Maximum Curving Speed

Maximum Curving SpeedNazmul Hasan1

Abstract: Full equilibrium superelevations are rarely installed on tracks. Consequently, trains run on unbalanced superelevations. Currently,the maximum curving speed is modeled with blanket unbalanced superelevations. Blanket unbalanced superelevations are not desirable forall actual superelevations, regardless of their values. In this paper, the maximum curving speed is modeled to ensure comfort and safety,considering two primary parameters: the curve parameters and the vehicle characteristics. The curve parameters include radius and actualsuperelevation, and the vehicle characteristics include the height of the center of gravity (c.g.) on top of the rail and the suspension factor. Theproposed model has interesting features. It suggests that unbalanced superelevation is proportional to actual superelevation. The maximumcurving speed is evaluated against the normal operating speed as the base reference. The model is applied and validated from the point of viewof both comfort and safety. The paper suggests a method to divide equilibrium superelevation into actual and unbalanced superelevations.DOI: 10.1061/(ASCE)TE.1943-5436.0000648. © 2013 American Society of Civil Engineers.

Author keywords: Actual superelevation; Unbalanced superelevation; Curve radius; Suspension factor; Track width; Comfort criteria;Maximum safe unbalanced superelevation; Height of center of gravity (c.g.) of vehicle on top of rail level.

Introduction

In practice, full equilibrium superelevations are rarely installed ontracks, regardless of the service—passenger or freight. Reasons forthis practice include the following:• Full equilibrium superelevations could produce discomfort for

passengers on a train that is moving much slower than the speedit is designed for, or for passengers on a train that is stopped inthe middle of the curve [Esveld 2001; International Union ofRailways (UIC) Code 703 (UIC 1989)].

• At equilibrium superelevations, lateral acceleration does notexist, so self-wheel steering is not an ideal way to take advan-tage of wheel iconicity.

• Very high superelevations can cause load displacement, possiblyjeopardizing the stability of work vehicles and special loadingswith a high center of gravity (c.g.) [Esveld 2001; UIC Code 703(UIC 1989)].

• In very slow-moving trains, very high superelevations cannotprevent the derailment of tall cars on the low side of curves,nor can it prevent low-rail rollover derailments of slow-moving,high-axle-load rolling stocks [UIC Code 703 (UIC 1989)].

• With high superelevation, ballasted tracks can move insidewhile tamping in cold weather [Esveld 2001; UIC Code 703(UIC 1989)].Consequently, trains run with unbalanced superelevations.

Currently, the maximum curving speed is modeled with a blanketunbalanced superelevation. This type of superelevation is notdesirable for actual (installed) superelevations, regardless of theirvalues, because both actual and unbalanced superelevation runoffsoccur simultaneously. Moreover, a blanket unbalanced supereleva-tion must be conservative in terms of meeting the requirements

of all types of vehicles, regardless of their height and suspensiondesign. Thus, a blanket unbalanced superelevation cannot exploitspeed’s potential effectively. The operating speed on a curve is al-ways more than the equilibrium speed. When the lateral componentof vehicle weight balances the centrifugal force, the equilibriumspeed is safe as both wheels bear equally on the rail, and the result-ant force acts at a right angle to the plane on the top of the rails.The preceding facts do not mean that speeds exceeding the equi-librium speed are unsafe, as 60% wheel offloading is usually per-missible [Federal Railroad Administration (FRA) 2008]. Noadditional unbalanced superelevations on top of design unbalancedsuperelevation are considered when estimating the normal operat-ing speed. The curving speed is usually even higher than the normalspeed. The maximum curving speed is modeled and evaluated, withthe normal speed as the base reference. The maximum curvingspeed is validated from both comfort and safety points of view.

Modeling Speed by Equilibrium SuperelevationEquation

The equilibrium superelevation equation [Esveld 2001; TransitCooperative Research Program (TCRP) 2000, 2012] is given asfollows:

Eq ¼ Eaþ Eu ¼ 11.8V2

Rð1Þ

From Eq. (1), speed can be expressed as

V ¼ 1ffiffiffiffiffiffiffiffiffi11.8

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðEaþ EuÞ

pð2Þ

Putting Eu ¼ 0 into Eq. (2), equilibrium (balanced) speed maybe written as

VEq ¼ 0.29ffiffiffiffiffiffiffiffiffiREa

pð3Þ

For the abovementioned reasons, equilibrium superelevationsare never installed, and trains always run under unbalanced super-elevations. It is possible to express Eu in terms of Ea, and Eq. (2)may be rewritten as

1Senior Trackwork Engineer, SNC-Lavalin, Transportation Division,1800-1075 West Georgia St., Vancouver, BC, Canada V6E 3C9. E-mail:[email protected]

Note. This manuscript was submitted on August 12, 2013; approved onNovember 14, 2013; published online on December 26, 2013. Discussionperiod open until May 26, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Transportation En-gineering, © ASCE, ISSN 0733-947X/04013023(7)/$25.00.

