P. T. pro bl ems. increase as R decreases assuming that the velocity remains constant.

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1 PART VI COMPOUND CURVES Any simple circular curve used in a highway, railway or canal presents pro bl ems. At the P. C. there is an instantaneous change of direction and the magnitude of this change increases as the radius decreases. In the case of a highway or railway, a vehicle or a train cannot change direction instantaneously. In the case of the vehicle, it does not follow a truly circular path in the first part of the curve. In the case of the train it imposes forces on the rails which may cause them to shift. The magnitude of the forces will increase as R decreases assuming that the velocity remains constant. There is a1 so the problem of super-elevating or " bankingu a simp1 e circular curve. The greater the design speed the greater will be the need for superelevation. Theoretically, there should be no superelevation on one side of the P. C. and there should be the required superel evation on the other, with no transition in between. Because such a situation is impractical we have to compromise and perhaps begin our superelevation before the P.C. is reached so that the first part of the curve has, at least, part of the required superelevation. The situation becomes worse as R is decreased, for a constant design speed. Obviously what has been said in the preceding two paragraphs applies a t the P.T. as well. On canals, too rapid a change in direction could lead to erosion problems. Because the magnitude of the problems of change of direction and superelevation increase as R decreases we can, at least partially, relieve the problem by combining arcs of simple curves of different radius as shown. The curve shown in Ffg. I consi sts of two outer arcs, both having the same long radius and the same central angle. The central portion has s short radius. In this case the curve is symmetrical about the dotted Q and is called a Symmetrical ly Compounded Curve. The long radius portion between the P.C. and the P.C.C. (Point of Compound Curvature) a1 lows for a more gradual change in direction and a more gradual (and smoother) appl icat ion of superel evation so that the short radius portion will be safe. The second long radius portion (from the second P.C.C. to the P.T.) allows for a gradual change back to straight-line conditions beyond the P. T.

2 Figure 1

3 We can have a multi-centred curve as illustrated below* This is a multi-centered In terns of riding and driving qualities it would be an improvement over the symnetri call y-compounded curve in Figure 1. Later we will discuss a better solution to the problem-the so-called transition spiral Figure 2 There are many applications of mu1 ti-centered curves which are not symmetrical-particularly in various types of grade-separation structures.

4 Calculate the sub-tangent distance, T, for the symnetrically-compounded curve shown above. In this case we can make use of symnetry to set up a simple solution. We can treat A00 and B0C each as separate self-contained curves. Through B we can draw a 1 ine V, BV2, I to BO, which is tangent to both curves. Figure 4

5 AV, = VIB = TI BVp = V2C = TZ VIVp = TI + T2 180-A Angle AVV2 = Angle VV V - A1 = 15' 1 2- By deduction, Angle V, V2V = 180' - 60'-15' = 105' T, = 500 x Tan 7'30' = We can use the sin law to solve for V1 Y in the triangle VVIVZ We can now try an example where the curve is not symmetrical.

6 2. Example Two Figure 5 Figure 6

7 AVl = VIB = TI = 400 Tan 10' = BV2 = V2C = T2 = 300 Tan lo0= CV3 = V D = T = 200 Tan lo0= In Figure 6 Angle We must now calculate the 1engthsAV and VD by the method of latitudes and departures. The approach to all such problems will be the same. As the number of different arcs is increased the only change will be the length of the latitudes and departures problem. 4 '5; VV1 V1 V2 vzv3 V3V Traverse Data Lenqth Bearing. - Ass. West ~70'~ ~50'~ ~30'~ 0 Functions Cos Sin Latitudes -I Departures + I (4) (3) 1

8 * 1. c Southings = = c Northings All Northings are in V3V = V3V x Cos 30' = V3v = pp = Cos Dep. in V3V = x = 5' 4. c Eastings = = c Westings Balance of Westings is VV1 VV, = =v Back Tangent = AV = VV1 + T1, Forward Tangent = VD = VV3 + T3 = VD = , Note that, in any compound curve, the curve must be run in sections, the instrument occupying the beginning or end of each section of different radius. It should also be noted that it is necessary to put in control independent of the curve in order to isolate any errors that may be made. In the case of the assymetrical compound curve in Figures 5 and 6 it i s highly desirable (in fact, essential) that points V1, B, V2, C and V3 be 1 ocated before one begins running the curve. 3. Example Three Two tangents intersect at n = 70, P. I. = Stn Set up the field notes for the following compound curve (symmetrical). 1st curve A = 20, D = 8'~ (20 rn chords) 2nd curve A = 300, D = 120R (10 m chords) 3rd curve A = 200, D = 80R (20 m chords)

9 Solution 1st curve R1 = 2nd curve R2 = 20 8 = m 2 sin (Z-) TI = 'to R1 Tan A 1 20 = (143.36) tan (-T) = m 2 = m 2 2 sin (1) 2 R2Tan A2 T2 = = 30 2 (47.83) tan rn 3rd curve = same as 1st curve

10 Figure 7

11 Figure 8 AV, TI = 25.28m In A V1V2V VIV2 = TI + T using sin law V,V = m.. Back tangent = AV1 + VIV = Since curve is symmetrical the forward tangent = back tangent = rn

12 P.I. = Stn 32 t B.T P.C Check P.C L CC P,T, P.T Figure 9 Def 1 Station Angle P.T P.C.C P.C.Cml '40 ' 40 " r '40 ' 40 " P.C. oooo' 00" = (18.39)(8)(40 = ~ d2 = (11.61)(8)(40 = 2 19'20" 3 5 ~ ~ ' =bt/2 ~ ~ " curve 82 D = 12'.*.d = 6 0 3'40 ' 40" dl = (8.39)(12)/20 = 5 02t00" 29'40 ' 40" d2 = (6.61 )~12)/20 = 3'58'00" 25'40 ' 40 " 25 00' '02 '00" I 5'02' 00" 10~00'00" Curve #1 D = 8'.'. d = 4' = ("1+*~~/2 V' curve #3 D = 8'.'.d=4' dl = (3.39) (8)/40? 0~40'40" d2 = (6.61)(8)/40= " = *1/2 4 NOTE: The checks of P.C.Cm1, P.C.C. and P.T. are essential! 2 '

13 PROBLEMS Field procedure for compound curves is essentially the same as for simple curves. The transit has to 6e set up at P.C.C.1, and P.C.C.2 in order to run in the curves. 7. Determine the radius of the central curve of a symmetrical compound curve which passes through point "Au, and determine the stationing of point "A".