Introduction
The geometric design of a highway deals with the dimensions and layout of visible features of the highway such as alignment, sight distances and intersections. The geometrics of highway should be designed to provide optimum efficiency in traffic operations with maximum safety at reasonable cost. Geometric design of highways deals with the following elements:
1. Cross section elements. (Width of pavement, formation and etc.)
2. Sightdistance considerations.
3. Horizontal alignment details.
4. Verticals alignments details.
5. Intersection elements.
Factors which Govern the Geometric Elements
1. Design speed : This is most important factor controlling the geometric design elements. It is decided on the basis of type of road or class of road such as NH, SH, MDR etc., and topography of the region.
Do you know ?
Design of almost every geometric design element of a road is dependent on the design speed.
2. Topography : Design speed or ruling speed of NH and SH on different terrains is as follows:
(a) Plain terrian (cross slope up to 10%)100 kmph.
(b) Rolling terrian (cross slope of 10% to 25%)80kmph
(c) Mountainous terrain (cross slope of 25% to 60%)50 kmph.
3. Traffic Factors : These include
(a) Vehicular characteristics.
(b) Human characteristics.
4. Design Hourly volume and capacity : The traffic flow or volume keeps fluctuating with time, from a low value during offpeak house to the highest value during the peak hour. It will be uneconomical to design the roadway facilities for the peak traffic flow or the highest hourly traffic volume.
Therefore a reasonable value of traffic volume is decided for the design and this is called the design hourly volume.
5. Environmental and other factors : Such as aesthetics, landscaping, air and noise pollution should be given due consideration in in the design of road geometrics.
Highway Crosssection Elements Friction : Friction determines the operating speed and distance requirements in stopping and accelerating the vehicles. Skid : Skid occurs when the vehicles slide without revolving or when the wheels partially revolve i.e., when path travelled along the road is more than the circumferential movement of the wheel due to their rotation. The lateral skidding may also take place at curves. The lateral skid is more dangerous as the vehicle goes out of control.
The maximum lateral skid coefficient is generally equal to or slightly higher than the forward skid coefficient in braking test. Slip occurs when wheel revolves more than the corresponding movement along the roads.
Pavement Unevenness : The pavement surface conditions are commonly measured by using an equipment called “Bump integrator” in terms of unevenness index, which is the cumulative measure of vertical undulations of the pavements surface recorded per unit horizontal length of the road. Unevenness index should be kept below 150 cm/ km for good pavement surfaces of high speed highways. A value of 250 cm/km is satisfactory up to speed of about 100 kmph. Value more than 350 cm/km is considered very uncomfortable even at speed of 50 kmph.
Cross Slope or Camber : Camber is the slope provided to the road surface in the transverse direction to drain off the rain water from the road surface. The requirement of camber of a pavement depends upon
1. The type of pavement surface.
2. The amount of rainfall.
A flat camber of 1.7 to 2.0% (Approximately half of the ruling gradient) is sufficient on relatively impervious pavement surface like cement concrete or bituminous concrete. The camber may be given a parabolic, elliptic or straight line shape. Parabolic and elliptic are preferred by fast moving vehicles, because they require frequent crossing the crown line during over taking operations. When very flat cross slope is provided as in cement concrete pavement, straight line shape of camber may be provided. The cross slope of shoulder should be 0.5% steeper than the cross slope of adjoining pavement, subjected to minimum of 3% and maximum of 5% (for earth shoulders).
Types of Road surface Heavy Rainfall Light Rainfall
Camber
1. Cement concrete and high type bituminous surface 0%  1.7%
2. Water bound macadam and gravel pavement 0%2.5%
3. Thin bituminous surface 2.5% 2.0%
4. Earth 4.0% 3.0%
Do you know ?
The chamber is given a parabolic elliptic or straight line shape in the crosssection.
Width of Pavements or Carriageway . The pavement or carriageway width depends on the width or traffic lane and number of lances.The carriageway intended for one line of traffic movement may be called a traffic lane.
Class of Road Width of carriageway
1. Single 3.75 m 3. Two lane with raised kerb 7.50 m
2. Two lance without raised curve 7.00 m
3. Intermediate carriage way 5.5 m
4. Multilance pavement 3.5 m
Do you know ?
Width of single lane for village roads = 3 m.
The minimum width recommended for kerbed urban road is 5.5 m to make allowance for a stalled vehicle.