© ASCE 04013023-1 J. Transp. Eng.

J. Transp. Eng. 2014.140.

Dow

nloa

ded

from

asc

elib

rary

.org

by

PEN

N S

TA

TE

UN

IV o

n 11

/07/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

http://dx.doi.org/10.1061/(ASCE)TE.1943-5436.0000648




V ¼ kffiffiffiffiffiffiffiffiffiREa

pð4Þ

in which k > 0.29Changing the unit of speed from km=h to m=s, Eq. (1) is

expressed as

Eq ¼ 153v2

Rð5Þ

Differentiating both sides of Eq. (5) with respect to time, t,results in the following:

dEqdt

¼ 153ddt

�v2

R

�¼ 153 × jerk ð6Þ

The desirable equilibrium superelevation runoff is now relatedto jerk by Eq. (6). An acceptable jerk limit of 0.03 g=s (TCRP2000, 2012) suggests a desirable runoff of 45 mm=s for equilib-rium superelevation:

dEqdt

¼ 45 mm=s ð7Þ

The desirable actual superelevation runoff is calculated from adesirable rotational runoff of 1 deg =s. The calculation is given asfollows:

dEadt

¼ ddt

ðBθÞ ¼ 1,500dθdt

dθdt

¼ 1 deg =s

dEadt

¼ 1,500 ×1 × π180

¼ 26.18 mm=s ð8Þ

Thus, the desirable actual superelevation runoff is 26.18 mm=s.The desirable unbalanced superelevation runoff is calculated asfollows:

Eq ¼ Eaþ EudEqdt

¼ dEadt

þ dEudt

45 ¼ 26.18þ dEudt

dEudt

¼ 18.82 mm=s ð9Þ

The desirable unbalanced superelevation runoff is 18.82 mm=s.The relation between actual and unbalanced superelevation is

obtained by dividing Eq. (9) by Eq. (8):

Eu ¼ 0.72Ea ð10Þ

Putting Eq. (10) in Eq. (2), the following equation results:

V ¼ffiffiffiffiffiffiffiffiffi1.7211.8

r ffiffiffiffiffiffiffiffiffiREa

p¼ 0.38

ffiffiffiffiffiffiffiffiffiREa

pð11Þ

The designers usually label the speed of Eq. (2) or (11) asthe normal operating speed because no extra unbalanced superel-evation is allowed on top of design unbalanced superelevation.The normal operating speed is allowed without a vehicle qualifi-cation test. Thus, the k value of 0.38 can be used without a vehiclequalification test. With the current code, some extra unbalancedsuperelevation, ΔEu, is allowed, so Eq. (2) can be modified tomodel the maximum curving speed, as follows:

Vmax ¼1ffiffiffiffiffiffiffiffiffi11.8

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðEaþ EuþΔEuÞ

pð12Þ

The Federal Railroad Authority (FRA) allows a total unbalancedsuperelevation (EuþΔEu) of 100 and 75 mm, with and without avehicle qualification test, respectively, in Eq. (12) (FRA 2008).

Thus, the maximum curving speed is greater than the normaloperating speed.

The maximum curving speed may be modeled as

Vmax ¼ kffiffiffiffiffiffiffiffiffiREa

pð13Þ

in which k > 0.38The unbalanced superelevation under the maximum curving

speed is given by

Eu ¼ ð11.8 × k2 − 1Þ × Ea ð14ÞThe value of k depends on the comfort limit, which is usually

0.1 g (TCRP 2012, 2000); the suspension design; and the heightof the vehicle. It may be determined by a series of test runs oncurves with different k values. This paper determines the kvaluetheoretically.

Interesting Aspects of the Model

The model Vmax ¼ kffiffiffiffiffiffiffiffiffiREa

phas some interesting features that do

not depend on the k value:• The model would ensure variable unbalanced superelevations

in proportion to actual superelevations, which seems logical.The disadvantages of blanket unbalanced superelevation arediscussed in detail in an earlier paper of mine (Hasan 2011).

• If some curves of comparable curvature are closely spaced, thena common speed may be implemented by designing curveradiuses and superelevations so that the products of the radiusand the actual superelevation of all curves would be equal. Thiswould ensure uniform, smooth rides at a constant speed over astretch of track.

• The model can be a good tool for testing curving speed becauseit is expressed as a function of the product of two vital fixedparameters of a curve—radius and installed superelevation. Themodel reduces the selection of test curves. Test results on threerepresentative curves of three different curvatures—sharp, mild,and shallow—are sufficient to determine the maximum speedfor all curves in a system.

