Horizontal Curve

As discussed earlier, 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. For a certain speed of a vehicle, the centrifugal force is dependent on the radius of the horizontal curve. To keep the centrifugal ratio, $\frac {P}{W}$, within a low limit, the radius of the should be kept correspondingly high.

The centrifugal force which is counteracted by the superelevation and lateral friction is given by,

$e+f=\frac{v^2}{gR}$

If the design speed is fixed for a highway, then the minimum radius to be provided can be found from the above relation.
The ruling minimum radius of a curve for ruling speed $\text{v m/s}$ is given by,

${R}_{ruling}=\frac{v^2}{g(e+f)}$

If minimum speed $v'$ of vehicle is adopted instead of design speed $v$, then minimum radius of horizontal curve ${R}_{min}$ is given by,

${R}_{min}=\frac{(v')^2}{g(e+f)}$

Minimum Radii of horizontal curves for different terrain conditions in $m$

Classification of Roads	Plain Terrain		Rolling terrain		Mountainous Terrain
					Area not affected by snow		Snowbound areas
	Ruling min	Absolute min	Ruling min	Absolute min	Ruling min	Absolute min	Ruling min	Absolute min
NH & SH	360	230	230	155	80	50	90	60
MDR	230	155	155	90	50	30	60	33
ODR	155	90	90	60	30	20	33	23
VR	90	60	60	45	20	14	23	15

Classification of Roads	Steep Terrain
	Area not affected by snow		Snowbound areas
Ruling min	Absolute min	Ruling min	Absolute min	Ruling min
NH & SH	50	30	60	33
MDR	30	14	33	15
ODR	20	14	23	15
VR	20	14	23	15

Widening of Pavement on Horizontal Curves

When a vehicle travelling at high speed negotiates a horizontal curve, there is a tendency of a vehicle to move outward. To have a safe negotiation of vehicles sometimes the width of the pavement at the curve is widened because of the following reasons:

1. Whenever a vehicle goes around a horizontal curve, it goes off track due to rigid wheelbase of the vehicle.
2. If the vehicle is moving at a very high speed, superelevation and lateral friction developed won't be able to fully counteract the outward thrust due to the centrifugal force which may lead to transverse skidding and rear wheel may take the outward path instead following the front wheel path.
3. In the case of trailer units, the path traced by the units is also likely to be on either side of the central path of towing vehicle.
4. While negotiating a horizontal curve, a driver has a tendency to use the outer side of the curve to have a greater visibility at the curve.
5. While two vehicle cross or overtake at horizontal curve there is a psychological tendency to maintain a greater clearance between the vehicles.

The extra widening of pavement on horizontal curves is divided into two parts:

1. Mechanical Widening: The widening required to account for the off-tracking due to the rigidity of wheel base is called mechanical widening.
   ${W}_{m}=\frac{nl^2}{2R}$
        $n=\text{number of lanes}$
        $l=\text{Length of the wheel base}$ (normally taken as 6 m for commercial vehicles, if not known)
        $R= \text{Radius of the horizontal curve, m}$


2. Psychological widening: Extra widening is also provided for psychological reasons such as, to provide for greater manoeuvrability of steering at higher speeds, to allow for the extra space requirements for the overhangs of vehicles and to provide greater clearance for crossing and overtaking vehicles on the curves.
   
   ${W}_{ps}=\frac{v}{9.5\sqrt{R}}$
   

If radius is more than 300 m than there is no need to provide extra widening.

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