The width of vehicle is = 2.44 m.
Traffic Separators or Medians: These are used to prevent headon collision between vehicles moving in opposite directions. The IRC recommends a minimum desirable width of 5.0 m for medians of rural highways, which may be reduced to 3.0 m where land is restricted. On long bridges the width of the medians may be reduced up to 1.2 to 1.5 m. The minimum recommended width of medians at intersections of urban roads are 1.2 m for pedestrian defuse, 4.0 to 7.5 m for protection for vehicles making right turn and 9.0 to 12 m for protection of vehicles crossing at grade. The absolute minimum width of medians in urban area is 1.2 m and desirable is 5.0 m.
Kerbs : Kerb indicates the boundary between the pavement and shoulder, or sometimes islands or footpath or kerb parking space.
1. Low or mountable types kerb has the height of about 10 cm.
2. Semibarrier types kerb is provided where pedestrian traffic is high and has a height of 15 cm above the pavement edge width a batter of 1 : 1 on the top 7.5 cm.
3. Barrier type kerb is provided it built up area and the height is of the order of 20 cm above the pavement edge with a steep batter of 1.0 vertical 0.25 horizontal.
Road Margins : The various elements included in the road margins are should, parking lane,frontage road, driveway, cycle track, footpath, guard rail and embankment slope.
1. Shoulders : They are provided along the road edge to serve as an emergency lane for vehicle compelled to be taken out of the pavement or road way. The minimum width of shoulders is 4.6 m. So that a truck stationed at the side of the shoulder would have a clearance of 1.85 m from the pavement edge.
2. Parking Lanes : They are provided on urban roads to allow kerb parking. 3.0 m width is required for parallel parking.
3. Laybyes : They are provided near public conveniences with guide maps to enable drivers to stop clear off the carriage way. They should be 3.0 m wide of at least 30 m length with 15 m end tapers on both side.
4. Bus Bays : They are provided avoid conflict with moving traffic. They should be located 75 m away from the intersection.
5. Dirve Ways : These are used to connect the highway with commercial establishment like fuel stations, service stations etc. They should be away from the intersections.
Do you know ?
The radius of the drive way curve should be kept as large as possible, but the width of the drive way shoulder be minimised to reduce the length of cross walks.
6. Cycle Track : A minimum width of 2.0 m is provided and width may be increased by 1.0 m for each additional cycle lane.
7. Foot Path : A minimum width of 1.5 m is provided.
8. Guard Rails : They are provided at the edge of shoulder when the road is constructed on a fill so that vehicles are prevented from running off the embankment especially when the height of the fill exceeds 3 m.
Width of Roadway or Formation: Width of formation or roadway is the sum of widths of pavements or carriageway including separators if any; and the shoulders.
SI. No. Road Classification
Plain and Rolling Terrain Mountainous and Steep Terrain
Roadway with, m at
1. National & State Highways (a) Signal lane (b) Two lane
2. Major District Roads (a) Signal lane (b) Two lane
12.0
12.0
6.25
8.80
3. Other District Roads (a) Signal lane (b) Two lane
9.0
9.0
4.75
—
7.5
9.0
4.75
—
4. Village roadssingle lane 7.5 4.75
Do you know ?
In multilance highways, roadway width should be adequate for the requisite number of traffic lanes besides shoulders and central median.
The minimum roadway width on single lance bridge is 4.25 m.
Right of Way : It is area of land acquired for the road, along its alignment. The width of this land is known as land width and it depends on the importance of the road and possible future development. The normal width required for NH and SH on open plain terrain is 45 m and maximum width required is 60 m; the corresponding width between building lines is 80 m and that between the control lines is 150 m, thus allowing a set back distance of 10 and 45 m beyond the road boundary lines. The recommended land for different classes of urban roads are : Class road Land width Arterial 50 to 60 m Sub arterial 30 to 40 m Collector streets 20 to 30 m Local streets 10 to 20 m Sight Distance. It is the length of road visible ahead to the driver at any instance.
Following are the three situations.
1. Stopping or Absolute minimum sight distance.
2. Safe overtaking or Passing sight distance.
3. Sage sight distance for entering into uncontrolled intersections. Apart from above three situations, the following sight distances are considered by the IRC in Highway design.