First Method of Deriving k Value

Unbalanced superelevation should be limited because of the suddenchange in curvature when passing from straight to curved track.The American Railroad Engineering and Maintenance-of-WayAssociation (AREMA) recommends a 75-mm unbalanced superel-evation to determine the maximum curving speed on a turnout. TheInternational Union of Railways (UIC) Code allows unbalancedsuperelevation between 80 and 100 mm on a turnout [UIC Code711 R (UIC 1981)]. In Europe, the maximum curving speed ona turnout (Esveld 2001) is given by

Vmax ¼ 2.79ffiffiffiffiR

pð15Þ

Eq. (15) is based on a 90-mm [¼ ð2.792 × 1; 500Þ=ð3.62 gÞ]unbalanced superelevation.

On trackage that is subject to FRA jurisdiction, the maximumallowable unbalanced superelevation is 75 mm, except for allAmtrak equipment, for which the FRA allows up to 100 mm(FRA 2008; Ahlf 2003). The FRA suggests a 75-mm unbalancedsuperelevation without any vehicle qualification test (FRA 2008).In practice, evidence exists of passenger trains operating in NorthAmerica at an unbalanced superelevation of 178 mm, and thesehave been tested to more than 300 mm without exceeding safety



Dow

nloa

ded

from

asc

elib

rary

.org

by

PEN

N S

TA

TE

UN

IV o

n 11

/07/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

limits (Hasan 2011). In Sweden, Norway, Germany, and France,intercity railways commonly employ from 100 to 150 mmof unbalanced superelevation and occasionally use an even higherunbalance value for specific applications (Esveld 2001). Thus, anunbalanced superelevation of 75 mm is a conservative value fromthe point of view of comfort and safety. Putting Ea ¼ 0 and Eu ¼75 mm in Eq. (2), the maximum curving speed may be written as

Vmax ¼1ffiffiffiffiffiffiffiffiffi11.8

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRð0þ 75Þ

p¼ 2.52

ffiffiffiffiR

pð16Þ

Eq. (13) does not apply to a nonsuperelevated curve because ofthe internalization of unbalanced superelevation in terms of actualsuperelevation. Eq. (16) can be modified into Eq. (17), shown next,to make it apply to a nonsuperelevated curve by the following rea-soning: The accepted practice in the industry is to disregard super-elevation if the calculated value is less than the minimum value.The minimum value varies for different railways. For example,if the calculated superelevation is less than 20 mm, it can be dis-regarded (Esveld 2001). If the calculated superelevation is 25 mmor less, actual superelevation is not usually installed (TCRP 2012).For the sake of analysis, it is suggested that a minimum valueof 25 mm for actual superelevation be disregarded. Thus, themaximum curving speed on a curve may be written as

Vmax ¼ kffiffiffiffiffiffiffiffiR25

p¼ 5k

ffiffiffiffiR

pð17Þ

Equating Eqs. (16) and (17) produces the following:

k ¼ 2.52=5 ¼ 0.5

The maximum curving speed is given by

Vmax ¼ 0.5ffiffiffiffiffiffiffiffiffiREa

pð18Þ

It can be inferred from a k value of 0.5 that the unbalancedsuperelevation is 1.95 [cf. Eq. (14)] times the actual superelevationunder the maximum curving speed. For an actual superelevationover 40 mm, the unbalanced superelevation would exceed 75 mm.The maximum allowable speed is around 30% (¼ 1 − 0.5=0.38)more than the normal operating speed.

Absolute Maximum Safe Lateral Acceleration

In this section, the maximum lateral accelerations are estimatedand evaluated from different perspectives to suggest an absolutemaximum safe value.

In practice, evidence exists of passenger trains operating inNorth America at an unbalance of 178 mm, and these trainshave been tested to more than 300 mm without exceeding safetylimits (Hasan 2011). An unbalanced superelevation of 300 mmcorresponds to a noncompensated lateral acceleration of 0.2 g(¼ 300 g=1,500). The lateral acceleration would be ð0.2þ Ea=1,500Þ g, which suggests that a lateral acceleration above 0.2 gdoes not exceed the safety limit.

In Europe, evidence exists of passenger trains operating at anunbalance of up to 228 mm, which corresponds to a noncompen-sated lateral acceleration of 0.152 g (¼ 228 g=1,500). The lateralacceleration would be ð0.152þ Ea=1,500Þ g, which shall not posea threat to ride safety. For actual superelevation over 72 mm, thelateral acceleration would be above 0.2 g.

