NAZMUL HASAN

SNC-LAVALIN INC.

Transportation Division

1800-1075 West Georgia Street

Vancouver, British Columbia

Canada V6E 3C9

Tel: 604-662-3555

Fax: 604-662-7688

e-mail:nazmul.hasan@snclavalin.com

RAIL CONFERENCE, 2011

INTRODUCTION

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The maximum values of superelevation are often qualified as desirable,

allowable and absolute. Different railways limit the maximum superelevation

to different values. So different values are qualified by the same terms

desirable, allowable and absolute. The reasons behind the variation are:

1. There is no definition of these qualifying terms,

2. There is no theoretical basis to calculate the qualifying values.

This paper addresses the aforesaid issues and suggests the maximum

desirable, allowable, and absolute values of superelevation with a

theoretical basis for calculation.

NOTATIONS

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Ea = Actual superelevation (mm)

Eu = Unbalanced superelevation (mm)

Eq = Equilibrium superelevation (mm)

V = Speed (km/h)

L = Spiral length (m)

s = Track width, 1500 mm

x = Shift of c.g. towards high rail (mm)

h = Height of c.g. of vehicle above rail level (mm)

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MAXIMUM DESIRABLE SUPERLEVATION

In literature spiral length based on 3.33 sec’s run (=0.925V[km/h]) [

This in turn is based on a radial acceleration of 0.1g and a jerk of

0.03g/s ] is proportioned by the maximum allowable actual and

unbalanced superelevation to get spiral length for any Ea and Eu as

under:

EaAllowableMax

EaL 925.0

EuAllowableMax

EuL 925.0

Use of different maximum allowable values by different railways lead

to different spiral length for the same comfort criteria. Moreover the

above two equations lead to two different spiral lengths for the same

comfort criteria. This is not acceptable. This means the above

proportioning should be done with some values that would lead to an

acceptable single spiral length. These values are referred to as the

“maximum desirable values”.

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MAXIMUM DESIRABLE SUPERLEVATION

R

V

R

VEq

222 *153*6.3*8.11

R

V

dt

d

dt

dEq 2

*153

JerkR

V

dt

d

2

Differentiating both sides with respect to time, t

By definition,

smmsgsgJerksmmdt

dEq/45/03.0*153)/(*153)/(

Equation for equilibrium superelevation is given by:

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MAXIMUM DESIRABLE

ACTUAL SUPERELEVATION

Industry wide acceptable rotational run-off of 1 deg/sec leads to an actual

superelevation run-off of 26 mm/s as under:

dt

dS

dt

d

dt

dEa 1500)(

sec/18.26180

*1*1500 mm

The maximum desirable superelevation comes out to be: 3.33 s*26 mm/s

=87 mm.

VEaEa

VL1000

6.10

87925.0

The spiral length should be given by:

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MAXIMUM DESIRABLE

UNBALANCED SUPERLEVATION

sec/82.1818.2645 mmdt

dEa

dt

dEq

dt

dEu

The maximum desirable unbalanced superelevation is:

3.33 s * 18.82 mm/s = 63 mm

The spiral length should be given by:

VEuEu

VL1000

6.14

63925.0

Please note that spiral length given by proportioning with 87mm actual

superelevation and 63 mm unbalanced superelevation lead to same

spiral length unlike the formulae given by proportioning with the maximum

allowable values.

The desirable values are based on a radial acceleration of 0.1g, a jerk of

0.03g/s and a rotational run-off of 1 deg/sec.

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MAXIMUM ALLOWABLE

UNBALANCED SUPERELEVATION

TCRP in its report # 57 recommends 115 mm as the absolute maximum

allowable unbalanced superelevation. Absolute maximum value should be

some value that is limited by safety limit. In practice, there is evidence of

passenger trains operating in North America at an unbalance of 178 mm.

So the author recommends 115mm as the maximum allowable unbalanced

superelevation.

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MAXIMUM ALLOWABLE

ACTUAL SUPERELEVATION

The least desirable ratio between unbalanced (Eu) and actual (Ea)

superelevation is given by:

72.087

63

18.26

82.18

Ea

Eu

A ratio below 0.72 would increase the actual superelevation and spiral

length and a ratio above 0.72 would increase the unbalanced force on

track. To avoid excess of both actual and unbalanced superelevation and to

allow more speed on curve, an upper limit of the aforesaid ratio is

suggested to be unity.

