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Geometric Design: Theory and Concepts

Vertical Curves

In highway design, most vertical curves are equal-tangent curves, which means that the horizontal distance from the center of the curve to the end of the curve is identical in both directions. Unequal-tangent vertical curves, which are simply equal-tangent curves that have been attached to one another, are used only infrequently. Because of its overwhelming popularity, we will limit our discussion to the geometry of the equal-tangent parabolic curve.

In highway design, the grades of the disjointed segments of roadway are normally known before any vertical curve calculations are initiated. In addition, the design speed of the roadway, the stopping sight distance, and the decision sight distance are also well established. The first step in the design of a vertical curve is the calculation of the curve length, which is the length of the curve as it would appear when projected on the x-axis. (See figure 1.0 below). Because the stopping sight distance should always be adequate, the length of the curve is normally dependent upon the stopping sight distance. Occasionally, as with any other section of a highway, the decision sight distance is a more appropriate sight distance. In these instances, the decision sight distance governs the length of the vertical curve. The curve length calculations are slightly different for sag and crest vertical curves, and are covered separately in those sections of this chapter. 

Let’s assume that you have already calculated the appropriate length (L) for your curve. At this point you would probably want to develop the actual shape of the curve for your design documents. Refer to figure 1.0 throughout the following discussion.

The first step in developing the profile for your curve is to find the center of your curve. The location of the center-point is where the disjointed segments of the roadway would have intersected, had they been allowed to do so. In other words, draw lines tangent to your roadway segments and see where those lines intersect. This intersection is normally called the vertical point of intersection (VPI). 

Diagram of a vertical curve

Figure 1.0:  Vertical Curve

The vertical point of curvature (VPC) and the vertical point of tangency (VPT) are located a horizontal distance of L/2 from the VPI. The VPC is generally designated as the origin for the curve and is located on the approaching roadway segment. The VPT serves as the end of the vertical curve and is located at the point where the vertical curve connects with the departing roadway segment. In other words, the VPC and VPT are the points along the roadway where the vertical curve begins and ends. 

One you have located the VPI, VPC, and VPT, you are ready to develop the shape of your curve. The equation that calculates the elevation at every point along an equal-tangent parabolic vertical curve is shown below.

Y = VPCy + B*x + (A*x2)/(200*L)

Y = Elevation of the curve at a distance x from the VPC (ft)
VPCy = Elevation of the VPC (ft)
B = Slope of the approaching roadway, or the roadway that intersects the VPC
A = The change in grade between the disjointed segments (From 2% to -2% would be a change of -4% or -4)
L = Length of the curve (ft)
x = Horizontal distance from the VPC (ft) (Varied from 0 to L for graphing.)

At this point, you have everything that you need to develop the shape of a simple equal-tangent vertical curve. The procedure above will work for both sag and crest vertical curves.