The Transit Cooperative Research Program (TCRP 2000, 2012)suggests that the middle third criterion gives the maximum safeunbalanced superelevation. It is given by

Eu ¼ B½ðB=6Þ − x�h

ð19Þ

Eq. (19) also can be written as

Eu ¼ B2

6h− xh

ð20Þ

The term x=h in Eq. (20) can be disregarded because of itssmall value. This simplification reduces the requirement of vehicledata. Thus

Eu ¼ B2

6hð21Þ

The usual range of h is from 1,016 to about 2,134 mm(TCRP 2000). A typical transit car has a typical h value of1,270 mm (TCRP 2000). For a typical transit car, the maximumsafe unbalanced superelevation is 295 mm. For a typical transitcar, a lateral acceleration of 0.2 g (≈ 0.196 g) should not posea safety threat. The middle third unbalanced superelevation of295 mm corresponding to the extreme outer end of the middle thirdof the track, supports the two abovementioned real-life tests inEurope and North America.

TCRP (2000) recommends an absolute maximum of 140 mm.The City of Calgary (2009) recommends 140 mm, as does TCRP(2000). The design maximum actual superelevation on passengerand transit railroads is 150 mm (TCRP 2000). Superelevation valuesof up to 180 mm are used with just one type of either passenger orfreight traffic (Esveld 2001).When appropriate, railways not admin-istered by the FRAmayuse up to 200mmof actual superelevation ona curved track. This has been applied to at least two North Americantransit systems (TCRP 2000). This suggests that with unbalancedsuperelevations of over 100 mm, the lateral acceleration wouldbe over 0.2 g. However, it is more common to limit the maximumactual superelevation to 150 mm on Light Rail Transit (LRT) sys-tems because it becomes more difficult to maintain ride comfort lev-els consistently at higher actual superelevations (TCRP 2000).

Alignment parameters and speeds on some internationalhigh-speed lines (HSLs) are taken from the Internet, and lateralacceleration is calculated in Table 1 to search for its upper limit.It appears from Table 1 that the German Railway (DB) applieda lateral acceleration of 0.2 g in 2002.

Application of a lateral acceleration of 0.2 g is now examinedagainst vehicle and track points of view:

Vehicle: In fact, the lateral design load for the suspension andtruck/bogie structure is above the lateral load corresponding to0.2 g. For example, the lateral design load of the MK II vehicleis as follows:• Lateral design static load: 0.36 g of car mass at crush

load (AW4)

Table 1. Examples of Lateral Acceleration on HSL

Line Opening Traffic

Maxspeed(km=h)

Minimumradius(m)

Maximumlateral

acceleration,a (g)

TGV-Atlantik SNCF 1990 Passenger 300 6,000 0.12TGV-SodostParis-Lyon

SNCF 1983 Passenger 270 4,000 0.14

Joetsu Shinkansen JR 1982 Passenger 260 4,000 0.13Tohuku Shinkansen JR 1982 Passenger 260 4,000 0.13Tokaido-Shinkansen JR 1964 Passenger 220 2,500 0.15NBS Frankfurt/Main-Koln

DB 2002 Passenger 300 3,350 0.20



Dow

nloa

ded

from

asc

elib

rary

.org

by

PEN

N S

TA

TE

UN

IV o

n 11

/07/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

• Lateral design dynamic load: �0.187 g of car mass of fullyloaded vehicle weight (AW2).Thus, lateral acceleration up to 0.2 g is not a safety concern

for vehicle trucks and suspension, and it does not apply to ridesafety either.

Track: A practical value for the lateral resistance required ofa loaded track to guarantee stability was determined in the 1950sby the French National Railway (SNCF) and is known as thePrud’homme formula (Esveld 2001), which reads as follows:

Htr > 10þ 1

3P ð22Þ

In general, the empirical coefficients appearing in Eq. (22) (here,10 and 1=3) depend on the type of track and its maintenance con-dition. This case is about a shovel-packed track with crushed stoneand wooden sleepers. Although measurements in a track packedwith concrete sleepers suggested a high value for the coefficient,this formula is generally adopted as a design standard (Esveld2001). So far as rolling stock is concerned, it is required thatthe horizontal wheel load Hrs exerted by a wheel be restricted,according to Esveld (2001):

Hrs < 0.85

�10þ P

3

�ð23Þ

The lateral wheel load is limited to less than 85% of the lateralstrength of track; in other words, 28% (¼ 0.85=3) of axle load plus8.5 kN (¼ 0.85 × 10). It is equivalent to lateral acceleration over0.28 g. The lateral acceleration of 0.2 g is not a concern in trackstability either.

In view of this discussion, from a safety point of view, a lateralacceleration of 0.2 g may be considered as the absolute maximumlateral acceleration on the track’s reference system given by v2=R.Comfort and safety under the speed suggested by Eq. (18) areanalyzed in the next section.

