Horizontal Curves (circular, spirals)


References:

Circular Curve Notation (see figure below):


Review: Circular Curve Stakeout

From CE 211 Lab Manual


Moving up on a curve

From CE 211 Lab Manual


Spiral Curves in Design:

References:


General Information on Spirals

The introduction of the circular curve at the PC takes place at a point but drivers and vehicles do not make directional changes instantaneously.

It is also common practice in constructing curves on highways to tip or superelevate the pavement downward toward the inside of the curve to aid in the riding quality and safety for vehicles navigating the curve. Again it is not practicable or advisable to introduce the superelevation instantaneously. If introduced on the tangent where it is not needed, the driver must steer into it slightly with a negative steering angle. If introduced all on the curve some area of negative superelevation will generally result or the introduction will be done so quickly that both the riding quality and the visual attractiveness of the highway suffer.

A solution is to introduce both the curvature and superelevation at a gradual rate using an easement curve that gradually changes in radius from infinity to some finite value where the associated circular curve begins. In short, a spiral curve is required. There are a number of identifiable curves that spiral, but their mathematical differences do not affect their usefulness on highways.

The geometry of the spiral curve is more rigorous that that of the circular curve and handbook tables are the usual way of working out the deflection angles needed to lay out a spiral curve in the field. The discussion has been worked out with reference to Route Location and Design, 5th ed., Hickerson, Thomas F., New York: McGraw-Hill, 1964, for the appropriate tables.

The spiral curve element generally selected by the designer is the length of the spiral "ls". The choice is usually made to introduce superelevation slowly enough so as not to exceed certain relative slopes between pavement edge and centerline grades. As a minimum, spiral curve lengths should not be shorter than the distance covered in two seconds at highway design speed.


Spiral Notation


Selected formulas for spiral curves:




Example Problem - Regular Deflection Angles:

Example Problem - Moving up on a Spiral:

What is delta backsight? It is the angle to turn from a backsight point (TS in this case) to get tangent to the curve (the red line is tangent to the curve). ...

What is delta forward? It is the angle to turn after you have become tangent to the curve, to get to the next stake point (SC in this case). ...