(i) Intermediate sight distance : This is defined as twice the stopping sight distance. When overtaking sight distance can not be provided, intermediate sight distance is provided to give limited overtaking opportunities to fast vehicles. (ii) Head light sight distance : Distance visible to a driver during night driving under the illumination of vehicle head lights. This is important for up gradients and ascending stretch of the valley curves.
1. Stopping Sight Distance (SSD): The minimum sight distance available on a highway at any spot should be of sufficient length to stop a vehicle traveling at design speed, safely without collision with any other obstruction. The absolute minimum sight distance is therefore equal to the stopping sight distance, which is also some times called nonpassing sight distance. The sight distance available on a road to a driver at any instance depend on:
1. Features of the road ahead
2. Height of the driver's eye above the road surface.
3. Height of the object above the road surface.
Do you know ?
IRC has suggested the height of eye level of driver as 1.2 m and the height of object as 0.15 m above the road surface.
Stopping distance depends upon following factors :
(a) Total reaction time of driver
(b) Speed of the vehicle
(c) Efficiency of brakes
(d) Frictional resistance between road and tyres.
(e) Gradient of the road if any.
Total Reaction Time : It consist of 1. Perception time 2. Brake reaction time. Perception time is the time required for driver to realize that break must be applied. Break reaction time depends upon skill and type of problem.
PIEV Theory : The total reaction time is split into 4 parts.
1. Perception : Time required to receive an object.
2. Intellection : Time required to understand the situation.
3. Emotion : Time elapsed during emotional sensation.
4. Volition : Time taken for final action.
Reaction Time and PIEV Process: Total reaction time varies 0.5 to 4 sec. Stopping distance of vehicle is the sum of A. Lag distance; distance travelled by vehicle during total reaction time. IRC recommends a value of 2.5 sec. for total reaction time Lag distance = v.t....(v = m/sec) = 0.278.V.t.....(V = km/hr).
Where t is total reaction time in seconds.
ii). Braking distance: Distance travelled by vehicle after application of brakes.
The coefficient of friction decreases with increase in speed.
I.R.C. recommends that f = 0.40 for speed of 20 to 30 kmph
= 0.34 for speed of 100 kmph.
Work done against friction = Initial KE f .W. 1Wv2 2g l = v 2
2gf l = where, l = braking distance in m. v = speed in m/sec f = longitudinal friction coefficient (0.35 to 0.4)
If speed is expressed in Kmph then
V 2
254f l = (V in kmph) Total stopping side distance (SSD) = Lag distance + braking distance
SSD
V 2
V.t 2gf + (in meters) If speed is V km/hr then, SSD = 0.278
V 2
V.t 254f + (in meters)
Effect of Grandient SSD = ( )
v 2
V.t 2g f 0.01n +
±
If speed is V km/hr.
SSD = 0.278 ( )
v 2
V.t 254 f 0.01n +
±
Where, n is % gradient. For ascending use + sign and for descending use  sign.
Do you know?
1. On roads with restricted width or on single lane roads when two way movement of traffic is permitted, the minimum stopping sight distance should be equal to twice the SSD to stop both vehicles coming from opposite direction.
2. The SSD should be provided throughout the length of all roads and hence this is also known as absolute minimum sight distance.
3. When the stopping sight distance for the design speed is not available on any section of a road, the speed should be restricted by a warning sign and a suitable speed limit regulation sign.
Approximate values of SSD are given below: Design speed (kmph) SSD(m) 20
20
40
45
65
90
80
120
100
180
2. Overtaking Sight Distance (OSD) The minimum distance open to the vision of the driver of a vehicle intending to overtake slow vehicle ahead with safety against the traffic of opposite direction is know as the minimum overtaking sight distance (OSD) or the safe passing sight distance available. OSD depends upon;
(a) Speed of overtaking, overtaken and vehicle coming from opposite direction.
(b) The spacing between overtaking and overtaken vehicle.
(c) Skill and reaction time of driver
(d) Rate of acceleration of overtaking vehicle.
(e) Gradient of road, if any.
Let, d1 = distance travelled by overtaking vehicle A during the reaction time t sec. of the driver from position A1 to A2. d2 = distance traveled by the vehicle A from A2 to A3 during the actual overtaking operation in time T seconds. d3 = distance travelled by on coming vehicle C from C1 to C2 during the overtaking operation of A. i.e. T seconds.