Comfort and Safety Analysis

As a general rule, any speed and transition that provide a com-fortable ride through a curve are well within the limits of safety(TCRP 2012). Thus, the comfort issue is discussed first, followedby the safety issue. The maximum curving speed is calculated fromEq. (18) in Table 2.

Comfort

Generally, comfort refers to the lateral acceleration measured in acar body’s reference system, usually on the floor on the top of theleading and trailing bogie pivots. Theoretically, passengers feel the

noncompensated part of the axle base lateral acceleration aug-mented by cant loss, which is determined by the suspension factor.The maximum speed is now checked against the comfort criteria.The resulting noncompensated lateral acceleration on the vehiclein the track plane is given by Esveld (2001):

ad ¼v2max

R− gEa

B¼ g · Eu

Bð24Þ

With a noncompensated lateral acceleration, the centrifugalforce exerted at the c.g. will cause a moment of overturning.The spring system of the vehicle body is compressed unevenly,and the body tilts toward the outside (Esveld 2001). Since thiscancels out some of the cant, the vehicle body, and, hence thepassengers and load, are subjected to a higher lateral acceleration(Esveld 2001):

ap ¼ ð1þ εÞad ð25Þ

This paper proposes that providing a superelevation that exceedsthe designed actual superelevation—an excess measured by somefraction of anticipated superelevation loss—would offset the neg-ative effect of superelevation loss. This would increase comfortby decreasing noncompensated lateral acceleration. Of course, itwould increase the spiral length, too.

The noncompensated and perceived lateral accelerations arecalculated in Table 3.

As per the UIC Code, the maximum value of ap to be consid-ered acceptable should not be outside the 1.0–1.5 m=s2 range(¼ 0.1–0.153 g) [UIC Code 703 (UIC 1989)]. The accelerationsap must in all cases remain below 1.5 m=s2, and preferably below1 m=s2 (Esveld 2001). Thus, the maximum speed does not seem tobe a threat to comfort and, hence, not a threat to safety either.

The unbalanced superelevation must be adapted to the suspen-sion factor. For modern rolling stock, this value is in the order of0.4 and can be reduced to 0.2 by special measures (Esveld 2001).A back calculation is done next to compute the unbalanced super-elevation to ensure a comfort level of 0.153 g for a suspensionfactor, ε of 0.4 [cf. Eqs. (24), (25)]:

0.153 g ¼ 1.4 × ad ¼ 1.4EuB

g

Eu ¼ 1,500 × 0.151.4

¼ 160 mm

It appears that no improvement of the suspension factor over0.4 is needed for unbalanced superelevation below 160 mm toensure a comfort limit of 0.15 g. To use an unbalanced superele-vation over 160 mm, a suspension factor below 0.4 will be required;

Table 2. Computation of the Maximum Allowable Speed, Vmax

R (m)V

(km=h)Eq(mm)

Ea(mm)

Eu(mm)

Vmax(km=h)

100 35 145 84 61 46200 50 148 86 62 65300 60 142 82 59 79400 70 145 84 61 92500 80 151 88 63 105600 85 142 83 59 111700 90 137 79 57 118800 100 148 86 62 1311,000 110 143 83 60 144

Table 3. Noncompensated and Perceived Lateral Acceleration under theMaximum Speed

R Vmax Eu ad (g) ε ap (g)

100 46 164 0.109 0.4 0.153200 65 167 0.111 0.4 0.155300 79 161 0.107 0.4 0.150400 92 164 0.109 0.4 0.153500 105 171 0.114 0.4 0.160600 111 161 0.107 0.4 0.150700 118 155 0.103 0.4 0.144800 131 167 0.111 0.4 0.1551,000 144 162 0.108 0.4 0.151

Note: For a suspension factor of 0.2, ap will be less than 0.1 g.



Dow

nloa

ded

from

asc

elib

rary

.org

by

PEN

N S

TA

TE

UN

IV o

n 11

/07/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

i.e., a much-improved suspension design will be needed. Civilspeed limits for a curved track are set by determining the maximumlateral acceleration that passengers can comfortably endure. This isusually in the range of 0.1–0.15 g, which establishes the levelof unbalanced superelevation on curves (TCRP 2012). Thus, formodern rolling stock, a k value of 0.5 in Eq. (13) is acceptablefor a comfort limit of 0.15 g. In North America, the comfort limitis usually chosen at 0.1 g. The TCRP recommends an unbalancedsuperelevation of 115 mm for modern LRT vehicles. But to ensurea comfort level of 0.1 g, the suspension factor should be 0.3 (max)for a 115-mm unbalanced superelevation. A k value of 0.5 is notsuitable for a comfort limit of 0.1 g. Thus, the k value must bereduced to ensure a comfort limit at 0.1 g.

