Upload
doanlien
View
218
Download
0
Embed Size (px)
Citation preview
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
6
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.