The clear distance or set-back distance m (fig 4.7) required from the centre line of a horizontal curve to an obstruction on the inner side of the curve to provide adequate sight distance depend upon the following factors:
(i) required sight distance (ii) radius of horizontal curve (iii) Length of curve
Let ‘ ‘ be the angle subtended by the arc length equal to the required sight distance
‘S’ at the centre of the horizontal curve. Then
= 180S / RT degress.
Fig 4.7
The distance from the obstruction to the centre is R cos / 2.
Then the set back distance CD = m required from the centre line is given by
m = R - R cos / 2 4.23
In the case of wide roads with two or more lanes if ‘d’ is the distance between the centre line of the road and the centre line of the inside line in metres , the sight distance is measured along the middle of the inner side lane and the set back distance ‘m1’ is given by
m’= R - (R - d) cos / 2 4.24
where ‘= 180S / (R - d) degress
d is usually taken as total width of pavement / 4 = (w + We) / 4 4.25
The IRC has suggested equation 4.24 for finding the set back distance required at horizontal curves. The clearance of obstruction upto the set back distance is important when there is a cut slope on the inside of the horizontal cuve.
Worked example 4.1 illustrates complete design of the horizontal alignment.
Worked Example:
4.1: A national highway is passing through a plain terrain. Design all the geometric features of the curve assuming suitable data. Also specify the minimum set back distance from the centre line or the two lane highway on the inner side of the curve upto which the buildings etc ., obstructing the vision should not be constructed so that the overtaking sight distance is available through out the circular curve. Assume the length of the circular curve greater than overtaking sight distance.
Solution:
The various geometric elements to be designed are:
(i) Ruling minimum radius (ii) Super elevation (iii) Extra widening (iv) Length of transition curve (v) Overtaking sight distance and set back distance.
Data assumed:
For National Highway through Plain Terrain Ruling speed = 100 kmph ; No. of lanes 2 Width of pavement = 7m
Speed of overtaking vehicle = (100 - 16) = 94 kmph For overtaking reaction time of driver = 2 sec.
(i) Ruling minimum radius = V2 / 127 (a + f) = 1002 / 127 (0.07 + 0.15) = 357.9 m or say 360m.
(ii) Super elevation Design:
Super elevation is designed for 75% of the design speed.
e = (0.75 V)2/ 127 R = (0.75 x 100)2/ 127 x 360 = 0.123
as the value of e is greater than 0.07 , it is restricted to 0.07 and check for friction is made f = V2/ 127 R-e = 1002/ 127 x 360 - 0.07 = 0.149
As this value of f is less than 0.15 , superelevation of 0.07 may be provided.
(iii) Extra -width of pavement to be provided
We= nl2/ 2R + V / 9.5
R
= 2×62/ 2×360 + 100 / 9.5 360 = 0.65m Total width of pavement = 7 + 0.05 = 7.05m.(iv) Length of transition curve is calculated based on the following considerations and the longest of all the three lengths is adopted. (a) based on the rate of change of centrifugal acceleration , C , rate of change of centrifugal acceleration.
= 80 / (75 + V) = 80 / (75 + 100) = 0.457
The minimum specified value of C = 0.5 is to be adopted.
Length of transition curve = LS= V2/ 46.5 CR = 1003/ 46.5 ×0.5 ×360 = 123.46 m .
(b) Based on rate of introduction of superelevation - As the terrain is plain , the superelevation is introduced at a rate of 1 in 150 by rotating the pavement about the inner edge.
LS= N . e (W + We) = 150×0.07 (7 + 0.65) = 80.32 m (c) based on the minimum length requirements of IRC.
LS= 2.7×V2/ R = 2.7×1002/ 360 = 75 m.
The highest value of these three lengths is 123.46 m.
Hence adopt a transition curve of length 125m . (v) Overtaking Sight Distance:
(a) Distance covered during reaction time d1= 0.278 Vbt = 0. 278×(100 - 16) ×2 = 46.7 m.
Minimum spacing between the vehicles = S = (0.2 VB+ 6) = (0.2 ×84 + 6) = 22.8 m.
Time for overtaking Tsec =
14.4S / A
A - rate of acceleration for 100 kmph = 1.92 kmph / sec.
Tsec =
14.4 × 22.8 / 1.92
= 13.02(b) Then d2= b + 2S = 0.278 VBT + 2S = 0.278×84×13.82 + 2 ×22.8 = 351.04 = 352 m.
(c) Distance covered by the on coming vehicle
d3= 0.278 V T = 0.278 ×100×13.08 = 363.0 m Overtaking sight distance = d1+ d2+ d3= 761.2m (vi) Set back distance:
d = distance between the centre line of the road and the centre line of the inside pavement is taken as width of pavement / 4 = 7.65 / 4 = 11.91m.