Safety

The lateral accelerations (which are axle based) and unbalancedsuperelevation under the maximum speed are calculated inTable 4. The maximum lateral acceleration in Table 4 is less thanthe absolute maximum safe lateral acceleration of 0.2 g. Thus,the maximum speed does not seem to be a concern in terms ofvehicle safety and running. The maximum unbalanced superele-vation in Table 4 is 172 mm. The wheel offloading for a typicaltransit car (h ¼ 1,270 mm) under 172 mm unbalanced superele-vation is only 19% (¼ 2 × 172 × 1; 270=1; 5002), which is lessthan the limiting value of 60%. FRA (2008) allows a maximumoffloading of 60% that says derailment conditions are createdwhen a load drops to 0.4 of the nominal wheel load. In reality,no one in the rail industry uses such high unbalanced superele-vations as the ones in Table 4 to compute the operating speed toensure comfort. The analysis implies that all safe speeds may notbe comfortable.

Alternative and Better Method to Derive k Value

The k value is formulated from the comfort and safety pointsof view.

From the Comfort Point of View

Replacing k by kC and expressing the unit of speed in m=s, Eq. (13)is written as

3.6vmax ¼ kCffiffiffiffiffiffiffiffiffiREa

pð26Þ

Squaring both sides of Eq. (26) and rearranging:

v2max

R¼ Ea × k2C

3.62ð27Þ

By putting Eq. (27) into Eq. (24), noncompensated lateralacceleration is given by

ad ¼Ea × k2C3.62

− EaB

g ð28Þ

By putting Eq. (28) into Eq. (25), the lateral accelerationperceived by the passengers (noncompensated lateral accelerationaugmented by a suspension factor) is given by

ap ¼ ð1þ εÞ�Ea × k2C3.62

− EaB

g

�ð29Þ

From the ride comfort criteria, the following equation results:

ap ¼ Cg ð30Þin which

As per AREMA, C ¼ 0.1As per UIC Code, C ¼ 0.1 ∼ 0.15Equating Eqs. (29) and (30) results in the following:

ð1þ εÞ�Ea × k2C3.62

− EaB

g

�¼ Cg

kC ¼ 3.6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�C

Eað1þ εÞ þ1

B

�g

sð31Þ

When kC ¼ 0.39 ∼ 0.39 for Ea ¼ 150 mm, C ¼ 0.1, andε ¼ 0.2 ∼ 0.6; for Ea < 150 mm, the kC values will be higher.

When kC ¼ 0.44 ∼ 0.41 for Ea ¼ 150 mm, C ¼ 0.15 andε ¼ 0.2 ∼ 0.6; for Ea < 150 mm, the kC values will be higher.

From the Safety Point of View

Replacing k by kS in Eq. (14), the maximum safe kS value may beestimated by equating Eqs. (14) and (21):

ð11.8k2S − 1ÞEa ¼ B2

6hkS ¼ 0.29

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�B2

6h × Eaþ 1

�sð32Þ

When kS ¼ 0.43 ∼ 0.54 for Ea ¼ 150 mm, h ¼ 1,016 ∼2,134 mm (TCRP 2000); for Ea < 150 mm, the kS values willbe higher.

As a general rule, any speed and transition that provide a com-fortable ride through a curve are well within the limits of safety(TCRP 2012). Thus, all comfortable speeds are safe, but all safespeeds may not be comfortable. In general, the k value from com-fort criteria—kC—is less than it is from safety criteria, kS.

Maximum Curving Speed

A general equation for the maximum curving speed is given by


p

in which the maximum value of k is the minimum of the valuesobtained by Eqs. (31) and (32).

The unbalanced superelevation under the maximum curvingspeed is given by Eq. (14). The formula does not suggest a blanketunbalanced superelevation. With the increase of actual supereleva-tion, noncompensated unbalanced superelevation decreases, whichmakes room for more unbalanced superelevation. The formulasuggests an unbalanced superelevation proportional to the actualsuperelevation, which is sensible.

Table 4. Lateral Acceleration (Axle-Based) under the Maximum Speed

R Vmax

Ea(mm)

Eu(mm)

Max lateralAccn (g)

100 46 84 166 0.17200 65 86 163 0.17300 79 82 163 0.16400 92 84 166 0.17500 105 88 172 0.17600 111 83 159 0.16700 118 79 155 0.16800 131 86 167 0.171,000 144 83 162 0.16



Dow

nloa

ded

from

asc

elib

rary

.org

by

PEN

N S

TA

TE

UN

IV o

n 11

/07/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

In Tables 5–7, kC values are computed for different actual cantsand suspension factors (e.g., 0.6, 0.4, and 0.2), for a comfort limitof 0.1 g.

