The objective of a horizontal curve in a plan is to provide a change in the direction to the central line of a road. When a vehicle negotiates a horizontal curve, the centrifugal force acts horizontally outwards through the centre of gravity of the vehicle. This centrifugal force is counteracted by the transverse frictional resistance developed between the tyres and the pavement surface which enables the vehicle to change the direction along the curve and to maintain the stability of the vehicle.
Centrifugal Force, P
$W=$ Weight of the vehicle, KG
The ratio of the centrifugal force to the weight of the vehicle, $\frac {P}{W}$ is known as the centrifugal ratio or the impact factor which is equal to $\frac{v^2}{gR}$.
The centrifugal force acting on a vehicle negotiating a horizontal curve has 2 effects:
To counteract the effect of centrifugal force and to reduce the tendency of overturning and skidding of the vehicle, 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.
The forces acting on a vehicle while traversing a circular curve of radius $\text{R m}$ at a speed of $\text{v } m/sec$ are:
After analysing the forces shown in the figure, we will get result as,
The value of $f$ is recommanded by IRC is $0.15$
IRC has recommended the maximum value of superelevation to take care of the mixed flow condition and the loading condition of veḥicles as a truck/bullock cart can be loaded with less dense material such as cotton due to which the CG of the vehicle will be high and it will not be safe for it to move on a road with high rate of superelevation.
Minimum superelevation is required to drain off the surface water. If the calculated superelevation is comes out to be less than the minimum camber of the road surface then the minimum superelevation will be equal to the minimum camber to be provided.
IRC has given the values of radii beyond which normal cambered section may be maintained and no superelevation is required for curves,
The design of superelevation for a mixed traffic condition is complex problem because different vehicles run on the road with different speeds. If we rely fully on to counteract the centrifugal force and neglect the lateral friction and provide maximum superelevation, it would be benefical for fast moving vehicles but might cause problem for slow moving vehicles on the other hand if we rely fully on to counteract the lateral friction and provide minimum superelvation, it would be unsafe for fast moving vehicles.
As a compromise and from practical consideration it is suggested that the superelevation should be provided to fully counteract the centrifugal force due to 75 percent of the design speed (by neglecting the lateral friction developed).
Steps for superelevation design:
If the allowable speed is greater than the design speed, the design is adequate and provide a superelevation of 0.07. If speed is less than the design speed, the speed is limited to the allowable speed, ${v}_{a}$.
Appropriate warning sign and speed limit regulation sign are installed to restrict and regulate the speed at such curves when the safe speed ${v}_{a}$ is less than the design speed $v$.
The road cross section at the straight portion is cambered with the crown at the centre of the pavement and sloping down towards the edges. But the cross-section in the circular curve portion of the road is superelevated with uniform tilt sloping down from the outer edge of the pavement up to the inner edge.
The crowned camber section at the straight before the start of the transition curve should be changed to a single cross slope equal to the desired superelevation at the beginning of the circular curve.
The attainment of superelevation may be split up into two parts:
Centrifugal Force, P
$P=\frac{W\times{v^2}}{gR}$
$W=$ Weight of the vehicle, KG
$v=$ Speed of vehicle, $m/sec$
$R=$ Radius, $m$
$g=$ Acceleration due to Gravity $=9.81 m/{sec^2}$
The ratio of the centrifugal force to the weight of the vehicle, $\frac {P}{W}$ is known as the centrifugal ratio or the impact factor which is equal to $\frac{v^2}{gR}$.
The centrifugal force acting on a vehicle negotiating a horizontal curve has 2 effects:
- Overturning Effect:
The equilibrium condition for overturning will occur when $Ph=\frac{Wb}{2}$ or when $\frac{P}{W}=\frac{b}{2h}$. This means that there is a danger of overturning when the centrifugal ratio $\frac{P}{W}$ or $\frac{v^2}{gR}$ attains a value of $\frac{b}{2h}$. - Transverse Skidding Effect:
The centrifugal force developed has also the tendency to push the vehicle outwards in the transverse direction. If the centrifugal force $P$ developed exceeds the maximum possible transverse skid-resistance due to friction, the vehicle will start skidding in the transverse direction.
In other words, when centrifugal ratio $\frac{P}{W}$ attains a value equal to the coefficient of lateral friction there is a danger of lateral skidding.
$\frac{P}{W}=\frac{v^2}{gR} < \frac{b}{2h}$
$\frac{P}{W}=\frac{v^2}{gR} < f $
$\frac{P}{W}=\frac{v^2}{gR} < f $
To counteract the effect of centrifugal force and to reduce the tendency of overturning and skidding of the vehicle, 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.
$e=\frac{E}{B}=tan\theta$
The forces acting on a vehicle while traversing a circular curve of radius $\text{R m}$ at a speed of $\text{v } m/sec$ are:
- The centrifugal force, $P=\frac{W\times{v^2}}{gR}$ acting horizontally outwards through the centre of gravity, CG.
- the Weight, $W$ of the vehicle acting vertically downwards through the CG.
- The frictional force developed between the tyres and the pavement counteracts transversely along the pavement surface towards the centre of the curve.