‘/ 2 = 180 ×762 / 2 (360 - 1.91) = 60.6 Set back = 360 - (360 - 1.91) cos 60.6 = 187 m.
Answers:
Ruling Mini Radius = 360 m.
Super elevation e = 0.07 Extra widening = 0.65m
Length of Transition curve = 125m.
overtaking sight distance = 690 m.
Set back distance = 187m.
4.7. Gradients:
Gradient is the rate of rise or fall along the length of the road with respect to the horizontal. A rising or ascending gradient is designated by plus (+) sign while a descending gradient by negative (-) sign. Gradient is expressed as a ratio of 1 in n (one unit of vertical to n units of horizontal) and some times as a percentage that is n in 100. The angle which measured the change of direction at the intersection of two grades is called the ‘Deviation Angle’ and is represented by N , and is equal to the algebraic difference between the two grades. In Fig 4.8 , the deviation angle N = DBC = BAC + BCA = n1 - (- n2) = n1 + n2.
Where n1is the ascending gradient of AB and (-n2) is the descending gradient of BC.
Fig 4.8 MEASUREMENT OF GRADIENT
Gradient in roads should not be very steep. Steep grades not only make it difficult for vehicles to climb over them , but also increase operational cost of the vehicles. Designer should try to provide as easy gradient as possible provided earth work is not unnecessarily increased.
Gradients are classified into the following types:
(i) Ruling gradient (ii) Limiting gradient
(iii) Exceptional gradient , and (iv) Minimum gradient.
4.7.1. Ruling Gradient:
This is the maximum gradient normally adopted. Its value depends upon the type of terrain , length of grade , speed of vehicles , power of vehicles , type of traffic and presence of horizontal curves in the road alignment. Based on experience , for mixed traffic conditions , values of ruling gradients as recommended by the IRC are presented in Table 4.1.
4.7.2. Limiting Gradient:
Limiting gradient is steeper than ruling gradient and is provided at places , where by adopting slightly steeper gradient a lot of saving in earth work and other aspects can be affected. The length of the limiting grades should not be continuous but limited in length.
Every limiting gradient should be followed by a stretch of road having a very small gradient or levelled ground. In hilly areas these gradients are very common. Recommended values of limiting gradients of roads in differenct terrains are as presented in Table 4.1.
4.7.3. Exceptional Gradient:
Exceptional gradients are steeper than limiting gradients and have to be provided only under unavoidable circumstances. However the excceptional gradient should be strictly limited only for short stretches not exceeding about 100 metres at a stretches. Recommended values of this gradient for different terrains are also presented in Table 4.1.
Table 4.1. Gradients For Roads in Different Terrains:
Terrain Gradient in percent
Ruling Limiting Exceptional
1. Plane or Rolling 3.3 (1 in 30) 5.0 (1 in 20) 6.7 (1 in 15) 2. Mountainous terrain and steep
terrains having elevation more than 3000m above M.S.L.
5.0 (1 in 20) 6.0 (1 in 16.7) 7.0 (1 in 14.3)
3. Steep terrains upto 3000m height above M.S.L.
6.0 (1 in 16.7) 7.0 (1 in 14.3) 8.0 (1 in 12.5)
The maximum length of ascending gradient which a loaded truck can operate without undue reduction in speed is called ‘Critical Length of Grade’ for design. A reduction of speed of about 25 kmph , may be considered as reasonable limit. The critical length of ascending gradient should therefore be limited to lower values at steeper gradient.
4.7.4. Minimum Gradient:
It is desireable to have certain minimum graident on roads from drainage point of view , provided topogrophy favours this. The minimum gradient would depend on the rainfall , run off , type of soil , topography and other site conditions.
A gradient of about 1 in 500 may be enough to drain off water in concrete drains ; but on inferior surfaces of drains a slope of 1 in 200 may be needed where as on soil drains steeper slopes upto 1 in 100 may be needed.
4.7.5. Compensation in Gradients on Horizontal Curves:
If a horizontal curve is also located on an ascending road , vehicles while negotiating curve and grade will have to come across the joint resistance offered by the curve and grade effectively. It is necessary in such cases , the total resitance due to grade and curve should not
exceed the resistance due to the maximum value of the gradient specified. The gradient in such cases is slightly reduced so that the vehicle may deal with resistance offered by the curve and grade effectively. This reduction in gradient is called ‘Grade Compensation’ and is calculated from the following formula.
Grade compensation , (percent) = (30 + R/R) 4.26 Where R is the radius of curve in metres.
Maximum valve of the grade compensation is limited to 75 / R. According to the IRC , the grade compensation is not necessary for gradients better than 4% (1 in 25).