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For gradually introducing and increasing the centrifugal force acting downwards, the best shape that could be given for a valley curve is a transition curve. Cubic parabola is generally preferred
HIGHWAY GEOMETRIC DESIGN 10CV755 Dept. Of Civil Engg, SJBIT Page56 in vertical valley curves. See figure above during night, under headlight driving condition, sight distance reduces and availability of stopping sight distance under head light is very important. The head light sight distance should be at least equal to the stopping sight distance. There is no problem of overtaking sight distance at night since the other vehicles with headlights could be seen from a considerable distance. Length of the valley curve The valley curve is made fully transitional by providing two similar transition curves of equal length The transitional curve is set out by a cubic parabola y = bx3where b = 2N/3L2The length of the valley transition curve is designed based on two criteria: 1. Comfort criteria; that are allowable rate of change of centrifugal acceleration is limited to a comfortable level of about 0:6m\sec3. 2. Safety criteria; that is the driver should have adequate headlight sight distance at any part of the country. Comfort criteria The length of the valley curve based on the rate of change of centrifugal acceleration that will ensure comfort: Let c is the rate of change of acceleration, R the minimum radius of the curve, v is the design speed and t isthe time, then c is given as
HIGHWAY GEOMETRIC DESIGN 10CV755 Dept. Of Civil Engg, SJBIT Page57 For a cubic parabola, the value of R for length Lsis given by: Therefore, Where L is the total length of valley curve, N is the deviation angle in radians or tangent of the deviation angle or the algebraic difference in grades, and c is the allowable rate of change of centrifugal acceleration which may be taken as 0.6m\sec3. Safety criteria Length of the valley curve for headlight distance may be determined for two conditions: (1) length of the valley curve greater than stopping sight distance and (2) length of the valley curve less than the stopping sight distance. Case 1 Length of valley curve greater than stopping sight distance (L > S) The total length of valley curve L is greater than the stopping sight distance SSD. The sight distance available will be minimum when the vehicle is in the lowest point in the valley. This is because the beginning of the curve will have in infinite radius and the bottom of the curve will have minimum radius which is a property of the transition curve. The case is shown in figure below. From the geometry of the figure, we have
HIGHWAY GEOMETRIC DESIGN 10CV755 Dept. Of Civil Engg, SJBIT Page58 Where N is the deviation angle in radians, h1 is the height of headlight beam, is the head beam inclination in degrees and S is the sight distance. The inclination α is almost equal to 10 Case 2 Length of valley curve less than stopping sight distance (L < S) The length of the curve L is less than SSD. In this case the minimum sight distance is from the beginning of the curve. The important points are the beginning of the curve and the bottom most