Assumptions made during calculation of d1, d2 and d3 are. 1. It is assumed that the vehicle A is forced to reduce its speed to the speed Vb of the slow vehicle B and behind it allowing a space S, till there is an opportunity for safe over taking operation. The distance travelled by vehicle A during this reaction time is d1. This distance will be vb. t where t is the reaction time » 2 sec. i.e., d1 = 0.278 vb t
2. From position A 2 vehicle A star ts accelerating, shifts to adjoining land overtakes the vehicle B, and shifts back to its original lane ahead of B in position A3 in time T sec. This is distance d2 (Between A2 and A3). The minimum distance between A2 and B1 = distance between A3 and B2, which may be taken as minimum spacing S of the two vehicles while moving with vb (m/sec) S = (0.7 vb +6).......(vb = m/sec) = (0.0 2Vb + 6..........(Vb = kmph) If the time taken by vehicle A for the overtaken operation A2 to A3 is T sec. The distance covered by B in this time is b = vb.T d2 = b + 2S 2
b
1
v .T a.T 2
=+ (i) Where a is acceleration in m/sec2 From equation (i) 2S 1 a.T 2 2
=
4S
T
a
=
Hence, d2 = vb.T + 2S 3. The distance travelled by vehicle C moving at design speed v m/sec during the over taking operation of A. d3 = v.t V is in m/sec.
OSD = d1 + d2 + d3 OSD = vb.t + (vb.T + 2S) + v.T.
Where vb is in m/sec.
OSD = 0.278vb.t + 0.278vb.T + 2S + 0.278 v.tWhere Vb is in kmph. 4S
T
a = where, a is in m/sec2 14.4S
T
A
=
Where A is kmph/sec.
Do you know?
If Vb is not given then Vb = V – 16 (kmph) v = v – 4.5 (m/sec) can be taken.
OVERTAKING ZONES
It is desirable to construct highways in such a way that the length of road visible ahead at every point is sufficient for safe overtaking. This is seldom practicable and there may be stretches where the safe overtaking distance can not be provided. In such zones where overtaking or passing is not safe or is not possible, sign posts should be installed indicating" No Passing" or "Overtaking Prohibited" before such restricted zones starts. But the overtaking opportunity for vehicles moving at design speed should be given at frequent intervals. These zones which are meant for overtaking are called overtaking zones.
OSD = Overtaking sight distance = (d1 + d2) for one way traffic = (d1 + d2 + d3) for two way traffic SP1 = Singh post "Overtaking zone ahead"
SP2 = Sign post "End of overtaking zone" Overtaking Zones maximum length = 5 0SDDESIGN OF HORIZONTAL ALIGNMENTS
IRC suggests Ruling design speed for various Roads as follows: Road Permissible Speed 1. NH & SH 100 kmph for plain terrian 80 kmph for rolling terrain 2. MDR 80 kmph for plain terrain 65 kmph for rolling terrain 3. ODR 65 kmph for plain terrain. 4. VR 30 kmph for plain terrain.
IRC Recommendations for Urban Roads 5. Arterial roads 80 kmph 6. Sub arterial roads 60 kmph 7. Collector streets 50 kmph 8. Local streets 30 kmph.
Horizontal Curves The centrifugal force developed at curve is counter acted by the transverse frictional resistance developed between the tyres and the pavement.
Centrifugal force, Wv2 P
gR =´
Here, P = centrifugal force, N W = weight of the vehicle, kg R = radius of the circulare curve, m V = speed of vehicle, m/sec g = acceleration due to gravity = 9.8m/sec2Ratio of the centrifugal force (p) to the weight of the vehicle P W
is known as the centrifugal ratio or the impact factor i.e., centrifugal ratio = Pv2 W gR =
The centrifugal force acting on a vehicle negotiating a horizontal curves has two effects. 1. Tendency to overturn the vehicle 2. Tendency to skid the vehicle laterally outward. (i) Overturning Effect Inner Side of Curve
Outer Side of Curve
Overturning due to Centrifugal Force Equilibrium conditions for overturning will occur when, P.h = w b 2
´
Hence for no over turning P V 2b W gR 2h æö ç =÷£ èø
(iii) Transverse Skidding Effect Frictional force = f.W.
Centrifugal force = P = Wv2 gR
For No skidding Centrifugal force £ frictional force Pv2 f W gR æö ç =÷£ èø SUPPER ELEVATION
In order to counteract the effect of centrifugal force and to reduce the tendency of the vehicle to overturn or skid, the outer edge of the pavement is raised with respect to the inner edge, thus providing a transverse slope throughout the length of the horizontal curve. This transverse inclination to the pavement surface is known as superelevation or cant or banking. The superelevation 'e' is expressed as the ratio of the height of outer edge with respect to the horizontal width.