After analysing the forces shown in the figure, we will get result as,
$e+f=\frac{v^2}{gR}$
The value of $f$ is recommanded by IRC is $0.15$
Maximum Superelevation
IRC has recommended the maximum value of superelevation to take care of the mixed flow condition and the loading condition of veḥicles as a truck/bullock cart can be loaded with less dense material such as cotton due to which the CG of the vehicle will be high and it will not be safe for it to move on a road with high rate of superelevation.
Recommended Maximum Values of Superelevation
| For plain and rolling | 7.0% |
|---|---|
| Hills bound by snow | 7.0% |
| Hills not bound by snow | 10.0% |
| Urban roads with frequent intersections | 4.0% |
Minimum Superelevation
Minimum superelevation is required to drain off the surface water. If the calculated superelevation is comes out to be less than the minimum camber of the road surface then the minimum superelevation will be equal to the minimum camber to be provided.
IRC has given the values of radii beyond which normal cambered section may be maintained and no superelevation is required for curves,
| Design Speed | Radius (m) of horizontal curve for camber of | ||||
|---|---|---|---|---|---|
| KMPH | 4% | 3% | 2.5% | 2% | 1.7% |
| 20 | 50 | 60 | 70 | 90 | 100 |
| 25 | 70 | 90 | 110 | 140 | 150 |
| 30 | 100 | 130 | 160 | 200 | 240 |
| 35 | 140 | 180 | 220 | 270 | 320 |
| 40 | 180 | 240 | 280 | 350 | 420 |
| 50 | 280 | 370 | 450 | 550 | 650 |
| 60 | 470 | 620 | 750 | 950 | 1100 |
| 80 | 700 | 950 | 1100 | 1400 | 1700 |
| 100 | 1000 | 1500 | 1800 | 2200 | 2600 |
Superelevation Design
The design of superelevation for a mixed traffic condition is complex problem because different vehicles run on the road with different speeds. If we rely fully on to counteract the centrifugal force and neglect the lateral friction and provide maximum superelevation, it would be benefical for fast moving vehicles but might cause problem for slow moving vehicles on the other hand if we rely fully on to counteract the lateral friction and provide minimum superelvation, it would be unsafe for fast moving vehicles.
As a compromise and from practical consideration it is suggested that the superelevation should be provided to fully counteract the centrifugal force due to 75 percent of the design speed (by neglecting the lateral friction developed).
Steps for superelevation design:
- Calculate Superelevation for 75 percent of design speed $(\text{v } m/{sec})$ by neglecting lateral friction
$e=\frac{{0.75 \times {v}}^2}{gR}$ - If the value of $e$ calculated above comes out to be less than 0.07 or 7 percent then the value is so obtained is provided.
If it is greater than 0.07 then the maximum superelevation equal to 0.07 will be provided and proceed to step 3 or 4. - Calculate the value of lateral friction for the maximum value of $e=0.07$
$f=\frac{{v}^2}{gR}-0.07$
If the value of $f$ calculated above is less than 0.15, the superelevation of 0.07 is safe otherwise, calculate the restricted speed as shown in step 4. - Restricted/Allowable speed at curve,
$e+f=0.07+0.15=0.22=\frac{{{v}_{a}}^2}{gR}$
${v}_{a}=\sqrt {0.22gR}$
If the allowable speed is greater than the design speed, the design is adequate and provide a superelevation of 0.07. If speed is less than the design speed, the speed is limited to the allowable speed, ${v}_{a}$.
Appropriate warning sign and speed limit regulation sign are installed to restrict and regulate the speed at such curves when the safe speed ${v}_{a}$ is less than the design speed $v$.
Attianment of Superelevation
The road cross section at the straight portion is cambered with the crown at the centre of the pavement and sloping down towards the edges. But the cross-section in the circular curve portion of the road is superelevated with uniform tilt sloping down from the outer edge of the pavement up to the inner edge.
The crowned camber section at the straight before the start of the transition curve should be changed to a single cross slope equal to the desired superelevation at the beginning of the circular curve.
The attainment of superelevation may be split up into two parts:
-
Elimination of Crown of the cambered section:
This can be done by two methods:
- Rotation of outer edge about the crown: In this method, the outer half of the cross slope is rotated about the crown at a desired rate such that the surface falls on the same plane as the inner half and the elevation of the centre line is not altered.
- Diagonal Crown Method: In this method, the crown is progressively shifted outwards, thus increasing the width if the inner half of cross-section progressively. This method is not usually adopted as a portion of the outer half of the pavement has increasing values of negative superelevation on to a portion of the outer half, before the crown is eliminated.
- Rotation of pavement to attain full superelevation:
When the crown of the camber is eliminated, the superelevation available at the section is equal to the camber. But the superelevation to be provided at the beginning of the circular curve may be greater than the camber. Hence the pavement section is further rotated till the desired superelevation is obtained and this can be attained by two methods.
- By rotating the pavement cross section about the centre line. In this method, the vertical profile of the pavement remains unchanged and it also balanced the earthwork. But this method creates drainage problem if the subgrade is not in an embankment.
- By rotating the pavement cross section about the inner edge of the pavement section raising both the centre as well as the outer edge of the pavement. This method does not create drainage problem however additional earth is required as the entire shoulder is rotated along with the cross slope.
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