TRANSITION CURVES E.M.G.S. MANUAL ( 1) Transition Curves (or Easement Curves) SHEET 1 Issue Introduction. 1. Introduction.

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1 TRANSITION CURVES SHEET 1 Transition Curves (or Easement Curves) 1. Introduction. 2. Development of Transition Curves and the Adoption of Superelevation (Cant). 3. General Application of Transition Curves to Model Railways. 4. Designing Transition Curves for 4mm Scale Model Railways: 1. Introduction When the first railways were built the tracks consisted of a series of circular curves joined by straights, Fig.1. In model form this is found in the boxed toy train set and illustrated in the catalogues of the set track manufacturers. On railways constructed on these lines considerable damage would occur to both track and rolling-stock and the speed would have to be severely restricted. Table I. Table II. Table III. a) b) c) d) e) Tables IV-XII Length of Transitions. Tangent Shift. Setting out Transition Curves. Making Templates. Use of Templates. 5. Super-Elevation (Cant). Approximate Values of V (Speed) Equivalent Transition Lengths (approximate Metric -Imperial) Equivalent Radii (approximate Metric - Imperial) Transition Offsets and Tangent Shifts 6. Ideas for Forming Superelevation. i) For Open Top Baseboards. ii) For Solid Top Baseboards. 2. Development of the Transition Curve and the introduction of Superelevation. (Cant) The development of the transition curve was a gradual one and took more than 100 years to perfect. The first solution used to try to eliminate the Jolt when vehicles pass from straight track to a circular curve was to introduce superelevation or cant to the curve with the outer rail of the curved track raised as much as 6 inches above the inner one. The elevation was built up on the straight track on the approach to the curve, Fig.2. This method had only partial success as the train speed was limited due to the lurch which was still present with the sudden change in direction. The next step was to introduce a section of track with a larger radius than the main curve to act as a funnel. At first this larger radius curve was made twice the radius of the circular curve, Fig.3. The natural progression was to lay out a series of curves of gradually decreasing radii between the straight and the circular curve, which finally led to the introduction of the Spiral Transition of uniformly increasing sharpness. Eventually the Cubic Parabola y=kl 3 was adopted which gave an almost perfect transition provided only the centre of the parabola was used, Fig.4. It can now be seen that the elevation could be progressively increased over the length of the transition until the desired maximum cant was obtained at the beginning of the circular curve, Fig.5.

2 SHEET 2 3. General Application of Transition Curves to Model Railways It will be obvious that a model railway layout with transition curves is more desirable than one without. The simplest way of forming the required curve is to take a length of flexible track and let it form its own transition between the straight and the circular curve. If this is good enough for you, read no further, but for those who wish to know more and design the line of their tracks the following will be of interest: 4. Designing Transition Curves for 4mm Scale Model Railways (Millimetres are used for all calculations) a) Length of Transitions. The limiting factor is the amount of lurch experienced and this is called the Rate of change of Radial Acceleration = A. Where L = the length of the transition. v = the maximum train velocity. R = the radius of the circular curve. L = V 3 A.R Prototype practice is based on A = 300mm/sec 3 but it is suggested that for modelling purposes where lighter rolling stock is being used, a value of A = 150mm/sec 3 as a maximum should be adopted. This is not the overruling factor as there are other points to consider. Assume some values for an example: Let V = 300mm/sec (approx 60 mph) see Table l. R = 900mm (approx 3 ft) A = 100mm/sec 3 Therefore with the equation L = V 3 A.R L = 300 x 300 x 300 = 300mm 100 x 900 By juggling with the desired figures the length of the transition can be found.

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4 b) Tangent Shift. The next consideration is that by the introduction of the transition curve the whole of the circular curve must be shifted away from the tangent, see Fig.6 which also moves the tangent point further back. The shift of the curve is found from the equation S = L 2 24R and the shift of the tangent point is approximately = L 2 Thus for the transition of 300mm in length with a circular curve radius of 900mm the curve shift S = L 2 24R d) The Making of Templates SHEET 4 It may be found more convenient to construct templates for use when setting out on the layout baseboards, and these can be made using cardboard, plasticard, hardboard or plywood. If hardboard or plywood are used these materials should be given a coat of white emulsion paint. By using the equations or the information given in tables 4-12, the transitions can be drawn out. For example a piece of card approximately 250mm x 70mm will be required. Start with a straight line parallel to and 25mm from the longest side. Mark off at 25mm intervals along this line and setout at right angles the offset as shown in Fig.7. Drill the holes along the transition to accept a pencil point, also some 5mm o / Spy holes as indicated in Fig. 8. Therefore S = 300 x 300 = 100 = 4.2mm 24 x The Movement of the Tangent Point = L = 300 = 150mm 2 2 Having decided that the ideal curve, or such compromise as is necessary, can be fitted into the site, the setting out may proceed. c) Setting Out the Transition. With reference to Fig.7 it will be seen that it is possible to plot the curve by using the values of I in the equation y = I 3 6LR to derive the offsets y from the tangent. Again with transition length L = 300mm and radius of circular curve R = 900mm Then 6LR = 6 x 300 x 900 = 1,620,000 Therefore when I = 225 y = I 3 = 225 x 225 x 225 = 7mm 6LR y can be calculated for increasing values of I at 25mm intervals. l y l y * 2.1 * It will be seen that half way along the transition (at the tangent point of the circular curve) the offset is half the tangent shift. It will also be noted that at TP 2 where the transition meets the circular curve that I = L and therefore the offset y = I 3 = L 3 = L 2 6LR 6LR 6R e) Use of the Template. i) Decide on the circular curve and draw it on the baseboard. ii) iii) Lay the template for the chosen transition on the curve so that the points TP2 and T fall on the line of the circular curve. Side the template around the curve until the tangent A I - A - TP I points exactly in the direction required for the straight track. v) Holding the template in this position mark through the holes along the line of the transition, also mark the holes on the tangent. v) Remove the template and draw a line through the pencil marks previously made. Alternatively push common pins through the holes on the transition, snip off their heads with side cutters before removing the template. If you wish to locate the centre of the circular curve in relation to the straight line the following procedure may be followed: Lay the tangent TP I - A - A I of the template on the straight, mark out the transition points as well as T and TP 2. With the required radius and the centres T and TP 2 strike two arcs to give 0 the centre of radius of the circular curve. Now with the same radius and centre 0 draw in the required circular curve, then join up the points previously marked to form the transition. For those who require quick approximate Imperial/Metric conversions, Table 2 gives transition lengths and Table 3 gives equivalent Radii for Curves. If access to a calculator is difficult, Tables 4-12 give offsets, shifts and shift of tangent points for transitions from 200mm (8") to 600mm (24") in length with circular curves with radii ranging from 760mm (2'-6") to 1520mm (5'-0"). A total of 144 transitions.

