the cross head above the centre line of the driving wheel, and Q is the crank angle.
The maximum combined force of the hammer blow, the steam effect, and inertia for each driving wheel and the hammer blow effect of the coupled wheels do not act simultaneously due to the phase difference in the angular position of the counterweights in the coupled and driving wheels. The maximum combined effect of these forces is obtained by summing up the three curves for one complete revolution of the wheel.
Bending stresses on the rail due to vertical loads
The general theory of bending of rails is based on the assumption that the rail is a long bar continuously supported by an elastic foundation. Due to vertical loads, the rail is subjected to bending or flexural stresses. The bending stresses that a rail is subjected to as a result of vertical loads are illustrated in Fig. 5.3. The theory of stresses in rails takes into account the elastic nature of the supports. Based on this theory, the formula for bending moment is
where M is the bending moment, P is the isolated vertical load, l = (EI/ m)^{1/4} is the characteristic length, EI is the flexural stiffness of the rail, m is the track modulus, and x is the distance of the point from the load.
According to Eqn (5.7), the bending moment is zero at points where x = pl/4, 3pl/4 and maximum where x = 0, pl/2, 3pl/2, etc.
x 
0 
pl/ 8 
pl/ 4 
3pl/8 
pl/ 2 
5pl/8 
3plj 4 
7pl/8 
Pl 
9 pl/8 
5pl/4 
FMB as a percentage of the maximum 
100.00 
36.67 
+ 0.00 
+ 16.50 
+ 20.80 
+ 18.40 
+ 13.48 
+ 8.60 
+ 4.32 
+ 1.59 
+ 0.00 
For calculating the stresses acting on the rail, first the maximum bending moment caused due to a series of loads moving on the rail is calculated as per Eqn (5.7). The bending stress is then calculated by dividing the bending moment by the sectional modulus of the rail. The permissible value of bending stress due to a vertical load and its eccentricity is 23.5 kg/mm^{2} for rails with a 72 UTS.
Stresses on the Track
Stresses on the track due to the various kinds of forces applied on it are discussed in the following sections.
Lateral forces
The lateral force applied to the rail head produces a lateral deflection and twist in the rail. Lateral force causes the rail to bend horizontally and the resultant torque causes a huge twist in the rail as well as the bending of the head and foot of the rail. Lateral deflection of the rail is resisted by the friction between the rail and the sleeper, the resistance offered by the rubber pad and fastenings, as well as the ballast coming in contact with the rail.
The combined effect of lateral forces resulting in the bending and twisting of a rail can be measured by strain gauges. Field trials indicate that the loading wheels of a locomotive may exert a lateral force of up to 2 t on a straight track particularly at high speeds.
Longitudinal forces
Due to the tractive effort of the locomotive and its braking force, longitudinal stresses are developed in the rail. Temperature variations, particularly in welded rails, result in thermal forces, which also lead to the development of stresses. The exact magnitude of longitudinal forces depends on many variable factors. However, a rough idea of these values is as follows:
(a) Longitudinal forces on account of 3040% weight of locomotive of tractive effort for alternating current (ac).
(b) Longitudinal forces on account of 1520% of weight of braking force of the locomotive and 1015% weight of trailing load.
Tensile stresses are induced in winter due to contraction and compressive stresses are developed in summer due to compression. The extreme value of these stresses can be 10.75 kg/mm^{2} in winter and 9.5 kg/mm^{2} in summer.
Contact stresses between rail and wheel
Hertz formulated a theory to determine the area of contact and the pressure distribution at the surface of contact between the rail and the wheel. As per this theory, the rail and wheel contact is similar to that of two cylinders (the circular wheel and the curved head of the rail) with their axes at right angles to each other. The area of contact between the two surfaces is bound by an ellipse as shown in Fig. 5.4.
The maximum contact shear stress (F) at the contact point between the wheel and the rail is given by the empirical formula
(5.8)
where F is the maximum shear stress in kg/mm^{2}, R is the radius of the fully worn out wheel in mm, and P is the static wheel load in kg + 1000 kg for onloading on curves
Contact stress for the WDM2 locomotive Static wheel load (P) = 9400 + 1000 = 10,400 kg. Radius of worn out wheel for maximum wear of 76 mm (38 mm radius reduction):
Fig. 5.4 Contact stresses between rail and wheel
The contact stress for the WDM2 locomotive as such is 18.7 kg/mm^{2} The maximum value is, however, limited to 21.6 kg/mm^{2}, which is 30% of the UTS value (72 kg/mm^{2}) of the rail.
Surface defects
A flat on the wheel or a low spot on the rail causes extra stresses on the rail section. Empirical studies reveal that an additional deflection of about 1.5 times the depth of the flat or low spot occurs at the critical speed (about 30 km/h). Additional bending moment is caused on this account with a value of about 370,000 kg cm for the BG group A route with the WDM4 locomotive.
