1

Maximum Superelevation:

Desirable, Allowable, and Absolute

Nazmul Hasan, M. Eng.

SNC-Lavalin Inc.

Vancouver, ON

ABSTRACT

The maximum values of superelevation are often

qualified as desirable, allowable and absolute.

Moreover different railways limit the maximum

values of actual and unbalance super-elevation to

different values. There are two striking reasons

behind the variation in the maximum values

suggested across disciplines, associations, borders,

operators etc. Firstly, there are no definitions of the

aforesaid qualifying terms. These terms are loosely

used and hence confuse the readers in general.

Secondly, there is no theoretical basis to compute the

maximum values of superelevation. Thus the part of

literature narrating superelevation values is in a

chaotic state that should be disciplined. The

qualifying terms and the maximum values of

superelevation should be purpose specific. Three

purposes are identified in the paper. Then the analysis

and discussion is made on each purpose. The paper

addresses the aforesaid two issues and suggests the

maximum desirable, allowable and absolute values of

superelevation. As an extension to the work the

minimum operating speed and its significance is also

stated. The paper is intended for non-tilting

conventional train.

INTRODUCTION

Different railways limit maximum values of

actual and unbalance super-elevation to different

values. Generally, it is recognized that 75mm to

115mm of superelevation unbalance is acceptable for

light rail transit (LRT) operations, depending upon

the vehicle design [1]. With this recognition it is

accepted indirectly that allowable actual

superelevation should range from 35mm (=150-115)

to 75mm (=150-75). It suggests 35mm as the

minimum and 75 mm as the maximum actual super

elevation. All railroads administered by the Federal

Railroad administration (FRA) are limited to 150mm

of superelevation [1]. The City of Calgary limits the

actual superelevation to 110mm and unbalance to

65mm [2]. One can see three maximum allowable

values of actual superelevation in the above narration

- they are 75mm, 110mm and 150mm. One can also

find two maximum allowable values of unbalance –

they are 65mm and 115mm. Transit Co-operative

Research Program (TCRP) recommends 75mm as the

maximum desirable unbalance superelevation [1].

TCRP recommends 115mm as the absolute

maximum allowable unbalance superelevation [1]. In

the literature the maximum superelevation is often

qualified as desirable, allowable and absolute with no

definition of these terms. The maximum values and

its qualification should be purpose specific. At least

three purposes are identified. They are-

1. to proportion 3.33 sec’s ride to achieve spiral

length formula for any superelevation,

2. to design superelevation, and

3. to calculate the maximum allowable operating

speed on an existing curve.

Discussion on each purpose follows.

2

DISCUSSION

First Purpose:

The first purpose is to proportion 3.33 sec’s ride

i.e. 0.925V to derive an equation of the spiral length

for any superelevation. In the current literature [1]

this is done as under:

)(max925.0

allowableEa

EaVL

)(max925.0

allowableEu

EuVL

From above formulation it is obvious that

different maximum allowable values adopted by

different railways would lead to different spiral

lengths although the comfort criteria are same.

Besides two different lengths are given by the above

two equations although both are based on the same

comfort criteria. Same comfort criteria should lead to

a single spiral length whatever be the equation. So

the different maximum allowable values adopted by

different railways are not acceptable for this purpose.

The maximum allowable values of 150mm for

Ea and 115mm for Eu have been used by TCRP [1]

to derive spiral length equation for any value of Ea

and Eu as under:

)1(006.0150

925.0 VEaEa

VL

)2(008.0115

925.0 VEuEu

VL

Both the Eq. (1) and (2) are proven to be

theoretically wrong by the author [3]. The two

equations yield different spiral lengths although they

are based on same comfort criteria; they also

underestimate the length significantly. This above

exercise by TCRP showed that the maximum

allowable values of actual and unbalanced

superelevation do not work to achieve an acceptable

spiral formula. Different railways set their maximum

allowable values differently e.g. The City of Calgary

allows 110 mm for maximum allowable actual

superelevation and 65 mm for maximum allowable

unbalance superelevation. So proportioning of

0.925V will be different for different railways and

will lead to different spiral lengths. The spiral length

must not be different for different railways if the

comfort criteria are the same. The approach of

proportioning of 0.925V is not incorrect. The

maximum allowable values are not incorrect. Thus,

there must be something incorrect with the use of the

maximum allowable values.

Both 0.925V (km/h) and the maximum desirable

values are based on 3.33 sec’s (=0.1g/0.03g/s) ride.