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6 SHEET 6 5. Superelevation (or Cant) Superelevation is the amount by which the outer rail of curved track must be raised to overcome the Centrifugal force of a vehicle traversing that curve. Due to the gross reduction in radius of curves on model railways and the weight factor it is virtually impossible to translate prototype practice into model form as the cant required is the product of the train speed, vehicle weight and curvature. On full size railways superelevation does not exceed 6" (2mm in 4mm scale) see Fig.9. In these notes no attempt has been made to give any formulae for the calculation of the required cant. As is well known most model railways are without cant. This situation can occur in the prototype as a deficiency of cant, of 90mm (3½ inches) on jointed track and 110mm (41/4 inches) on continuously welded rail, is allowed. In model form this situation is obviated by wheels having overscale flanges (except 18.83mm gauge / P4). However, the graph in Fig.10 suggests the cant to be applied on model railway circular curves. 6. Ideas for Forming Superelevations i) For Open Top Baseboards. With this form of baseboard the superelevation can be applied to the supports of the track bases. Fig.11 shows a cross section of a baseboard and track supports. Curvature, transitions and cant must all be worked out before construction starts, although small subsequent alterations can be made with packings during construction. ii) For Solid Top Baseboards. Cant can be applied in two ways with this type of baseboard. a) Superelevation can be added between the baseboard top and the cork ballast underlay as shown in Fig.12. b) Alternatively packing may be introduced between the underside of the track and the cork ballast underlay as shown in Fig.13. The packing will not be apparent after ballasting. With both methods a) and b) the increase in cant over the length of the transition can be achieved as follows: Take some strips of notepaper approx 6mm wide and stack them to ascertain the number required for the full cant which is to be applied on the circular curve. Say you are using 1mm of cant which could need 10 strips of paper. Divide the transition length by the number of strips and then space them out, see Fig.14. The track will even out the gradual increase in cant. D.G.Yule (Hampshire)

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8 SHEET 8 Table 1 20mph 30mph 40mph 50mph 60mph 70mph 80mph 90mph 100mph Approximate Values of V (Speed) 100mm/sec 150mm/sec 200mm/sec 250mm/sec 300mm/sec 350mm/sec 400mm/sec 450mm/sec 500mm/sec Table 2 Equivalent Transitition Lengths Transition Length Approximate L mm Equivalent inches Table 3 Equivalent Radii Radius mm Equivalent Radius ft. in. Radius Equivalent Radius ft. in ' 6" '10" 810 2' 8" ' 0" 860 2'10" '2" 910 3' 0" '4" 960 3' 2" ' 6" ' 4" ' 8" ' 6" '10" ' 8" ' 0" TRANSITION LENGTH 200mm 2.0 TRANSITION LENGTH 250mm

9 SHEET 9 TRANSITION LENGTH 300mm TRANSITION LENGTH 350mm TRANSITION LENGTH 400mm

10 TRANSITION LENGTH 450mm SHEET 10 TRANSITION LENGTH 500mm

11 TRANSITION LENGTH 550mm SHEET 11 TRANSITION LENGTH 600mm IMPORTANT NOTE Cant Gradient When the Cant is increased over a transition curve a "TWIST" of the track takes place creating a potential derailment situation. It is therefore suggested that the rate at which the cant is increased (i.e. the rate at which the high rail climbs in relation to the low rail) should not exceed 1 in 250, and it is recommended that even with cant gradients of this magnitude equalised chassis for all rolling stock including locomotives should be used. A cant gradient of 1 in 250 means that the cant would rise to a height of 1mm in a length of 250mm. D.Yule (Hampshire)