Applications of Parabolas:Highway Overpasses using Type 1

Vertical Curves

John CatlettMathematics Teacher

North Star High School

What is a parabola?

• A parabola is the graph that results from an equation of the form:

• Parabolas are symmetric about a vertical line known as the Axis of Symmetry.

• The Axis of Symmetry runs through the maximum or minimum point of the parabola which is called the Vertex.

yAx2 Bx C

• Examples of parabolas:

Exploration of Parabolas

• How do the coefficients A, B and C affect the graph of the parabola?

• Let’s use Geogebra to find out:• http://isite.lps.org/dtravis2/jcat/geogebra/

parabolaexploration.html

Parabolas and Vertical Curves• In road design, vertical curves are designed

using parabolas. There are four types:

Focus: Type 1 Curves and Overpasses

Design Factors• Parabolas used in the overpass design are

based on a number of factors:– The entrance and exit grades, g1 and g2

– The length of the vertical curve, L (NDOR uses 600 ft as the minimum L)

– The design speed (speed the road is designed for)– The sight stopping distance, S (based on design

speed and line of sight)– The elevation at the start of the curve, ElevBVC

Equation of the parabola:yAx2 Bx C

Ag2 g1

2L, Bg1, C ElevBVC

Finding the Parabola for Overpass Design

• Lets find equations to fit some existing overpasses.

• http://isite.lps.org/dtravis2/jcat/geogebra/overpassexample1.html

• For our purposes today, we will use:– The NDOR minimum ( 600 ft ) for L– The distance above the roadway that runs below

the overpass at the start of the curve as the ElevBVC

(these calculations are more complex in real life)

Now lets find the equation of the parabola using our formula,

yAx2 Bx C

Ag2 g1

2L, Bg1, C ElevBVC

Example:The entrance grade, g1, is 3%, the exit grade, g2,

Is 2%, the ElevBVC is 8 feet and the horizontal length of the curve, L, is 900 feet.

Solution:

Equation: Check Solution:

http://isite.lps.org/dtravis2/jcat/geogebra/overpassequationchecker.html

A .02 .03

2900 0.00003 B0.03 C 8

y 0.00003x2 0.03x 8

Final ThoughtsVideo: http://isite.lps.org/dtravis2/jcat/

Other things to consider?