So 0.925V should be proportioned by the maximum

desirable values as shown below:

)(max925.0

desirableEq

EqVL

)(max925.0

desirableEa

EaVL

)(max925.0

desirableEu

EuVL

The maximum desirable values are determined

on the basis of following comfort criteria [4]:

Radial acceleration = 0.1 g

Jerk = 0.03 g/s

Roll run-off = 1 deg/sec

The maximum desirable values are given below [4]:

Equilibrium superelevation = 150mm

Actual superelevation = 87mm

Unbalanced superelevation = 63mm

The proportioning of 0.925V by the above maximum

desirable values will lead to an acceptable spiral

length [3, 4]. It is to be noted that the equation of

spiral length can be derived without using the above

proportioning approach [3]. The message of the

above exercise is: the maximum desirable values

should be same for all railways but the maximum

allowable values adopted by different railways may

be different.

Second Purpose:

3

The design of superelevation and the design of

spiral length should be seen holistically [3]. Thus for

the design of superelevation of the curve the

maximum desirable value of actual and unbalance

superelevation of 87mm and 63 mm should be used.

These two maximum desirable values give a way to

proportion equilibrium superelevation into actual and

unbalance superelevation [4]. Equilibrium

superelevation, Eq should be divided by 1.72 to

achieve actual superelevation. The unbalanced

superelevation is obtained by simply subtracting

actual superelevation from equilibrium

superelevation. It is desirable that the superelevation

should not exceed the maximum desirable values as

the equilibrium superelevation equation is based on a

radial acceleration of 0.1 g and there is a universal

consensus to design superelevation on the basis of 0.1

g radial acceleration. At the same time every railway

agrees to exceed a radial acceleration of 0.1 g to

implement the maximum speed. The third purpose

comes into play and in order to serve it, one must

know the maximum allowable and/or absolute

maximum value of superelevation.

Third Purpose:

In the industry the maximum allowable speed is

usually calculated by the following formula:

8.11

*

max

RxEaV

in which:

*x blanket unbalance.

The maximum allowable speed is greater than the

speed stipulated by the equilibrium equation based on

0.1 g radial acceleration.

To compute the maximum allowable speed, different

blanket unbalances usually greater than the maximum

desirable unbalance are used. For example FRA

suggests a 75 mm blanket unbalance [5]. With the

intension of gaining even more speed, the designer

may exceed the maximum desirable value of actual

superelevation i.e. 87mm. Clearly, the maximum

desirable values need to be exceeded to gain a higher

speed. The acceptable values beyond the maximum

desirable values are limited by the maximum

allowable values. Thus the maximum allowable value

should be somewhere in the range shown below.

The maximum allowable actual superelevation:

87mm ~ 150mm,

The maximum allowable unbalance superelevation:

63 mm ~ 150mm.

Any value (excluding the lower limits) used in the

range mentioned would exceed the desirable comfort

criteria. An exercise is carried out below to determine

the maximum allowable value.

The maximum allowable unbalanced superelevation

The spiral length is given by the formula [3]:

)3(161 Eu

VEaL

This formula is used to determine the maximum allowable unbalanced superelevation, Eu . Differentiating both

sides with respect to time, t:

2)161(

**)161(

Eu

dt

dEuVEa

dt

dEaVEu

dt

dL

2)161(

**)161(

Eu

dt

dEuVEa

dt

dEaVEu

V

4

]4[/18.26 smmdt

dEa

]4[/82.18 smmdt

dEu

2)161(

82.18**18.26**)161()/(

Eu

EaVVEusmV

Multiply both sides by 3.6 to change the unit of speed term on the left side from m/s to km/h

2)161(

82.18**18.26**)161(6.3)/(6.3

Eu

EaVVEusmV

2)161(

*82.18*6.3*)161(18.26*6.3

Eu

EaVVEuV

2)161(

*75.67**)161(248.94

Eu

EaVVEuV

Cancelling V from both sides

2)161(

75.67)161(248.941

Eu

EaEu

)150(75.67)161(248.94)161(

75.67)161(248.94)161(

2

2

EuEuEu

EaEuEu

150*75.67161*248.94)248.9475.67(*161*2161 22 EuEuEu

428.5011498.26*161*2161 22 EuEuEu

0428.5011498.26161322 22 EuEuEu

057.20909502.2952 EuEu

2

)57.20909*4502.295502.295 2 Eu

mmsaymmmmEu 115,117,178

It is observed that the desirable maximum run-off

value of actual and unbalance superelevation lead to

two maximum values of unbalance superelevation:

178 mm and 115 mm. However, 178mm is not

acceptable because Eq.(3) is based on a radial

acceleration of 0.1 g that corresponds to an

equilibrium superelevation of 150 mm. Moreover, it

exceeds the maximum safe unbalance value of

5

161mm [3, 6]. The maximum allowable unbalance is

also calculated to be 115 mm in another way [3, 6].

