Safe Speed on Curves
For all practical purposes safe speed means a speed which protects a carriage from the danger of overturning and derailment and provides a certain margin of safety. Earlier it was calculated empirically by applying Martin’s formula:
For BG and MG
Transitioned curves
V = 4.4Jr  70 (13.7) where Vis the speed in km/h and R is the radius in metres.
Nontransitioned curves Safe speed = fourfifths of the speed calculated using Eqn (13.7).
For NG
Transitioned curves
V = 3.65yjR  6 (13.8) (subject to a maximum of 50 km/h).
Nontransitioned curves
V = 2.92yjR  6 (13.9)
(subject to a maximum of 40 km/h).
Indian Railways no longer follows this concept of safe speed on curves or the stipulations given here.
13.3.1 New Formula for Determining Maximum Permissible Speed on Transitioned Curves
Earlier, Martin’s formula was used to work out the maximum permissible speed or safe speed on curves. This empirical formula has been changed by applying a formula based on theoretical considerations as per the recommendations of the committee of directors, chief engineers, and the ACRS. The maximum speed for transitioned curves is now determined as per the revised formulae given below.
On Broad Gauge
V = )/^ +_{3}/76 ^{XR} = 0.27#T+QI^{7}* (13.10)
where V is the maximum speed in km/h, C_{a} is the actual cant in mm, C_{d} is the permitted cant deficiency in mm, and R is the radius in m. This equation is derived from Eqn (13.6) for equilibrium superelevation and is based on the assumption that G = 1750 mm, which is the centretocentre distance between the rail heads of a BG track with 52kg rails.
On Metre Gauge
V = 0.347 VcqTQ)R (13.11)
This is based on the assumption that the centretocentre (c/c) distance between the rail heads of an MG track is 1058 mm.
Narrow Gauge (762 min.)
V = 3.65^/R  6 (subject to a maximum of 50 km/h) (13.12)
13.3.2 New Criteria for Determining Maximum Speed on Curves Without Transition
As per the procedure being followed at present, the determination of the maximum permissible speed on curves without transitions involves the concept of virtual transitions. The linear velocity of a train moving with uniform velocity on a straight track begins to change into angular velocity as soon as the first bogie reaches the tangent point. This change continues till the rear bogie reaches the tangent point, at which moment the train acquires full angular velocity. The change in the motion of the train from a straight line to a curve takes place over the shortest distance between the bogie centres and is considered a virtual transition. Normally, this distance is l4.6 m on BG, 13.7 m on MG, and 10.3 m on NG, commencing on a straight line at half the distance before the tangent point and terminating on the curve at half the distance beyond the tangent point. The deficiency of cant is considered as being gained over the length of the virtual transition and the cant has to be gained in a similar manner. The cant gradient must not be steeper than 1 in 360 on BG and 1 in 720 on MG and NG under any circumstance.
The safe speed should be worked out on the basis of the the cant that can be practically provided based on these criteria, and increased by the permissible amount of cant deficiency. In the case of nontransitioned curves, where no cant is provided, the safe speed for the curve can be worked out by calculating the permissible cant deficiency after taking into consideration the rate at which the cant deficincy is gained or lost over the virtual transition.
13.3.3 Maximum Permissible Speed on a Curve
The maximum permissible speed on a curve is the minimum value of the speed that is calculated after determining the four different speed limits mentioned here. The first three speed limits are taken into account for the calculation of maximum permissible speed, particularly if the length of the transition curve can be increased. For highspeed routes, however, the fourth speed limit is also very important, as cases may arise when the length of the transition curve cannot be altered easily.
(i) Maximum sanctioned speed of the section This is the maximum permissible speed authorized by the commissioner of railway safety. This is determined after an analysis of the condition of the track, the standard of interlocking, the type of locomotive and rolling stock used, and other such factors.
(ii) Maximum speed of the section taking into consideration cant deficiency
This is the speed calculated using the formula given in Table 13.5. First, the equilibrium speed is decided after taking various factors into consideration and the equilibrium superelevation (C_{a}) calculated. The cant deficiency (C_{d}) is then added to the equilibrium superelevation and the maximum speed is calculated as per this increased superelevalion (C_{a} + C_{d}).
Type of curve 

