Track Design Handbook for Light Rail Transit, Second EditionThe result should be compared against the minimum spiral lengths defined by the formulae that considered unbalanced superelevation and track twist and the longest spiral selected. Unless Eu has been artificially constrained so as to keep lateral acceleration well under 0.1 g, the formula considering unbalance will usually govern. As noted in the last paragraph of Article 3.2.5.4.2, the minimum lengths for deliberate track twist situations should be based on the formulae given in this chapter for minimum spiral lengths. Such situations include both changes in crosslevel in embedded track and twisted decks on aerial structures. In addition to the discussion above, there are a number of documents with good explanations of the derivation of runoff theory; the references at the end of this chapter contain extensive background on the subject.[8], [9], [10], [11] 3.2.6 Determination of Curve Design Speed The calculation of design speed in curves is dependent on vehicle design and passenger comfort. In addition to the preceding guidelines, curve design speed can be determined from the following principles if specific vehicle performance characteristics are known. This analysis is also necessary if the vehicle dimensions are significantly different than the LRT vehicles described in Chapter 2. 3.2.6.1 Categories of Speeds in Curves Speed in curves may be categorized as follows: • • • • Overturning Speed: The speed at which the vehicle will derail or overturn because centrifugal force overcomes gravity. Safe Speed: The speed limit above which the vehicle becomes unstable and in great danger of derailment upon the introduction of any anomaly in the roadway. Maximum Authorized Speed (MAS): The speed at which the track shall be designed utilizing maximum allowable actual superelevation and superelevation unbalance. Signal Speed: The speed for which the signal speed control system is designed. Ideally, signal speed should be just a little faster than the speed at which an experienced operator would normally operate the vehicle so that the automatic overspeed braking system is not deployed unnecessarily. 3.2.6.2 Determination of Eu for Safe and Overturning Speeds Figure 3.2.4 illustrates a typical transit car riding on superelevated track and the forces associated with the vehicle’s center of gravity. Due to the characteristics of the vehicle’s suspension system, as it negotiates the curve the center of gravity will shift outboard of a point over the centerline of the track. The resultant vector of the mass of the vehicle and centrifugal force will shift toward the outer rail. A typical high-floor transit car has a center of gravity shift (x) and height (h) of 2.50 inches [63.5 mm] and 50.00 inches [1270 mm], respectively. By contrast, a freight railroad diesel locomotive has typical ‘x’ and ‘h’ values of 3 inches [76 mm] and 62 inches [1575 mm], respectively. 3-32 Light Rail Transit Track Geometry Figure 3.2.4 Force diagram of LRT vehicle on superelevated track 3.2.6.2.1 Overturning Speed Overturning speed is dependent upon the height of the center of gravity above the top of the rail (h) and the amount that the center of gravity moves laterally toward the high rail (x). When the horizontal centrifugal forces of velocity and the effects of curvature overcome the vertical forces of weight and gravity, causing the resultant vector to rotate about the center of gravity of the vehicle and pass beyond the outer rail, derailment or overturning of the vehicle will occur. The formula for computing superelevation unbalance for ‘Overturning Speed Eu’ is derived from the theory of superelevation: Overturning Speed Eu = Be/h where B = rail bearing distance = 59.25 inches [1520 mm] as discussed earlier e = B/2 – x h = height of center of gravity = 50 inches [1270 mm], which is an average for a typical high-floor LRV If ‘x’ = 2 inches [50 mm], then e = [(59.25/2) – 2] = 27.625 inches [702 mm] 3-33 Track Design Handbook for Light Rail Transit, Second Edition then Overturning Speed Eu = and Overturning Speed V = (59.25 * 27.625) = 32.7 inches [831 mm] 50 (Eu + Ea) * R 3.96 For example, if ‘Ea‘ is given as 6 inches [150 mm] and curve radius is 1145.92 feet [349.3 meters] ( a 5o00’00” curve in arc definition), then Overturning Speed V = (32.7 + 6) * 1145.92 3.96 = 106 mph [170 km/h] Obviously, the overturning speed will always be far in excess of the curve’s maximum authorized speed. 