TCRP recommends 115 mm as the absolute

maximum allowable unbalanced superelevation [1].

In practice there is evidence of passenger trains

operating in N. America at an unbalance of 178 mm

and being tested to more than 300 mm without

exceeding safety limits [6]. Therefore, the absolute

maximum unbalance should exceed 115 mm. Thus,

the 115 mm unbalance should not be labeled as the

absolute maximum value. It may be seen as the

maximum allowable value, as it is in between the

maximum desirable and the absolute value.

The maximum allowable actual superelevation

There is a wide consensus on the allowable

maximum unbalance of 115 mm; however, there is

no consensus for the maximum allowable actual

superelevation. Railways that are not administered by

the FRA may, when appropriate, use up to 200 mm

of actual superelevation on curved track. This has

been applied to at least two North American transit

systems. However, it is more common to limit

maximum actual superelevation to 150 mm on LRT

systems, as it becomes more difficult to consistently

maintain ride comfort levels at higher actual

superelevations [1]. The city of Calgary recommends

a maximum actual superelevation of 140 mm [2].

TCRP suggests an absolute maximum actual

superelevation of 100 mm and at the same time uses

150 mm as the maximum allowable actual

superelevation to arrive at spiral length (Ref: Eq.(1))

[1]. At this point, the author does not prefer to use the

term “absolute”. The absolute term should go with

the values beyond the maximum allowable values.

The maximum allowable unbalance of 115 mm

suggests a minimum actual superelevation of 35 mm

(=150-115 mm). The minimum actual superelevation

of 35mm suggests a minimum unbalance of 25 mm

(=35*0.72). Current literature supports a value very

close to 25 mm. If the calculated cant is less than 20

mm it can be disregarded [7]. In a way the statement

supports a minimum unbalance of 20 mm.

The minimum allowable actual unbalance of 25

mm suggests the maximum allowable superelevation

of 125 mm (=150-25 mm). It is equivalent to a cross

gradient of 8.33% (=125/1500) and seems to be too

high to ensure comfortness when compared to usual

longitudinal gradient and the maximum desirable

cross-gradient of 5.8% (=87/1500) .

In fact, the installation of very high cant is

undesirable for many reasons. Some are noted below:

Longer spiral length is required

It could produce passengers discomfort on a

train that is moving much slower than the

design speed or stopped in the middle of the

curve

Very high super-elevation can cause load

displacement. Stability of work vehicle and

of special loading with a high centre of

gravity can be jeopardized

Very high super-elevation cannot guard

against derailment of tall cars on the low

side of curves for very slow moving trains,

and for low rail rollover derailments for

slow moving high axle load rolling stocks

With high super-elevation, ballasted track

can move inside while tamping in cold

weather

The author suggests a procedure to compute the

maximum allowable superelevation. It is established

that the least desirable ratio between unbalance and

actual superelevation is 0.72 [4]. To avoid excess of

both actual and unbalance superelevation, an upper

limit of the ratio between unbalance and actual

superelevation is suggested to be unity. It means

1EaallowableMax

EuallowableMax

1115

EaallowableMax

Thus the maximum allowable actual superelevation is

suggested to be 115 mm.

If a train moves much slower than the design

speed or is stopped in the middle of a curve elevated

to 115 mm, the unbalanced superelevation will be

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115 mm that is the maximum allowable unbalance.

The load experienced by the high rail under an

unbalance of 115 mm will be the same load

experienced by the low rail on a curve elevated to

115 mm during the static condition of a vehicle.

Moreover, 115 mm represents 7.6% cross gradient

which seems to be acceptable.

The absolute maximum value

The literal meaning of the absolute maximum value

is the maximum possible value. So the absolute

maximum values would exceed the maximum

allowable values. The purpose is to gain even more

speed. Thus the value should be based on the safety

limit. The middle third criteria may be used for this

purpose. The maximum safe unbalance [1] given by

the middle third criteria is

)4(

)6

(

h

xs

s

Eu

It is impossible to suggest a unique value of the

absolute maximum unbalance in general by the Eq.