Procedure for calculating max. permissible speed or safe speed 
Fully transitioned curve 
(i) 
For BG V = 0.27 ^/R (C_{a} + C_{d}) 

(ii) 
For MG V = 0.347 ^/R(C_{a} + C_{d}) 

(iii) 
For NG V = 3.65^/R _{}6 (subject to a maximum of 50 km/h) 
Nontransitioned curve with 
(i) 
Cant to be gained over virtual transition is 14.6 m 
cant on virtual transition 

on BG, 13.7 m on MG, and 10.3 m on NG, and the cant gradient is to be calculated accordingly 

(ii) 
The cant gradient is not to exceed 1 in 360 (2.8 mm/m) on BG and 1 in 720 (1.4 mm/m) on MG and NG. 
Nontransitioned curves 
(i) 
Calculate permissible cant deficiency that is to 
with no cant 

be gained or lost over the virtual transition 

(ii) 
The desirable value of rate of change of cant deficiency is 35 mm/sec for BG and 55 mm/sec for MG 
Curves with inadequate 
(i) 
Calculate the actual cant or cant deficiency which 
transition 

can be provided taking into consideration its limiting value 

(ii) 
The cant or cant deficiency has to be run over the transition length. The rate of change of cant or cant deficiency should not exceed its limiting value. For BG, the desirable value is 35 mm/sec and the maximum permissible value is 55 mm/sec. 
(iii) Maximum speed taking into consideration speed of goods train and cant
excess Cant (C_{a}) is calculated based on the speed of slow moving traffic, i.e., goods train. This speed is decided for each section after taking various factors into account, but generally its value is 65 km/h for BG and 50 km/h for MG.
The maximum value of cant excess (C_{e}) is added to this cant and it should be ensured that the cant for the maximum speed does not exceed the value of the sum of the actual cant + and the cant excess (C_{a} + C_{e}).
(iv) Speed corresponding to the length of the transition curves This is the least value of speed calculated after taking into consideration the various lenths of transition curves given by the formulae listed in Table 13.6.
The following points may be noted when calculating the maximum permissible speed on a curve.
(a) Criterion (iv) is to be used only in cases where the length of the transition curve cannot be increased due to site restrictions. The rate of change of cant or cant deficiency has been permitted at a rate of 55 mm/sec purely as an interim measure for the existing curves on BG tracks.
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Railway Engineering
(b) For highspeed BG routes, when the speed is restricted as a result of the rate of change of cant deficiency exceeding 55 mm/sec, it is necessary to limit the cant deficiency to a value lower than 100 mm in such a way that optimum results are obtained. In this situation, the maximum permissible speed is determined for a cant deficiency less than 100 mm, but gives a higher value of the maximum permissible speed. This concept is further explained with the help of the following solved problems.
Table 13.6 Various lengths of transition curves to be considered when calculating speed
Criteria for length of transition curve 
Desirable length of transition curve* 
Minimum length of transition curve 
When the rate of change of cant is taken as 35 mm/sec for normal cases and 55 mm/sec for exceptional cases 
C = F_{m}/125 (0.008 CV_{m}) 
C_{a} = Vm/198 
When the rate of change of cant deficiency is taken as 35 mm/sec for normal cases and 55 mm/sec for exceptional cases 
C_{A} = Vm/125 (0.008 CdV_{m}) 
Cd = Vn/198 
Taking the cant gradient into 
Cant gradient not to 
Cant gradient not to 
account 
exceed 1 in 720 
exceed 1 in 360 for BG and 1 in 720 for MG and NG 
* Notation used in the table: C_{a} is the value of actual cant in mm, V_{m} is the maximum permissible speed in km/h, and C_{d} is the cant deficiency in mm.
Example 13.1 Calculate the superelevation and the maximum permissible speed for a 2° BG transitioned curve on a highspeed route with a maximum sanctioned speed of 110 km/h. The speed for calculating the equilibrium superelevation as decided by the chief engineer is 80 km/h and the booked speed of goods trains is 50 km/h.
