Page 1 :
Curves are the geometrical arcs provided at the change in, , alignment of a, , road., , gradient or, , 21. Necessity of Providing Curves, Curves are provided at the change in, or gradient of a, alignment, due to the following reasons:, (a) To lay the road according to topography of the, country., (b), , To avoid costly land, , (C), , To avoid excessive cutting and, filling., To avoid certain, important structures., , (d), , (e), , To make use of the, existing, To provide access to the, , 2.61/Types of Curves, , road, , road, bridge etc., , particular place., , Curves on highways have been divided into, the following classes, (), Horizontal Curves, , Verical Cuves
Page 2 :
TRANSPORTATION ENGINEERING, , 33HORIZONTAL CURVES, A hortzontal curve is a curve in plan to provide change in direction to, , the central line of a road. The minimum radius of a horizontal curve depends, of minimum radii, on the permissible design speed for the road. The values, , tor vanous categories of roads in different areas, recommended, are given in Table 2.10., , by the I.R.C., , The following types of horizontal curves are used in the aligniment of, , highway, Simple curve, (6), , Compound curve, Reverse curve, , Transition curve, circular curve which consists of a single arc, of uniform radius. This curve is expresed in terms of degree of the curve,, This, which is the angle subtended at the centre by a chord of 30 m length., of curve is suitable for lare radius and for slow moving traffic., , Simple Curve: It is a, , type, , Circular curve, , angent, , I6ueL, Fig 28. Simple Curve., , circular curve which is comprised of, different radii which turn in the, a series of two or more simple curves of, hard, curve is used to avoid cuting through, of, direction., type, This, same, 2.9, 7,M and MT2 are two, rocks, heavy cutting or filling etc. Refer Fig., , eyCompound Curve: This is a, , tarigent at M.
Page 3 :
ROAD GEOMETRICS, , Fig. 2.9. Compound Curve., , Reverse Curve This is a circular curve consisting oftwo simpie, curves of same or different radii which tun in the opposite direction. These, Curves are suitable for highways lying in hilly regions. Refer Fig. 2.10, 77,, and T,T are two adjacent sinmple curves having a common tangent at 7,, d their centres ie on opposite side of the curve., , O2, -R2, , 12, -R, , Common tangent at T, , O, , Fig. 2.10. Reverse Cunve., , P38RANSTON CURVE, A, , infinity, , transition curve is the curve having, at, the, , tangent point, , type of curve is, , to, , a, , radius which decreases from, designed radius of the circular curve. This, a, , generally introduced on highways between, astraight and, ease and gradual change in direction, of a road, , circular curve to provide, , alignmerit, 2.8.1., , Objects of Providing Transition Curve, (a) To provide gradual and, easy transformation from, Carcular curve and from circular, straight, curve to the, , straight roads, , to
Page 4 :
TRANSPORTATION ENGINEERING, To provide a gradual change of curvature from zero at the, tangent point, to that of circular curve at their junction point., , (b), , (c)To enable gradual introduction of the designed superelevation, and extra widening of pavement at the start of the circular curve., (d), , To improve the aesthetic appearance of the road., , 2.8.2. Types of Transition Curves, , -Lemn-isc~at*-e -° -, , The following types of, transition curves are commonly, , adopted, , in, , horizontal, , alignment:t, Lemniscate, , (a), b), , Spiral, , (c), , Cubic parabola., , Cubic, , Spiral, , Parabola, , The shapes of three, curves are shown in Fig. 2.11., Out of the three transition, , curves., , the 1.R.C has-, , recommended the, use of spiral, in the horizontal aligntmenta, , Fig. 2.11. Types of transition curves., , highways., , 28.3. Length of Transition Curve, Generally, the length of transition eurve is detemined from the following, considerations and iarger of th vlues afould be adopted for, design, , purpose., , (a) Rate of chäfige öf ceñtrifugál acceleration: Rate, ofchange of, centrifugal accelerätiok should Bë such that it will not cause discomfort, for, the passengers of å vëhicle, moviñg with design speed. Based on this, consideration, thië lëñigth of transitiön urve can be caculated, by using the, equatioh, , 0.0215p3, , where, , CR, , L,= Lehgth of transition curve, V= Desigh speed in Km/ht, R, C, , Radius of circular curve (metre), Allowable rate of chañge öf centrifugal acceleration, (mvsec), , 75v, , 80, , m/sec (0.5<C<0.8]