NL
e tan ML
= =q
E
e tan sin B
= q» q= where, B = Pavement width E = Relative elevation of the outer edge.
Calculation of Superelevation Force acting on vehicle while moving on circular curve of radius R metres with speed Vm/sec. 1. Centrifugal force P acts horizontally out ward through C.G. 2. Weight 'W' of the vehicle acts vertically downward through C.G 3. Frictional force acts in ward to the centre of the curve.
For Equilibrium P cosq = W sinq + f(W cosq + p sinq) P [cosq –f sinq] = W sinq +f.W cosq P tanf ef W 1 f tan q+ = =+
q (since, f tanq <<1)
e + f Pv 2 W gR == where, v is in m/sec.
v 2
ef 127R
+= where, v is in km/hr Where V is kmph.
R = Radius in m e = Rate of super elevation = tanq f = Design valued of lateral friction = 0.15 Super Elevation Design Step 1: The super elevation for 75% of design speed (V kmph)is calculated neglecting the friction.
(
e 0.75V)2 v 2 127R 225R
==
Step2: If the calculated value of 'e' is less than 7% (or 0.07), the value so obtained is provided.
If the value of 'e' exceeds 0.07 then provided the maximum super elevation equal to 0.07 and proceed with steps (iii) and (iv). Step3 : Check the coefficient of friction f for maximum value of e= 0.07 at the full value of design speed.
v 2
f – 00.7 127R
=
If f < 0.15, the value of e = 0.07 is safe otherwise restrict the speed as given in 4th step.
Step4: Find the allowable speed Va (kmph) at the curve by e + f = 0.07 +0.15 2
Va
127R
=
Va = 27.94 R .......(Kmph) If the allowable speed Va is higher than the design speed V then design is adequate and provide e = 0.07 and f = 0.15. If the Va < V, then the speed is limited to Va kmph, and appropriate warning sign & speed limit sign are installed Maximum Superelevation IRC recommends emax = 0.07 (7%) for plain and rolling terrain = 0.1(10%) for Hilly terrain not bonded by snow.= 0.04 (4%) for urban roads with frequent inter sections.
Minimum Superelevation IRC recommends emin = Camber, from drainage consideration.
Attainment of superelevation The full SE is attained by the end of transition curve or at the begining of circular curve.
The attainment of SE can be split in to two parts: (a) Elimination of the crown of the cambered section. (b) Rotation of the pavement to attain full super elevation.
There are two methods of rotating after eliminating camber.
A. Rotation about Centre Line Depressing the inner edge by E 2
and raising the
outer edge by E 2
The disadvantage of this method is the drainage problem due to depressing the inner edge and advantage is that earth work can be balanced.
B. Rotation of Pavement about the Inner Edge Outer edge of the pavement is raised by E
This is suitable for areas of high rainfall and level erranin to avoid drainage problem.
Ruling Minimum radius of Horizontal Curve
()
2
ruling
V
R
127 e f =
+
V = ruling design speed (Kmph) When minimum design speed V' (Kmph) is adopted instead of ruling design speed V kmph, the absolute minimum radius of horizontal curve is
()
2
min
V
R
127 e f =
+
Ruling minimum radius 1. For NH and S.H. = 360m 2. For MDR = 230 m 3. For ODR = 155 m 4. For VR = 90 m Extra Widening of Pavement Widening is provided for the following reasons. (a) To avoid of tracking (b) At speed higher than design speed to encounter transverse skidding. (c) To account rigidity of wheel base. (d) To increase the visibility at curves (e) While overtaking to encounter psychological tendency. It has been a general practice that extra widening is provided. When the radius of horizontal curves is less than 300 m.
Extra widening (We) of pavement consist of 1. Mechanical widening (Wm) 2. Psychological widening. (Wps) We = Wm + Wps We = n.I2V 2R 9.5R
+
Where, n = No . of these l = Length of wheel base » 6 m R = Radius of H curve V = design speed of vehicle in kmph.
We » 0.6 m, for R = 100 to 300 m We = 0.9 m, for R = 60 to 100 m We » 1.5 m for R.
NOTE:
Psychological widening is more important in pavement with more than one lane.