(4) because it is a vehicle specific formula. This

formula gives a wide range of values, e.g. for x= 0 ~

100mm, h=1,016 ~ 2,134 mm, the maximum safe

unbalance comes out to be 109 ~ 250 mm [1]. It is

not likely to accept a value less than 150 mm as the

absolute maximum value. The author demonstrated

161 mm as the maximum safe unbalance in the paper

no. [3]. The absolute maximum value may be defined

as the maximum of the values given by Eq. (4) and

161 mm. This does not figure out a single value in

general. The author suggests a value of 161 mm

unbalance as the maximum absolute value because (i)

it is not a vehicle specific value, (ii) it is greater than

150 mm, and (iii) it is a conservative value compared

to high end values computed by the Eq. (4). The

author also suggested 161 mm as the absolute

maximum value [6].

It is necessary to know the upper bound of safe radial

acceleration to determine the absolute maximum

value of 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

[6]. It suggests that a radial acceleration above 0.2g

does not exceed the safety limit. So, the upper bound

of safe radial acceleration may be assumed to be

0.2g. This assumption leads to the absolute maximum

value of actual superelevation of 140 mm

(=0.2*1500-161=139mm). This is equivalent to a

cross gradient of 9.3%.

THE MINIMUM OPERATING SPEED

Current literature does not describe the

restriction on low speed. While unconventional, it is

necessary to impose a limit on the low speed as well.

The reason of this necessity is explained below.

Wide gauge conditions are often maintained

frequently by gauging the curve to correct gauge. Of

course, normal wide gauge correction work requires

the low rail to be unspiked, moved, plugged, and

respiked. Imagine several iterations of this until the

point is reached where the high side of the curve

includes holes that are very large from gauge

widening/spreading forces. The rail is still wearing.

Now the difference between static and dynamic

gauge exceeds the allowable limit. Effective ties in a

rail are counted. The single worst case in the curve is

used as the basis for the speed restriction. Now with

the slow order imposed, the excess elevation shifts

the load to the low rail. The holes with spikes

plugged several times and then respiked are now in

the equation. Currently, there is no such thinking on

slow speed, but there should be a limit on slow speed

too on ballasted track as well.

Since a minimum unbalance value has been

figured out, the minimum operating speed on a curve

should be:

8.11

)25(min

EaRV

It means the restricted speed should not reach below

the minimum speed dictated by the 25 mm

unbalance. The curve should be maintained so that it

does not require imposing a speed restriction that is

less than the minimum operating speed. No curve

should be designed for a speed below minimum

operating speed. Obviously, this minimum speed is

7

higher than the equilibrium speed and trains are not

generally operated at equilibrium speed.

CONCLUSION

The maximum values of superelevation with the

maximum radial acceleration are given in the table 1.

Parameter Desirable Allowable Absolute

Actual

superelevation

(mm)

87 115 140

Unbalance

superelevation

(mm)

63 115 161

Radial

acceleration,

g

0.1 g 0.15 g 0.2 g

Table 1: Maximum Values of Superelevation and Radial Accn

The minimum operating speed is given by:

8.11

)25(min

EaRV

NOTATIONS

Ea Actual superelevation (mm)

Eu Unbalance superelevation (mm)

Eq Equilibrium superelevation (mm)

h Height of center of gravity of vehicle above

rail level (mm)

L Spiral length (m)

R Radius of curve (m)

s Track width, 1500 mm

V Speed (km/h)

minV Minimum speed (km/h)

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

REFERENCES

[1] TCRP, 2000, “Track Design Handbook for

Light Rail Transit,” Report # 57, National

Academy Press, Washington, pp. 3-13, 3-22,

3-23,3-25.

[2] The City of Calgary; 2009, LRT Design

Guide Line, Section 3- Track Alignment

[3] Hasan, N.,”Spiral Length Design,” JRC

2010-36050, Proceedings of the 2011

ASME/ASCE/IEEE Joint Rail Conference,

Urbana, Illinois, USA.

[4] Hasan, N.,”Passenger Track Curve Design

Criteria: Comfort Criteria, Equivalent

Comfort Criteria, and Application,” JRC

2011-56012, Proceedings of the 2011

ASME/ASCE/IEEE Joint Rail Conference,

Colorado, Pueblo, USA.

[5] 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.

8

[6] Hasan, N.,”Maximum Allowable Speed On

Curve,” JRC 2011-56007, Proceedings of

the 2011 ASME/ASCE/IEEE Joint Rail

Conference, Colorado, Pueblo, USA.

[7] Esveld, C., 2001, Modern Railway

Technology, MRT-Productions, The

Netherlands, pp.37.

[8] Hasan, N.,”DFF Spacing and Stiffness

Design,” JRC 2011-56008, Proceedings of

the 2011 ASME/ASCE/IEEE Joint Rail

Conference, Colorado, Pueblo, USA.