3.2.6.2.2 Safe Speed It is generally agreed that a rail vehicle is in a stable condition while rounding a curve if the resultant horizontal and vertical forces fall within the middle third of the distance between the wheel contact points on the rails. This equates to roughly the middle 20 inches [500 mm] of the bearing zone ‘B’ indicated in Figure 3.2.4. Safe speed is therefore an arbitrarily defined condition where the vehicle force resultant projection stays within the one-third point of the bearing distance. That speed is entirely dependent upon the location of the center of gravity, which is the height above the top of rail ‘h’ and the offset ‘x’ of the center of gravity toward the outside rail. From the theory of superelevation, we derive the formula for computing superelevation unbalance for maximum safe speed ‘Eu.’ Safe Speed Eu = Be/h where B = rail bearing distance = 59.25inches [1520 mm]) e = B/6 – x If ‘x’ = 2 inches [50 mm], then e = (59.25/6) – 2 = 7.875 inches [200 mm] h = height of center of gravity = 50 inches [1270 mm] then Safe Speed Eu = and Overturning Speed V = square root (((Eu + Ea) x R) /3.96) (59.25 * 7.875) = 9.3 inches [237 mm] 50 3-34 ight Rail Tr ransit Trac ck Geometr ry Lig For example, e if ‘E Ea‘ is given as 6 inches [1 150 mm] and d curve radiu us is 1145.92 2 feet [349.3 o meter rs] ( a 5 00’00” curve in ar rc definition), then ed V = Overturning Spee (9.3 + 6) * 1145.92 2 3.96 = 66.6 mph h [107.1km/h] 3.2.7 Reverse Cir rcular Curves s e horizontal geometry ma akes it impo ssible to pro ovide sufficient Where an extremely restrictive tange ent length be etween revers sed superele evated curves s, the curves s may meet at a point o of revers se spiral (PRS S). As a guid deline, the PR RS should be s set so that LS1 x Ea2 = LS2 x Ea1 where e Ea1 = Ea2 = LS1 = LS2 = actual super relevation app plied to the fir rst curve in in nches or millim meters actual super relevation of the t second ci ircular curve iin inches or m millimeters the length of f the spiral lea aving the first t curve in feet t or meters the length of f the spiral en ntering the se econd curve in n feet or mete ers A sep paration of up to about 3 fe eet [1.0 meter r] of tangent tr rack between n the spirals is s acceptable in lieu of f meeting at a point of reve ersal. The superelevation s n transition between b reversed spirals iis usually acc complished b by sloping bot th rails of o the track throughout t th he entire tran nsition spiral, as shown in Figure 3.2 2.5. Note tha at throug gh the transit tion, both rail ls will be at an a elevation above the th heoretical profile grade line e. This method m of su uperelevation transition creates additio onal design c considerations s, including a an increa ased ballast section s width at the point of o the reverse spiral and po ossible increa ased clearanc ce requir rements. Suc ch issues mus st be investiga ated in detail before incorp poration into t the design. Figure 3.2.5 5 Superelev vation transit tions for reve erse curves It is entirely e possib ble to have re everse spirals and remain n within acce eptable ride comfort criteria a. This is indeed the practice fo or European interurban r railway alignm ments and is s occasionally incorp porated into North American practic ce.[6] However, because e the direct tion of later ral accele eration chang ges at the PR RS, the spiral l lengths requ uired for reve erse spirals to o maintain rid de comfo ort should be made appreciably longer r than the abs solute minimu um by limiting g the jerk rate e, 3-35 Track Design Handbook for Light Rail Transit, Second Edition with 0.03 g/s as a suggested absolute maximum. See Article 3.2.4 for additional discussion on jerk rate and lateral acceleration. Refer to Article 3.2.1 for additional discussion on desirable minimum tangent distances between curves. 