Horizontal Transition Curve Functions (1) To introduce gradually centrifugal force to avoid jerk. (2) Comfort and security of driver. (3) For gradual introduction of super elevation and extra widening. (4) To improve aesthetic appearance of the road. The ideal shape of the transition curve should be such that the rate of introduction of centrifugal force should be constant. In ideal transitional curve the length Ls should be inversely proportional to radius R. ie., ss 1
L or L .R const.
R
¥=
The spiral transition fulfils the above condition. Common type transition curve used are: 1. Spiral (also called clothoid) 2. Lemniscate. 3. Cubic parabola.
All these curve follow the same path up to 4o and practically no difference up to 9°
Calculation of Length of Transition Curve The length of transition curve is designed to fulfill three conditions. 1. Rate of Change of Acceleration Lenth of Transition curve according to this condition is given by 3
s
v
L
C.R
= (Where v = m/sec) 3
s 0.0215V
L
C.R
=
Where, V = Designed speed of vehicle in kmph.
C = Allowable rate of change of centrifugal acceleration. 80
.......[0.5 C 0.8]
75V
= <<
+
2. By Rate of Introduction of Super Elevation Let outer edge is raised at a rate of 1 i n N (N varies from 150 in plain to 60 in hilly terrain) Ls = e. N.(W + We)...When pavement in rotated about inner edge.
Ls =
(
e.N. WWe ) 2
+
....When pavement is rotated
about centre line Where, W = original width of pavement We = extra width of pavement e = super elevation provided. 3. By empirical Formulae Length of horizontal transition curve is should not be less than the value given by following equation (IRC recommendations) (a)
2
s
2.7V
L ...
R
= For plain and rolling terrain
(b)
2
s
V
L ...
R
= For moutainous and steep slopes Where, Ls is in meters V is in kmph R is in meters Length of transition curve Ls should be highest above three.
CURVE RESISTANCE
The loss of tractive force due to turning of a vehicle on a horizontal curve, which is termed as curve resistance will be equal to (T – T cos a) or T (I – cos a) and will depend on the turning angle a...
Curve resistance = T – T cos a T = Tractive effort a = Turning angle GRADIENT
Gradient is the rate of rise or fall along the length of the road with respect to the horizontal.
Gradients are diveided into the following categories: Ruling gradient Limiting gradient Exceptional gradient and Minimum gradient 1. Ruling gradient It is that maximum gradient within which the designer attempts to design the vertical profile of the road hence it is known as design gradient. 2. Limiting Gradient These are steeper than ruling gradient used in view of enormous increase in cost in constructing roads with gentler gradients. 3. Exceptional Gradient This is steeper than above two gradients and is provided if it is unavoidable. The exceptional gradient should be strictly limited only for short stretches not exceeding about 100 m at a stretch. The maximum length of ascending gradient which is loaded truck can operate without undue reduction in speed is called Critical length of grade for design. 4. Minimum Gradient Minimum gradient on the roads is provided from drainage point of view. A minimum gradient of about 1 in 500 may be sufficient to drain water in concrete drains of gutter but on inferior surface of drains a slope of 1 in 200 (0.5%) may be needed where as for Kutcha drains steeper drains slopes up to 1 in 100 (1%) may be provided.
Gradient for Roads in Different Terrains Ruling Limiting Exceptional
Terrain
gradient gradient gradient 3.3 per cent 5 per cent 6.7 per cent Plain or rolling (1 to 30) (1in 20) (1in 15) Mountainous terrain, and steep terrain having elevation more than 3.000 m above the mean aea lev
5 per cent 6 per cent 7 per cent (1in 20) (1in 16.7) (1in 14.3)
el
Steep terrain upto 3, 000 m 6 per cent 7 per cent 8 per cent height above mean sea level (1in 16.7) (1in 14.3) (1in 12.5) COMPENSATION IN GRADIENT ON
HORIZONTAL CURVES
On sharp horizontal curves gradient should be decreased to compensate for the loss of tractive effort due to the curve. This reduction in gradient given by Grade compensation, % 30R R
+
= Subject to a
maximum value of 75 R
, where R is radius of
curve. IRC recommends that grade compensation is not necessary for gradients is flatter than 4.0% Vertical Curves Vertical curves can be sumit curve is valley curve. Due to changes in grade in the vertical alignment of highway, it is necessary to introduce vertical curve at the intersections of different grades to smoothen out the vertical profile and thus ease off the changes in gradients for the fast moving vehicles.
