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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:[email protected]
RAIL CONFERENCE, 2011
INTRODUCTION
2
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
3
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)
4
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”.
5
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:
6
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:
7
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.
8
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.
9
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.
10
ABSOLUTE MAXIMUM
UNBALANCED SUPERELEVATION
h
xs
s
Eu
6
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.
11
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.
12
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.
13
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)
14
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