1Ea

Eu

With the maximum allowable unbalanced superelevation of 115mm, the

maximum allowable actual superelevation comes out to be 115mm. Now if a

train stands on a curve with elevated to 115mm, the load on low rail would

be same as on the high rail if the train runs with 115 mm unbalanced S.E.

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ABSOLUTE MAXIMUM

UNBALANCED SUPERELEVATION

h

xs

s

Eu

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TCRP in its report # 57 determines the maximum safe unbalance by using

middle third criteria as under:

in which

Eu = maximum safe unbalanced superelevation,

s = track width,

x = lateral shift,

h = height of c.g. of vehicle on top of rail.

The formula suggests a range of values depending on the vehicle

characteristics. It does not give a single value to be used as an absolute

maximum value. For typical values of x and h, maximum safe unbalanced

superelevation varies from 109mm ~ 250mm.

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ABSOLUTE MAXIMUM

UNBALANCED SUPERELEVATION

In practice, there is evidence of passenger trains operating in North

America at an unbalance of 178 mm.

The author in paper no. JRC 36050-2010 showed theoretically that the

absolute maximum unbalanced superelevation should be 161 mm.

The proof is not dependent on vehicle characteristics and suggests a

single value. As the value is close to 178 mm and a compromise

between the value given by middle third criteria and the experimental

value of 178 mm, the author is inclined to suggest 161 mm as the

absolute maximum unbalanced value.

The absolute maximum unbalanced superelevation is 161 mm.

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ABSOLUTE MAXIMUM

ACTUAL SUPERELEVATION

In practice, there is evidence of passenger trains operating in North

America at an unbalance of 178 mm and being tested to more than 300 mm

without exceeding safety limits. It suggests that a radial acceleration above

0.2g (=300g/1500) does not exceed the safety limit. Thus the upper

boundary of safe radial acceleration is assumed to be 0.2g.

A radial acceleration of 0.2g represents an equilibrium superelelvation of

300 mm (=1500*0.2).

Thus the absolute maximum superelevation comes out to be 140 mm

(=300-161 mm).

The absolute maximum actual superelevation is 140 mm.

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SUMMARY

Parameter Desirable Allowable Absolute

Actual

superelevation 87

3.4”

115

4.5”

140

5.5”

Unbalanced

superelevation 63

2.5”

115

4.5”

161

6.3”

Table 1: The maximum values of superelevation (mm)

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REFERENCES

[1] TCRP, 2000, “Track Design Handbook for Light Rail Transit,” Report # 57,

National Academy Press, Washington, pp. 3-13, pp.3-22,pp 3-23,pp.3-25.

[2] City of Calgary; 2009, LRT Design Guide Line, Section 3- Track Alignment

[3] Hasan. N; 2010,”Spiral Length Design,”

JRC 2010-36050, Proceedings of the 2010 ASME/ASCE/IEEE Joint Rail Conference, Urbana-Champaign, USA

[4] Hasan. N; 2011,”Passenger Track Curve Design Criteria: Comfort Criteria, Equivalent Comfort Criteria, and Applications”,

JRC 2011-56012, Proceedings of the 2011 ASME/ASCE/IEEE Joint Rail Conference, Colorado, Pueblo, USA

[5] Hasan. N; 2011,”Maximum Allowable Speed on Curve,”

JRC 2011-56007, Proceedings of the 2011 ASME/ASCE/IEEE Joint Rail Conference, Colorado, Pueblo, USA

[6] Esveld, C., 2001, Modern Railway Technology, MRT-Productions, The Netherlands, pp.37.

[7] Hasan, N.,”DFF Spacing and Stiffness Design”, JRC 2011-56008,

Proceedings of the 2011 ASME/ASCE/IEEE Joint Rail Conference, Colorado, Pueblo, USA

[8] U.S. Department of Transportation, Federal Railroad Administration-Office of Safety, 2008,”Code of Federal Regulations Title 49”,

The Railway Educational Bureau, Omaha, NE, USA ,pp. 21

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