The vertical curves used in highway may be classified into two categories: (a) Summit curve or crest curves with convexity upwards (b) Valley or sag curves with concavity upwards Summit Curves Summit curves with convexity upwards are formed in any one of the case illustrated in figure.
The deviation angle between the two interacting gradients is equal to the algebraic difference between them. Of all the cases, the deviation angle will be maximum when an ascending gradient meets with a descending gradient i.e., N = n1 –(–n2) = (n1 + n2)
There is no problem of discomfort to passengers on sumit curves. The main problem in summit curves is to provide adequate sight distance. The SSD should be in variably present at all sections. Circular summit curve is ideal as the sight distance available throughout the summit curve is constant.
But in actual practice simple parabolic curve is used as simmit curve instead of circular arc. Discomfort problem may be important in case of small hump passing over culvert. Parabolic summit curves are generally adopted and the equation is given by y = a.x2 Where, a N 2L
=
O.S.D
Where N is deviation angle and L is the length of the cuve.
Length of Summit Curves Length of summit curve is necessary to consider SSD and OSD.
A Length of the summit curve for stopping sight distance (SSD) Case1 : When the length of the curve is greater than the sight distance (L > SSD) Length of parabolic curve is given by
(
)Where, L = Length of the summit curve (m) S = SSD (m) N = Deviation angle = Algebraic difference of gradesH = Height of eye level of driver above road way surface. ( = 1.2 m)h = Height of object above the pavement surface. ( = 0.15m)H = 1.2 m and h = 0.15 m, then NS2
L
4.4
=
Case2 When L < SSD
(
)
2
2H 2h L 2S
N
+
=
4.4
2S
N
=
Do you know?
The minimum radius of parablic summit curve is given by R L N
=
B. Length of Summit curve for safe overtaking sight distance OSD or Intermediate sight distance (ISD). Case–1 : When L > S NS2
L
8H
= (since H = h) According to IRC H = h = 1.2 m NS2
L
9.6
=
Where, L = Length of parabolic summit curve.
N = Deviating angle S = Overtaking or intermediate sight distance. Case 2: When L < S L = 2S 8H
N

L = 2S 9.6
N

Valley Curves Valley curves or sag curves are formed i n any one of the cases illustrated in figure. In all the cases the maximum possible deviation angle is obtained when a descending gradient meets with an ascending gradient.
Following factors are important in valley curves. 1. Impact free movement of vehicles at design speed or the comfort to the passengers. 2. In case of the valley curve the rate of change of centrifugal curve governs the design, obviously the best shape of valley curve is transition curve for gradually introducing and increasing the centrifugal force acting down wards. Cubic parabola is generally preferred in valley curves. The head light sight distance available at valley curves should be at least equal to the SSD, however there is no problem of OSD at night as other vehicles with head lights can be seen from a considerable distance.
Length of Valley Curves Length of Valley curve is designed based on the two criteria. 1. The allowable rate of change of centrifugal acceleration of 0.6m/sec3. 2. The head light sight distance.
+n2
–n1
Higher of two values is adopted. Usually head light sight distance is higher. The valley curve is made fully transitional by providing Ls= L2 on each side. When L is total length of transitional curve. CaseI : The length of Transition curve Ls for comfort condition is given by 3
s
v
L
CR
=
s
s
L
L sinR N
= æç=ö÷ èø Total length of curve Nv3 1/2 L2
C
éù = êú ëû Where N is deviation angle, v is speed in m/sec and C is the allowable rate of change of CF acceleration = 0.6 m/sec3.
L = 2ls = 0.38 (NV3)1/2 Case2 : When L > SSD
(
)
2
1
N.S
L
2h 2S tan =
+a
(
)
N.S2
1.5 0.035.S
=
+
Where, L = Total length of valley curve.
S = SSD (m) N = Deviation angle a = beam angle » 1° h1 = Avg. height of head light CaseIIB: When L < SSD
(
)
2h1 2S. tan N
L 2S
+a =
(
2S 1.5 0.035S)
N
+
=
The lowest point on the valley curve will be on the bisector of the angle between the grades, if the gradients on the either side are equal When the gradients are not equal the lowest point lies on the side of flatter grades at distance 1
0
n
X
2N
from the tangent point of the first grade
n1.
Equation of cubic parabola y = b.x3 Where,
3
2N
b. 3