Solution
(vi) The maximum permissible speed on the curve is the least of the following:
¦ maximum sanctioned speed, i.e., 110 km/h.
¦ maximum or safe speed over the curve based on theoretical considerations, i.e., 110.1 km/h.
¦ Also, there is no constraint on speed due to the transition length of the curve.
Therefore, the maximum permissible speed over the curve is 110 km/h and the superelevation to be provided is 100.8 mm or approx. 100 mm.
Simplified method of calculating permissible cant and speed
Often a simplified method is used for calculating the permissible cant and the maximum permissible speed in the field. This simplified method is applicable to most cases except those involving very flat curves.
Step 1 Calculate the cant for the maximum sanctioned speed of the section, say, 110 km/h, using the standard formula C = GV^{2}/127R . This is C_{110}.
Step 2 Calculate the cant using the same standard formula as for the slowest traffic,
i.e., for a goods train which may be running at, say, 50 km/h. This is C_{50}. To this add cant excess. This becomes C_{50} + C_{e}.
Step 3 Calculate the cant for equilibrium speed (if decided) using the same standard formula. Let it be 80 km/h. This value is C_{80}.
Step 4 Adopt the lowest of the three values obtained from the preceding steps and that becomes the permissible cant (C_{a}). The three values are C_{110}, C_{50} + C_{e}, and C_{80}.
Step 5 Taking this cant value (C_{a}), add the cant deficiency and find the maximum permissible speed using the Eqn (13.10).
Solution to example 13.1 Step 1
Step 4 The lowest of the three values calculated in the preceding steps is 100.8 mm. Therefore, 100 mm is adopted as the actual cant.
Step 5 Cant to be provided 100 mm, cant deficiency = 75 mm V = 0.27^(C_{a} + C_{d}) X R = 0.27^/(100 + 75) x 875 = 110.1 = 110 km/h approx.
Therefore, the maximum cant to be provided 100 mm and the maximum permissible speed is 110 km/h.
Example 13.2 Calculate the superelevation, maximum permissible speed, and transition length for a 3° curve on a highspeed BG section with a maximum sanctioned speed of 110 km/h. Assume the equilibrium speed to be 80 km/h and the booked speed of the goods train to be 50 km/h.
Solution
However, actual cant is to be limited to 165 mm, and, therefore, this value will be adopted.
(v) Equilibrium superelevation for a goods train with a speed of 50 km/h
(vi) Cant excess = actual cant  59 mm
= 165  59 = 106 mm
which is in excess of 75 mm—the permitted value. With 75 mm taken as cant excess, the actual cant to be provided now is 75 + 59 mm = 134 mm. Therefore, a cant of 135 mm should be provided (rounding off to the higher multiple of 5).
(vii) Safe speed or speed potential (for highspeed route)
yl(C_{a} + C_{d}) x R ^/(135^100)^5833
= 13.76 = 13.76
= 99.6 km/h (or approx. 100 km/h).
(viii) Maximum permissile speed on the curve is the least of the following:
¦ maximum permissible speed of the section, i.e., 110 km/h
¦ safe speed on the curve, i.e., 100 km/h
The maximum permissible speed on the curve is, therefore, 100 km/h.
(ix) The length of transition is the maximum value from among the following:
¦ When taking the rate of change of cant into consideration (35 mm/sec),
L = 0.008 (C_{a} x V_{m}) = 0.008 x 135 x 100 m = 108 m
¦ When taking the rate of change of cant deficiency into consideration (35 mm/sec),
L = 0.008 (Cd x V_{m})
= 0.008 x 100 x 100 m = 80 m
¦ When taking the cant gradient into consideration (1 in 720),
L = 0.72 x e = 0.72 x 135 m = 97.2 m Therefore, the superelevation to be provided is 135 mm, the maximum permissible speed over the curve is 100 km/h, and the length of transition curve is 108 m.
Example 13.3 Calculate the maximum permissible speed on a curve of a high speed BG group A route having the following particulars: degree of the curve = 1°, superelevation = 80 mm, length of transition curve = 120 m, maximum speed likely to be sanctioned for the section =160 km/h.