Page 5 :
ROADGEOMETRICS, I.R.C., the length oftransition, per, (bBy, not be less than that calculated, should, road, of, the, design, geometric, L, for, curve, , Empirical formula: As, , from the following equations, For plain and, , (i), , rolling terrain, 2.7v2, , R, , ()For mountainous and steepterrain, v2, , R, (c) Rate of introduction of superelevation: Based on this, length of, transition curve is calculated from the equation., , eN(W +W), where e = rate of superelevaton =, , VZ, 225R, , W, , Normal width of pavement, , W, , Extra widening providd, , 1 in N, , Rate of change of süperelevation, , 2.9NERTICAL CURVEs, Vertical curves are the curves provided at the intersections, different, grades in the vertical alignment of highway. This is introduced to of, smoothen, out the vertical profile and thus to eáse off the, in, changes gradients for the, fast moving vehicles., , 2.9.1. Types of Vertical Curves, Vertical curves are of two types, Summit Curves, , Nalley Curves, YSummit Curves: Summit curves are vertical curves having their, , convexity upward. This type of curve is provided in any one of the cases, illustrated in Fig. 2.12. The centrifugal force will act, against gravity, when a fast moving vehicle travels along a summitupwards, curve and there will be, no problem of discomfort to the, passengers. At the time of, the, , length, , of summit curve the, stopping, distance are considered separately., , sight distance, , and, , designing, overtaking sight
Page 6 :
TRANSPORTATION ENGINEERING, , 32, , N=(n12), , (+ve sign shows, , ascending gradient, wheras ve sign, shows decending, , +, , (a), , gradient), , (b), , v-n, ), , (c), , N=n+n), (d), , Fig. 2.12. Summit Curve., , V a l l e y Curves : Valley curves are vertical curves having their, , Convexity downward. This is also called _ag curves and are provided in any, one of the cases illustrated in Fig. 2.13. At the valley curvesthecentrifugal, force acts downwards adding to the pressureon the suspen_ions in addition, to the self weight of a vehicle moving on the curve. Hence the design of, valey curve is governed by the allowable rate of change of centriugal, , acceleration, , -1M-n,n,, (a), , (, (ve sign shows, , ascending gradient, , wheras-ve sign, , -, , n,+n, , shows decending, gradient), , /%, , .4N=nn), , (b), Fig. 2.13. Valley Curves., , UPERELEVATION, Superelevation is the inward transverse slope provided throughout, the length of the horizontal curve by raising the outer edge of the pavement
Page 7 :
ROAD GEOMETRICS, , 33, , with respect to the inner edge. This is also called cant or banking and is, , generally denoted by 'e, Itis provided to counteract the effect ofcentrifugal force and to reduce, , the tendency of the vehicle to overturn or skid, when it is moving on the, , horizontal curve., The superelevation is expressed as the ratio of the height of outer, edge with respect to the horizontal width of the pavement. From Fig. 2.14, , - t a n0, , Superelevation,, , W, , Fig. 2.14. Superelevation., , In practice the value of 0 is so small that tan, , From Fig.2.14,, , e, , =, , tan0, , », , is equal to sin, , sin6B=, , donsAnalysis of Superelevation, , When a vehicle (Refer Fig. 2.15) is moving on a circular curve of radius, R' at a speed V, then the forces acting on the vehicle are :, weight W of vehicle acting vertically downward., frictional force F and F acting along the pavement surface, towards the centre of the curve., , Cii), , the, , centrifugal force, , P, , w2, , aRacting horizontally outwards, , as shown in Fig. 2.15., , Resolving, condition., , the forces, , along, , the, , pavement AB and for equilibrium, , P cos- W sin + FA+Fs
