Improvement on Site Distance Equation for Crest Vertical Curves - Research Paper Example

Improvement on Site Distance Equation for Crest Vertical Curves Introduction Though the natural profile of the land would be expected to be level and it would have slopes of varying magnitudes at most of the locations. In order to ensure smooth movement of vehicles on the road, the changes in the slope or grades has to be smoothened by designing appropriate the vertical curves. The gradient is the measure of rise and fall in the profile of the road with respect to the horizontal and is expressed as ratio of the unit vertical change for the corresponding horizontal distance (Garber and Hoel, 2001). The figure 1 gives an illustration of the vertical curve. The elevation of various points of the curve gives the relative difference in the level at different points on the curve. Also shown in the figure are length of the curve, start of the curve and also the end of the curve. To provide easy movement of vehicles and also to smooth out the vertical profile the vertical curves are introduced at the intersection of the grades. Usually two type of vertical curves are used in the geometric design of roads. They are crest curves or summit curves and sag curves or valley curves. The crest curves have the convexity upwards and when a fast moving vehicle travels along the curve, upward action of the centrifugal force against the gravity and would relieve a part of the pressure on the tyres. This phenomenon would eliminate the discomfort experienced by the passengers wouldn't feel the discomfort while passing over these curves (Garber and Hoel, 2001). The process of aligning the vertical curves along the road is influenced by various factors like the vehicle speed, acceleration, stopping sight distance and comfort in travel (Wright and Dixon, 2004). Figure 1 : Vertical Curve (http://www.tpub.com/inteng/11h.htm) Equation for vertical crest curve The design of the summit curve is governed only by sight distance considerations. Though the circular crest curve is an ideal choice as the sight distance available throughout the length of a circular curve is constant most of the designs prefer parabolic curve. This is because the deviation angles in the vertical curves of highways are very small and between the same tangent points a simple parabola is congruent with a circular arc. In addition, easiness in computation of the ordinates besides the better rising comfort given by crest curves gives preference to parabolic curves (DRMB, 1993). When the parabolic crest curves are adopted the equation is given as y=ax2 , where a = N /2L. The N in the equation is the deviation angle and L is the length of the curve. Since the crest curves are long and flat , the length of the curve L is taken as equal to the horizontal projection. During the process of the design of the parabolic crest curves it is necessary to consider the stopping sight distance and overtaking sight distance separately. As indicated earlier, it is essential to provide sight distances atleast equal to the stopping distances at all points on the highways to avoid the accidents due to inadequate sight distance (Garber and Hoel, 2001). Figure 2 : The length of the crest curve is greater than the stopping side distance Length of the summit curves for stopping side distance. The two situations that need to be considered in the determination of length of the curve for stopping side distance (SSD) are (i) When the length of the curve is greater than the side distance (L > SSD) and (ii) When length of the curve is less than the side distance (L < SSD) (Mannering et al, 2005). For L > SSD (Figure 2), the length of the vertical crest curve is given as ------------------ (1) Where, L is the length of the vertical crest curve in metre, S is the stopping side distance in metre, N is the deviation angle which is equal to algebraic difference in grades, radians or tangent of the deviation angle, H is the height of the eye level of the driver above the road surface in metre, h is the height of the object above the pavement surface in metre. The value of H and h above the road surface is usually given by the transport agency as per the guidelines prevailing in a particular region. For L < SSD , the length of the vertical crest curve is given as (Mannering et al, 2005). ------------------- (2) The description of L, S, H, N and h are as explained earlier. The sketch of the vertical crest curve when the length of curve is less than the stopping side distance is given in the Figure 3. Figure 3 : The length of the curve is less than the stopping side distance. To design the length of the vertical crest curve for stopping sight distance, the first step is to determine the stopping sight distance (SSD) using the following equation (Mannering et al, 2005). --------------------------- (3) Where V is the speed in Kmph, t is the time, f is the friction coefficient, g is he acceleration due to gravity and n is the gradient of the surface. Then substituting the SSD in the equations discussed earlier ( equation 1 and 2) the length of the curve is estimated. Also, the minimum curve radius of the parabolic curve could be calculated from the relation R = L/N. Length of the summit curves for safe overtaking sight distance. To determine the length of vertical crest curves for safe overtaking sight distance (S) also two cases are considered (i) When the length of the curve is greater than overtaking side distance (L>S) and (ii) When the length of the curve is less than overtaking side distance (L < S) (Mannering et al, 2005). For L > S , the equation (1) is used for here with slight modification of substituting H = h. Then the length of the crest curve , L is given as For L < S , the equation (2) is used with slight modification of substituting H = h. Then the length of the crest curve is given as (Mannering et al, 2005). When the deviation angle is very small, the length of the crest curve works out to be less than the sight distance, in very small deviation angles, the length required works out to be negative value which indicates there is no problem in the sight distance. References Department of Transport (1987) Highway Construction Details, Department of Transport, Scottish Development Department, Welsh Office, Department of the Environment for Northern Ireland, HMSO, London. DMRB (1993), Design Manual for roads and bridges, Vol 6, The highways agency, London. Garber, N.J and Hoel, L.A (2001), Traffic and highway engineering, CL Engineering, 3rd Edition. Highway Link Design (2006), Design Manual for Roads and Bridges [Online] Available at http://www.standardsforhighways.co.uk/dmrb/vol6/section1.htm [18 February 2009] Mannering , F.L., Kilareski, W.P and Wasburn S.S (2005) Principles of highway engineering and traffic analysis, John Wiley & Sons. Roadway Design (1995), Roadway Fundamentals for Municipal Officials a workshop presented by the Maine Local Roads Center, Maine Department of Transportation, 1995 [Online] Available from < http://www.memun.org/SchoolsProject/Resources/Roads/Fundamentals.htm > [5 February 2009] Wright, P H and Dixon, K.K (2004), Highway engineering, 7th Edition, John Wiley & Sons. Read More