Sag vertical curves are curves that connect descending grades, forming a
bowl or a sag. Designing them is is very similar to the design of crest vertical
curves. Once again, the sight distance is the parameter that is normally employed to
find the length of the curve. When designing a sag vertical curve, however, the
engineer must pay special attention to the comfort of the drivers. Sag vertical
curves are characterized by a positive change in grade, which means that vehicles
traveling over sag vertical curves are accelerated upward. Because of the inertia of
the driver's body, this upward acceleration feels like a downward thrust. When this
perceived thrust and gravity combine, drivers can experience discomfort.
The length of sag vertical curves, which is the only parameter that we need for design,
is determined by considering drainage, driver comfort, aesthetics, and sight distance.
Once again, the aesthetics and driver comfort concerns are normally automatically resolved
when the curve is designed with adequate sight distance in mind. Driver comfort, for
example, requires a curve length that is approximately 50% of the curve length required
for the sight distance. Drainage may be a problem if the curve is quite long and flat, or
if the sag is within a cut. For more information on these secondary concerns, see
your local design manuals.
The theory behind the sight distance calculations for sag vertical curves is only
slightly different from that for crest vertical curves. Sag vertical curves normally
present drivers with a commanding view of the roadway during the daylight hours, but
unfortunately, they truncate the forward spread of the driver's headlights at night.
Because the sight distance is restricted after dark, the headlight beams are the focus of
the sight distance calculations. For sight distance calculations, a 1° upward
divergence of the beam is normally assumed.
In addition, the headlights of the vehicle are assumed to reside 2 ft above the roadway
surface. As with crest vertical curves, these assumptions lead to two possible
configurations, one in which the sight distance is greater than the curve length, and one
in which the opposite is true. The figure below illustrates these possibilities.
Where:
L = Curve length (ft)
S = Sight distance (ft) (normally the stopping sight distance)
B = Beam upward divergence (°) (normally assumed as 1°)
H = Height of the headlights (ft) (normally assumed as 2 ft)
A = Change in grade (|G2-G1| as a percent)
The stopping sight distance is normally the controlling sight distance for sag vertical
curves. At decision points, the roadway should be illuminated by other means so that
the sight distance of the driver is extended. Where possible, increased curve length
may also be provided.
Highway overpasses or other obstacles can occasionally reduce the sight distance on sag
vertical curves. In these instances, separate equations should be used to determine
the correct curve length. These equations are readily available in design manuals.
At this point, you have all of the information that you need to develop the precise
layout of your vertical curve. The parabolic curve calculations are identical for sag
and crest vertical curves. Just remember to use the appropriate positive or negative
values for the participating grades.