Stresses on a sleeper
The sleepers are subjected to a large number of forces such as dead and live loads, dynamic components of tracks such as rails and sleeper fastenings, maintenance standards, and other such allied factors. Based on the elastic theory, the maximum load on a rail seat is given by the following formula:
where P is the wheel load, m is the track modulus, S is the sleeper spacing, l is the characteristic length, and Z is the modulus of the rail section.
The maximum load on the rail seat is 30%50% of the dynamic wheel load, depending on various factors and particularly the packing under the sleeper.
The distribution of load under the sleeper is not easy to determine. The pattern of distribution depends on the sleeper as well as on the firmness of the packing under the sleeper. As the ballast yields under the load, the pressure under the sleeper is not uniform and varies depending on the standard of maintenance. The following two extreme conditions may arise.
Endbound sleeper The newly compacted ballast is well compacted under the sleeper and the ends of the sleepers are somewhat hard packed. The deflection of the sleeper at the centre is more than that at the ends.
Centrebound sleeper As trains pass on the track, the packing under the sleeper tends to become loose because of the hammering action of the moving loads. The sleeper thus tends to be loose under the rail seat. Alternatively, due to defective packing, the sleeper is sometimes hard packed at the centre.
Stresses on ballast
The load passed onto the sleeper from the rail is in turn transferred to the ballast. The efficacy of this load transmission depends not only on the elasticity of the sleeper but also on the size, shape, and depth of the ballast as well as the degree of compaction under the sleeper. Professor A.N. Talbot has analysed the pressure distribution in the ballast under the sleeper and investigations reveal that the pressure distribution curve under the sleeper would be shaped like bulbs as shown in Fig. 5.5. The following are the important conclusions drawn from Fig. 5.5.
(a) The pressure on the sleeper is maximum at the centre of its width. This pressure decreases from the centre towards the ends.
(b) The vertical pressure under the sleeper is uniform at a depth approximately equal to the spacing between the sleepers.
Pressure on formation or subgrade
The live as well as dead loads exerted by the trains and the superstructure are finally carried by the subgrade. The pressure on the subgrade depends not only on the total quantum of the load but also on the manner in which it is transferred to the subgrade. The spacing between the sleepers; the size, depth, as well as compaction of the ballast under the sleeper; and the type of subgrade play an important role in the distribution of pressure on the subgrade.
The values of maximum formation pressure permitted on Indian Railways are the following:
For motive power 3.5 kg/cm^{2} for BG
2.5 kg/cm^{2} for MG For goods wagons 3.0 kg/cm^{2} for BG
2.3 kg/cm^{2} for MG
Fig. 5.5 Stresses in the ballast
Relief of stresses
A train load consists of a number of wheel loads close to each other which act simultaneously on the rail. A single isolated wheel load creates much more bending moment in the rail as compared to a group of wheel loads, which on account of the negative bending moment under adjacent wheels provide what known as a 'relief of stresses'. The rail stresses in this case are comparatively smaller. The value of relief of stresses depends upon the distance of the point of contraflexure of the rail and the spacing between the wheels, but its value can be as high as 50%.
Permissible stresses on a rail section
The permissible bending stresses due to vertical load and its eccentricity, and lateral load on a rail section on Indian Railways is given in Table 5.5.
Table 5.5 Permissible bending stress

kg/mm^{2} 
t/irf^{2} 
Permissible stress due to bending 
36.0 
23.00 
Minimum ultimate tensile strength 
72.0 
46.0 
The stresses on a rail are measured by any of the following methods depending upon the facilities available.
(a) Photoelastic method
(b) Electric resistance strain gauge method
(c) Method employed using special test frame
At present, Indian Railways mostly uses the electric resistance strain gauges for measuring rail stresses.
Whenever a new locomotive or rolling stock design is introduced on the Railways, a detailed study is carried out followed by field trials to ensure that the permitted speed of the new locomotive or rolling stock does not cause excessive stresses on the track. The same stipulations are made whenever there is an increase in the speed or axle load of the existing locomotive or rolling stock design.
The various parameters and their limiting values required to be checked for BG are given in Table 5.6.
Table 5.6 Limiting values of stresses on BG
Parameter 
Permissible value 
Bending stress on the rail 
36.0 kg/mm^{2} 
Contact stress between the rail and the wheel 
21.6 kg/mm^{2} 
Dynamic overloads at rail joints due to 
27 t for locomotives and 
unsuspended masses 
19 t for wagons 
Formation pressure 
3.5 kg/cm^{2} for locomotives and 3.0 kg/cm^{2} for wagons 
Fish plate stresses 
30 kg/mm^{2} 
Bolt hole stresses 
27 kg/mm^{2} 
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