In Tables 8–10, kC values are computed for different actual cantsand suspension factors (e.g., 0.6, 0.4, and 0.2), for a comfort limitof 0.15 g.

For normal operating speeds calculated by Eq. (11), a suspen-sion factor is back-calculated in Table 11 for kC ¼ 0.38 to checkif any vehicle qualification test is required. Table 7 shows thatunbalanced superelevation is less than 75 mm. The FRA does notrecommend a vehicle qualification test for 75-mm unbalancedsuperelevations (FRA 2008). The requirement of a suspension fac-tor over unity does not suggest a vehicle qualification test for acomfort limit of 0.1 g. Thus, a vehicle qualification test is not re-quired to implement normal operating speed. In other words, for a kvalue of 0.38, a vehicle qualification test is not required. The FRArecommends a vehicle qualification test to implement unbalancedsuperelevations of 100 mm. The analysis implies that for somevalues greater than 0.38, a vehicle qualification test may not berequired either. Obviously, for an upper comfort limit of 0.15 g,a vehicle qualification test will not be required. It is to be noted

that a vehicle qualification test is required only when the suspen-sion factor is unknown and/or unbalanced superelevation is100 mm or more.

Validation

The interesting features of the model may make the case worthy ofexperimental validation. Experimental validation may be difficult,

Table 5. kC Values for Suspension Factor of ε ¼ 0.6 and Comfort Limit of0.1 g (C ¼ 0.1)

V R Eq Ea kC Vmax v2max=R (g)

50 250 118 69 0.45 59 0.1170 400 145 84 0.42 78 0.1275 500 133 77 0.43 85 0.1180 550 137 80 0.43 90 0.1285 600 142 83 0.43 95 0.1290 650 147 85 0.42 99 0.1290 700 137 79 0.43 101 0.12100 800 148 86 0.42 110 0.12



50 250 118 69 0.46 61 0.1270 400 145 84 0.44 81 0.1375 500 133 77 0.45 88 0.1280 550 137 80 0.45 93 0.1285 600 142 83 0.44 98 0.1390 650 147 85 0.44 103 0.1390 700 137 79 0.45 105 0.12100 800 148 86 0.44 114 0.13



50 250 118 69 0.49 64 0.1370 400 145 84 0.46 84 0.1475 500 133 77 0.47 93 0.1380 550 137 80 0.47 98 0.1485 600 142 83 0.46 103 0.1490 650 147 85 0.46 108 0.1490 700 137 79 0.47 110 0.14100 800 148 86 0.46 120 0.14



50 250 118 69 0.51 67 0.1470 400 145 84 0.48 87 0.1575 500 133 77 0.49 96 0.1580 550 137 80 0.48 101 0.1585 600 142 83 0.48 107 0.1590 650 147 85 0.47 112 0.1590 700 137 79 0.48 114 0.15100 800 148 86 0.47 124 0.15



50 250 118 69 0.53 70 0.1570 400 145 84 0.50 91 0.1675 500 133 77 0.51 100 0.1680 550 137 80 0.51 106 0.1685 600 142 83 0.50 111 0.1690 650 147 85 0.49 116 0.1690 700 137 79 0.51 119 0.16100 800 148 86 0.49 129 0.16

Table 10. kC values for Suspension Factor of ε ¼ 0.2 and Comfort Limitof 0.15 g (C ¼ 0.15)


50 250 118 69 0.56 74 0.1770 400 145 84 0.52 96 0.1875 500 133 77 0.54 106 0.1880 550 137 80 0.53 112 0.1885 600 142 83 0.53 117 0.1890 650 147 85 0.52 123 0.1890 700 137 79 0.53 126 0.18100 800 148 86 0.52 136 0.18

Table 11. Suspension Factor of ε for kc ¼ 0.38 and Comfort Limit of 0.1 g(C ¼ 0.1)

V R Eq Ea Eu ε

50 250 118 69 49 2.170 400 145 84 61 1.575 500 133 77 56 1.880 550 137 80 57 1.785 600 142 83 59 1.690 650 147 85 62 1.590 700 137 79 57 1.7100 800 148 86 62 1.5



Dow

nloa

ded

from

asc

elib

rary

.org

by

PEN

N S

TA

TE

UN

IV o

n 11

/07/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

but validation with Vampire, Nucars, Adams Rail, Medyna, or sim-ilar vehicle track interaction software would inspire confidence inthe model. In absence of such validation, the derivation process orthe background of the formula is discussed in great detail. The sug-gested formula takes the suspension factor and comfort limits intoaccount. Thus, the speed given by the model is limited by both thesuspension design and the comfort limit. For speeds in a higherrange, all comfortable speeds are likely to be safe, but all safespeeds may not be comfortable. The model is validated theoreti-cally from safety and comfort points of view, too.