3.2.8 Compound Circular Curves A transition spiral should be used at each end of a superelevated circular curve and between compound circular curves. Between compound curves, the spiral segment, instead of having an infinite radius at one end, will match the radius of the larger curve. The remainder of the spiral between that radius and the theoretical spiral-to-tangent point, where the radius would be infinity, is effectively not used. The minimum compound curve spiral length is the greater of the lengths as determined by the following: L L L S S S = f1 (E a2 − E a1 ) = f2 (E u2 − Eu1 ) V = f3 (E a2 − E a1 ) V where LS = f1 = minimum length of spiral, in feet [meters] the factor used in the corresponding equations for ordinary spiral length based on track twist (i.e., “desirable,” “acceptable,” and “absolute,” minima as appropriate to the design circumstances) actual superelevation of the first circular curve in inches [millimeters] actual superelevation of the second circular curve, in inches [millimeters] the factor used in the corresponding equations for ordinary spiral length based on unbalanced superelevation and speed superelevation unbalance of the first circular curve, in inches [millimeters] unbalanced superelevation of the second circular curve, in inches [millimeters] the factor used in the corresponding equations for ordinary spiral length based on actual superelevation and speed design speed through the circular curves, in mph [km/h] Ea1 = Ea2 = f2 = Eu1 = Eu2 = f3 V = = Ride comfort in spiraled compound curves is optimized if Eu is the same value in both circular curve segments. 3.2.9 Track Twist in Embedded Track When LRT tracks are embedded in pavement and particularly where they are in a shared mixed traffic lane, in many cases the track geometry will be dictated by the roadway agency’s criteria for pavement surface. These are typically dictated by the need to drain storm water off of the pavement surface. As a consequence, there will often be some cross slope in tangent lanes to which the track will need to conform. If this cross slope changes when the street (and track) 3-36 Light Rail Transit Track Geometry enters a curve, twist will occur over some distance. The track designer must verify that this rate of twist does not exceed the criteria specified in this chapter. It is also important to note that it is unlikely that the street alignment will be spiraled. The spiral lengths in the track must be carefully coordinated with the roadway design so as to both match the pavement surface and keep the horizontal track alignment in an optimal position relative to the traffic lanes. See Chapter 12 for additional discussion on this topic. 3.3 LRT TRACK VERTICAL ALIGNMENT The vertical alignment of an LRT alignment is composed of constant grade tangent segments connected at their intersection by parabolic curves having a constant rate of change in grade. The nomenclature used to describe vertical alignments is illustrated in Figure 3.3.1. The percentage grade is defined as the rise or fall in elevation, divided by the length. Thus a change in elevation of 1 foot over a distance of 100 feet is defined as a 1% grade. When using European reference sources, it is fairly common to see gradients defined in terms of the rise or fall in meters per kilometer. This ratio is known as “per mille” (literally, “per thousand” in Latin) and is usually abbreviated as 0/00. The similarity between that symbol and the more familiar “percent” symbol (%) can result in much confusion. The profile grade line in tangent track is usually measured along the centerline of track between the two running rails and in the plane defined by the top of the two rails. In superelevated track, the inside rail of the curve normally remains at the profile grade line, and superelevation is achieved by raising the outer rail above the inner rail. One exception to this recommendation is in circular tunnels, such as might be created by a tunnel-boring machine, In such cases, the superelevation may be rotated about the centerline of track in the interest of minimizing the size of the tunnel without compromising clearances. Note that circular rail transit tunnels follow a different mathematized alignment than the track. The tunnel’s profile grade line (PGL) effectively is coincident with the geometric center of the boring machine. In curved segments, the relationship between the tunnel PGL, the track PGL, and the rails will be complex as the tunnel PGL shifts inboard of the track centerline through curves so that clearances can be maintained. The vehicle’s performance, dimensions, and tolerance to vertical bending stress dictate criteria for vertical alignments. The following criteria are used for proposed systems using a modern lowfloor vehicle. It can be used as a basis of consideration for general use. 3.3.1 Vertical Tangents The minimum length of constant profile grade between vertical curves should be as follows: Condition Main Line Desired Minimum Length 100 feet [30 meters] or 3 V [0.57 V] where V is the design speed in mph [km/h], whichever is greater 40 feet [12 meters] Main Line Absolute Minimum 3-37