Curve Resistance and Grade Compensation

Curve Resistance for Turning Vehicle: 

Vehicles are steered by the turning of front wheel on horizontal curve, the direction of rotation of rear and front wheels are different, so direction of rotation of rear and front wheels are different, there is some loss in the tractive force. (Traction, or tractive force, is the force used to generate motion between a body and a tangential surface, through the use of dry friction, though the use of shear force of the surface is also commonly used.) 

The tractive force given by the rear wheels is T and tractive force available in the direction of movement is T cos α. 

Loss = T - T cos α

Therefore curve resistance means loss of traction force due to turning of vehicle on  horizontal curve is,   T - T cos α 

More curve resistance is on the sharpe curve. 

Grade Compensation:

When there is a horizontal curve in addition to gradient then there will be increased resistance to traction due to both curve and gradient, therefore it is necessary to compensate the gradient at horizontal curve. 

IRC suggested to  compensate the grade so that pulling power of vehicle cannot be compromise, therefore grade compensation taken is minimum of, 

where,

R = radius of curve

Compensated Grade = Gradient - Grade Compensation 

According to IRC grade compensation is not necessary for the gradient flatter than 4 % 

Extra Widening

Increase the width of pavement by increasing the number of lanes, this is done just to resolve the problem of increase in traffic volume is called widening. Additional width of carriageway that is required on horizontal curve is referred as extra widening. The rear wheels follow the inner path on the curve as compared with the front wheels, this phenomena is known as off tracking. 

Basic reasons to provide extra widening are, 

• To avoid off-tracking due to rigidity of wheel base
• To counteract transverse skidding
• To increase the visibility at curve
• To encounter psychological tendency of driver negotiating the curve

Extra-widening is dividing into two parts,

1. Mechanical Widening 
2. Psychological Widening 

1. Mechanical Widening: It is provided due to the rigidity of wheel base,when a vehicle travels on a horizontal curve, the path followed by front wheel is not same as the path followed by rear wheel on the curve this phenomena is known as off-tracking. Mechanical widening is provided to avoid off-tracking due to rigidity of wheel base.

calculation of mechanical widening,

where,

n = Number of lanes

R = Radius of curve

l  =  Length of wheel base

2. Psychology Widening: There is tendency for the drivers to drive closer to the edges of the pavements on curves and to maintain side gap between another vehicles, so psychological widening is also required to provide . IRC proposed an empirical formula for the psychological widening, 

Calculation of psychological widening, 

where,

V is in kmph

therefore, the total extra-widening (We) needed to be provided on horizontal curve will be given as,

 

IRC Recommendation,

IRC recommended values of extra widening for single and two lane pavement are given below,

1. If R < 50 m, then extra widening is provided at inner edge
2. If R > 300 m, then extra widening is not provided
3. If 50 < R < 300 m, then extra widening is provided at both the edge

Transition Curve

Transition curve is provided to change horizontal alignment from straight to circular curve gradually. Transition curve has a infinite radius at straight road and gradually changes to designated radius at circular curve. As the length of transition curve increases radius decreases,  i.e.

 

from straight to circular curve centrifugal force introduce gradually. i.e. 

Objectives or Necessity of transition curve:

• To gradually introduce the centrifugal force in between straight and circular curve.
• To avoid sudden jerk.
• Transition curve will insure easy to follow path for drivers.
• To gradually introduction of super elevation. 
• To introduce extra widening gradually.
• Improve aesthetic appearance of the road.  

Ideal shape of transition curve or Requirement of ideal transition curve:

• Rate of introduction of centrifugal force or rate of change of centrifugal acceleration should be constant or uniform. 
• Length of transition curve is inversely proportional to radius of curve.
• Spiral type of curve or clothoid type of transition curve fulfil the conditions of ideal transition curve as per IRC.      

Length of transition curve:

As per the IRC minimum length of transition curve should be determined as maximum of the following three criteria's,

1. As per rate of change of centrifugal acceleration. 
2. As per rate of change of super-elevation.
3. Minimum length of transition curve by empirical relation.  

1. As per rate of change of centrifugal acceleration :

2. As per rate of change of super elevation: 

• 1 in 150 for plain and rolling terrain
• 1 in 60 for mountainous and steep terrain 

where, 

1 in N means, 

1 = rise height of outer edge which is achieved at N length of transition curve.

therefore, 

length of transition curve can be written as,

Lt = 150. x            for plain and rolling terrain 

Lt = 60. x              for steep and mountainous terrain              

Calculation of x :

1. Rotation about inner edge: 

2. Rotation about central axis:

3. Minimum length of transition curve as per empirical relation:

for plain and rolling terrain, 

for steep and mountainous terrain,

Where, 

V in kmph

R in meter

Lt in meter