Summary and Conclusions

Using the available formulas and safety and comfort limits, thispaper presents an approach to calculate a higher speed for railroadcurves. The approach can be useful for railroad administrations tooperate at higher speeds on curves.

The maximum curving speed is given by


p

in which k is the minimum of the two values from1. The comfort point of view:

kC ¼ 3.6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�C

Eað1þ εÞ þ1

B

�g

s

2. The safety point of view:

kS ¼ 0.29

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�B2

6h × Eaþ 1

�s

Thus, the model ensures both comfort and safety. In gen-eral, kC < kS.

Much testing occurs in the rolling stock supplier’s own researchand development labs, but it is not publicized, primarily becausemany manufacturers have their own proprietary designs. Suspen-sion factor data may not be readily available from vehicle manu-facturers. In the absence of such data, the following formulas arerecommended for modern rolling stocks by using the lowest k valuefrom a high suspension factor of 0.6.

For a comfort limit of 0.1 g:


p

The suggested maximum speed is 11% higher than the normaloperating speed. The maximum unbalanced superelevation underthe maximum allowable speed is given by

EuðmaxÞ ¼ 1.08 × Ea

For a comfort limit of 0.15 g:


p

The suggested maximum speed is 24% higher than the normaloperating speed. The maximum unbalanced superelevation underthe maximum curving speed is given by

EuðmaxÞ ¼ 1.6 × Ea

A blanket unbalanced superelevation is not desirable for all ac-tual superelevations, regardless of value. Proportioning equilibriumsuperelevation may be based on a ratio of 0.72 of unbalanced toactual superelevation; i.e., Ea ¼ Eq=1.72 and Eu ¼ 0.72 × Ea.

Notation

The following symbols are used in this paper:a = axle-based lateral acceleration (m=s2);ad = noncompensated lateral acceleration (m=s2);ap = perceived lateral acceleration (m=s2);B = track width, 1,500 mm;C = coefficient of comfort limit, 0.1 ∼ 0.15;

Ea = actual superelevation (mm);Eq = equilibrium superelevation (mm);Eu = design unbalanced superelevation (mm);

EuðmaxÞ = unbalanced superelevation (mm) under the maximumcurving speed;

g = acceleration due to gravity, 9.81 m=s2;Htr = minimum lateral force (kN) that the track should be

able to resist without deformation;Hrs = horizontal wheel load (kN);h = height of c.g. (loaded) above the top of the rail

level (mm);k = speed model coefficient, min (kc, ks);kc = speed model coefficient from the comfort point of

view;ks = speed model coefficient from the safety point of view;P = axle load (kN);R = radius of curve (m);V = normal operating speed (km=h);

VEq = equilibrium speed (km=h);Vmax = maximum curving speed (km=h);

v = normal operating speed (m=s);vmax = maximum curving speed (m=s);

x = lateral shift of c.g. (mm);ΔEu = excess unbalanced superelevation on top of the design

unbalanced superelevation (mm); andε = suspension factor or roll coefficient, 0.2 ∼ 0.4; and

θ superelevation (cant) angle (radian).

References

Ahlf, R. (2003). “Characteristic of railway location and operation, Funda-mentals of railway engineering, light rail transit, rapid transit, andcommuter rail systems, Univ. of Wisconsin-Extension (UNEX), PyleCenter, Madison, WI, 38.

City of Calgary. (2009). “LRT design guide line.” Section 3-Trackalignment: Exhibit 3.3.4a, Calgary, Canada.

Esveld, C. (2001). Modern railway technology, MRT-Productions,The Netherlands, 38, 39, 61.

Federal Railroad Administration (FRA). (2008). Title 49, code of federalregulations, Railway Educational Bureau, Omaha, NE.

Hasan, N. (2011). “Maximum allowable speed on curve.” Proc. ofJoint Rail Conf., American Society of Mechanical Engineers,New York, 8.

International Union of Railways (UIC). (1989). “Layout characteristics forlines used by fast passenger trains.” Code 703, 2nd Ed., Paris, 8–9.

International Union of Railways (UIC). (1981). “Geometry of points andcrossings with UIC rails permitting speeds of 100 km=h or more on thediverging track.” Code 711 R, Paris, 5.

Transit Cooperative Research Program (TCRP). (2000). “Track designhandbook for light rail transit.” Rep. 57, National Academy Press,Washington, DC.

Transit Cooperative Research Program (TCRP). (2012). “Track designhandbook for light rail transit.” Rep. 155, National Academy Press,Washington, DC.



Dow

nloa

ded

from

asc

elib

rary

.org

by

PEN

N S

TA

TE

UN

IV o

n 11

/07/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Documents

Maximum Curving Speed