Maximum Superelevation: Desirable, Allowable, and Absolute Nazmul Hasan, M. Eng. SNC-Lavalin Inc. ancouver, 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 []. 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,. to design superelevation, and 3. to calculate the maximum allowable operating speed on an existing curve. Discussion on each purpose follows. 1
DISCUSSION First Purpose: The first purpose is to proportion 3.33 sec s ride i.e. 0.95 to derive an equation of the spiral length for any superelevation. In the current literature [1] this is done as under: L 0.95 (max allowable ) superelevation and 65 mm for maximum allowable unbalance superelevation. So proportioning of 0.95 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 0.95 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 0.95 (km/h) and the maximum desirable values are based on 3.33 sec s (=0.1g/0.03g/s) ride. So 0.95 should be proportioned by the maximum desirable values as shown below: L 0.95 (max allowable ) L 0.95 Eq Eq (max desirable ) 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 and 115mm for have been used by TCRP [1] to derive spiral length equation for any value of and as under: L 0.95 0.006 150 L 0.95 0.008 115 (1) () Both the Eq. (1) and () 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 L 0.95 L 0.95 (max desirable ) (max desirable ) The maximum desirable values are determined on the basis of following comfort criteria [4]: Radial acceleration Jerk Roll run-off = 0.1 g = 0.03 g/s = 1 deg/sec The maximum desirable values are given below [4]: Equilibrium superelevation Actual superelevation Unbalanced superelevation = 150mm = 87mm = 63mm The proportioning of 0.95 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:
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 1.7 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: max x 11.8 * R The maximum allowable unbalanced superelevation 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 spiral length is given by the formula [3]: L 161 (3) This formula is used to determine the maximum allowable unbalanced superelevation,. Differentiating both sides with respect to time, t: dl d (161 * * (161 d d d (161 * * (161 3
d 6.18mm/ s [4] d 18.8mm/ s [4] (161 * *6.18 * *18.8 ( m / s) (161 Multiply both sides by 3.6 to change the unit of speed term on the left side from m/s to km/h (161 * * 6.18 * *18.8 3.6 ( m / s) 3.6 (161 3.6* 6.18(161 * 3.6*18.8 * (161 94.48(161 * * 67.75 * (161 Cancelling from both sides 94.48(161 67.75 1 (161 (161 (161 161 161 94.48(161 67.75 94.48(161 67.75(150 *161* *161* 3 161 95.50 0909.57 0 (67.75 94.48) 94.48*161 67.75*150 6.498 5011.48 6.498 5011.48 0 95.50 95.50 4* 0909.57) 178mm,117mm, say115mm 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 4
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 00 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 []. 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 5 mm (=35*0.7). Current literature supports a value very close to 5 mm. If the calculated cant is less than 0 mm it can be disregarded [7]. In a way the statement supports a minimum unbalance of 0 mm. The minimum allowable actual unbalance of 5 mm suggests the maximum allowable superelevation 5 of 15 mm (=150-5 mm). It is equivalent to a cross gradient of 8.33% (=15/1500) 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 ery high super-elevation can cause load displacement. Stability of work vehicle and of special loading with a high centre of gravity can be jeopardized ery 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 0.7 [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 Max Max Max allowable allowable 115 allowable 1 1 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
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 s s( x) 6 (4) h 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 ~,134 mm, the maximum safe unbalance comes out to be 109 ~ 50 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 [6]. It suggests that a radial acceleration above 0.g does not exceed the safety limit. So, the upper bound of safe radial acceleration may be assumed to be 0.g. This assumption leads to the absolute maximum value of actual superelevation of 140 mm (=0.*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: min R( 5) 11.8 It means the restricted speed should not reach below the minimum speed dictated by the 5 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
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 NOTATIONS Actual superelevation (mm) 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 wih, 1500 mm Speed (km/h) min Minimum speed (km/h) x Shift of c.g. towards high rail (mm) REFERENCES [1] TCRP, 000, Track Design Handbook for Light Rail Transit, Report # 57, National Academy Press, Washington, pp. 3-13, 3-, 3-3,3-5. Actual superelevation (mm) Unbalance superelevation (mm) 87 115 140 63 115 161 [] The City of Calgary; 009, LRT Design Guide Line, Section 3- Track Alignment [3] Hasan, N., Spiral Length Design, JRC 010-36050, Proceedings of the 011 ASME/ASCE/IEEE Joint Rail Conference, Urbana, Illinois, USA. Radial acceleration, g 0.1 g 0.15 g 0. g Table 1: Maximum alues of Superelevation and Radial Acc n The minimum operating speed is given by: min R( 5) 11.8 [4] Hasan, N., Passenger Track Curve Design Criteria: Comfort Criteria, Equivalent Comfort Criteria, and Application, JRC 011-5601, Proceedings of the 011 ASME/ASCE/IEEE Joint Rail Conference, Colorado, Pueblo, USA. [5] U.S. Department of Transportation, Federal Railroad Administration-Office of Safety, 008, Code of Federal Regulations Title 49, The Railway Educational Bureau, Omaha, NE,USA, pp. 1. 7
[6] Hasan, N., Maximum Allowable Speed On Curve, JRC 011-56007, Proceedings of the 011 ASME/ASCE/IEEE Joint Rail Conference, Colorado, Pueblo, USA. [7] Esveld, C., 001, Modern Railway Technology, MRT-Productions, The Netherlands, pp.37. [8] Hasan, N., DFF Spacing and Stiffness Design, JRC 011-56008, Proceedings of the 011 ASME/ASCE/IEEE Joint Rail Conference, Colorado, Pueblo, USA. 8