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Evaluation of Accident Black Spots on Roads using Geographical Information Systems (GIS)
Principles of Horizontal Curves
A horizontal highway curve is a curve in plan to provide change in direction to the central line of a road. Horizontal curves are part of the geometry of a roadway. They are geometrically sections of circles. The geometry is thus quite simple. As there is no universal notation, the most used are shown in the following figures.  Fig 1.1 Notations for Simple Circular Curves Beginning of Curve - This is represented as BC and corresponds to the point where the road enters the curve. At this point, the road becomes tangent to the curve. This point is also known as TC (Tangent to Curve) or Point of Curvature (PC). End of Curve - This is represented as EC and corresponds to the point where the road exits into the second road. AT this point, the curve meets the tangent road. This point is also known as CT (Curve to Tangent) or Point of Tangency (PT). Point of Intersection - This is represented as PI and corresponds to the point of intersection of the extended entering and exit roads to the curve. Tangent - This is represented by T and corresponds to the distance between the BC and the PI. The exiting tangent corresponds to the distance between the PI and the EC, which is similar to the entering tangent, due to the symmetrical characteristics of an arc of circle. Radius - This is represented as R and corresponds to the radius of the arc that is tangent to both the entering and exiting roads. Interior Angle - This is represented as I and corresponds to the angle measured at the origin of the arc (O), from the direction of the BC to the EC. This can also be represented by ? It is very important to notice that, by geometry, the angle between the extension of the entering tangent and the exiting tangent is also ?. The angle between the direction of BC (or EC) and the PI is half of the interior angle Ñ/2. Length of Arc - This is represented by L and corresponds to the distance along the curve between the BC and the EC. Length of Long Chord - This is represented by LC and corresponds to the distance between the BC and the EC in an Euclidian straight line. Middle ordinate - This is represented by M and corresponds to the distance between the chord and the arc of the curve, measured along the bisector of the arc. The bisector of the arc is the line from the origin O to the PI. External distance - This is represented by E and corresponds to the distance measured along the arc bisector, from the arc to the PI. Radius of Horizontal Curve When a vehicle traverses a horizontal curve, the centrifugal force acts horizontally outwards through the center of gravity of the vehicle. The centrifugal force developed depends on the radius of the horizontal curve and the speed of the vehicle negotiating the curve. This centrifugal force is counteracted by the transverse frictional resistance developed between the tyres and the pavement which enables the vehicle to change the direction along the curve and to maintain the stability of the vehicle. Centrifugal force P is given by the equation: P = (W x v2 ) / (g x R), where P = centrifugal force, kg W = weight of the vehicle, kg R = radius of circular curve, m v = speed of vehicle, m/sec g = acceleration due to gravity = 9.8 m / sec2 The ratio of the centrifugal force to the weight of the vehicle, P/W is known as the centrifugal ratio or the impact factor. Thus, centrifugal ratio = v2 / gR For a certain speed of vehicle, the centrifugal force is dependent on radius of the horizontal curve. To keep the centrifugal ratio within a low limit, the radius of the curve should be kept correspondingly high. The centrifugal force, which is counteracted by the super elevation and the lateral friction is given by the relation where, e = rate of super elevation; the maximum value of e is taken as 0.07 at all the regions except at hill roads without snow where it is taken as 0.1; f = design value of transverse skid resistance or coefficient of friction, taken as 0.15 Hence, Thus the ruling minimum radius of the curve for ruling design speed v m/sec or V Kmph is given by: When the minimum design speed V' Kmph is adopted instead of V Kmph, the absolute minimum radius of horizontal curve Rmin is given by: The absolute minimum values of radii of horizontal curve of various classes of roads in different terrain (as per the latest IRC specifications) is given in the following table. 
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