Solution
(i) Radius of curve = 1750/D = 1750/1 = 1750 m
(ii) Safe speed over the curve as per theoretical considerations, this being a highspeed route,
V = 0.27V(C_{a} + C_{d}) X R
V = 0.27^(80 + 100) X 1750 = 151.3km/h
(iii) Speed based on transition length:
(a) Rate of gain of cant (not to exced 55 mm/sec)
(which is not steeper than 1 in 720).
(iv) Maximum permissible speed is the least of the following:
¦ maximum sanctioned speed of the section, i.e., 160 km/h
¦ safe speed based on theorectical considerations, i.e., 151.3 km/h
¦ speed based on the transition length, i.e., 237.6 km/h
Therefore, the maximum permissible speed over the curve is 151.3 km/h or about 150 km/h. As the controlling factor in this case is the safe speed based on theoretical considerations (and not the rate of change of C_{d}), hence no further analysis is necessary.
Example 13.4 Calculate the maximum permissible speed on a 1° curve on a Rajdhani route with a maximum sanctioned speed of 130 km/h. The superelevation provided is 50 mm and the transition length is 60 m. The transition length of the curve cannot be increased due to the proximity of the yard.
, . „ .. 60 X 1000 mm , .
(c) Cant gradient == 1 in 1200
50 mm
(which is not steeper than 1 in 720).
(iv) Maximum sanctioned speed on the curve is the least of the following:
¦ Maximum speed sanctioned for the section, i.e., 130 km/h
¦ Safe speed based on theoretical considerations, i.e., 138.3 km/h
¦ Speed based on transition length, i.e., 118.3 km/h.
In this case, the speed has to be restricted to 118.8 km/h, because of the constraint of transition length. A cant deficiency of 100 mm has been assumed, which is its maximum possible value. On the field, the cant deficiency may be somewhat lower, giving a lower rate of change of C_{d} for the given transition length and a higher permissible speed. The optimum value of this maximum permissible speed can be found from the following equation:
Equilibrium superelevation = actual cant + cant deficiency for maximum
permissible speed for a given transition length or
Solving this equation, V = 133 km/h. This value however, cannot be more than the MSS of the section, i.e., 130 km/h. Therefore, the maximum permissible speed over the curve is 130 km/h.
= 91.4 mm
which is less than 100 mm. Therefore, the maximum permissible speed over the circular curve is 130 km/h and that over the transition curve is 118 km/h.
Example 13.5 A BG branch line track takes off as a contrary flexure through a 1 in 12 turnout from a main line track of a 3° curvature. Due to the turnout, the maximum permissible speed on the branch line is 30 km/h. Calculate the negative superelevation to be provided on the branch line track and the maximum permissible speed on the main line track (when it takes off from a straight track).
Solution
(i) For a branch line track, the degree of the curve is 4  3 = 1°
Radius = 1750/D = 1750/1 = 1750 m
After rounding it off to a higher multiple of 5, it is taken as 10 mm.
(ii) The value of negative superelevation for a branch line track,
x = e  Cd = 10 mm  75 mm = 65 mm (negative)
(iii) The superelevation to be provided on the main line track is 65 mm, which is the same as the superelevation of the branch line track, but in the opposite direction.
(iv) The maximum permissible speed is calculated by taking the actual superelevation of the main line track (65 mm) and adding it to the cant deficiency (75 mm), and then using this value of superelevation, i.e., 140 mm (65 + 75) in the formula for equilibrium speed. The main line track has a 3° curve, i.e., 1750/3 = 583.3 m radius.
Therefore, the maximum permissible speed on the main line track,
Alternatively, the maximum permissible speed can also be calculated as follows
Therefore, the maximum permissible speed on the main line track is 77.16 km/h. After rounding it off to a lower multiple of 5, it becomes 75 km/h.