Vertical Curves
In highway design, most vertical curves are equaltangent curves, which
means that the horizontal distance from the center of the curve to the end of the curve is
identical in both directions. Unequaltangent vertical curves, which are simply
equaltangent 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 equaltangent 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 xaxis. (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 centerpoint 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).
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
equaltangent parabolic vertical curve is shown below.
Y = VPCy + B*x + (A*x2)/(200*L)
Where:
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
equaltangent vertical curve. The procedure above will work for both sag and crest
vertical curves.
