Vertical Curves

While traveling on the roads, sometimes, we come across some hilly terrains where we drive uphill and downhill and sometimes, we somehow feel like sinking in our stomach. This can be cause with excessive speed or inappropriate design of the vertical curve (alignment) on which we are driving. Therefore, there is a standard design for these curves to provide a smooth driving on a hill (crest) or a valley (sag). Another type of bad feeling happens when we drive on curves with higher than safe speed, details of which is discussed on horizontal alignment subject.

So basically, whenever we have two opposite slopes on the roadway, we will need to design for a vertical curve. The different ways these curves can be classified is shown in the picture below borrowed from School of Engineering at UCONN website. So the the Types I and II vertical curves are of crest type and types III and IV are of sag type. As can be seen, the two tangents at the two sides of the curve intersect with each other at a virtual point outside the curve, which is called the vertical point of intersection (VPI). These two tangents have two different slopes, called grades of gradients, which can be positive (+G), or negative (-G) slopes. Grades are usually shown in percentages.

The point where the curve begins is called vertical point of curvature (VPC), and where it ends is called vertical point of tangency (VPT). The length (L) of the curve is measured from VPC to VPT and the closest distance from VPI to the curve is known as middle ordinate, vertical offset, shown as E or e, (probably coming from eccentricity). The overall change in grade is the absolute value of difference between G1 and G2, shown as A.

The well-known metric which is used to characterize curves is the change in curvature, shown as K, which is the ratio of curve length per gradient difference:

K = L / A

So, driving on a curve with small K more sensible or with a large K! That is to be known soon. In designing vertical curve, the main thing is to determine what is the minimum K, or the minimum length (as the grades don't change). For this, we need to know what speed we would like to drive on this land with specific slopes. Intuitively, we can say if we want to drive slowly, even a short length (lower K) would work fine, but if we'd like to drive fast and furious, we probably need a long stretch for that curve, so it won't fly out of the curve after we reach the crest! 

For the design of crest curves, the main factor is whether we can see behind the hill and how far we can see. It is all about safety, isn't it? The distance we can see in front of us is related to the stopping sight distance (SSD). That is, if we see something on the road, and we brake, will we come to a halt before we hit the obstacle? That's another aspect of the design, which is directly related to our speed, the faster we drive, the more distance (visible) we need to stop the car. The SSD is related to the following factors:

• Driver's speed, 

• The time he takes to react to an obstacle and push the brake, 

• The deceleration rate of the vehicle, and

• The slope of the road

The minimum required length (L), depends on the sight distance (S). The following equations are used to find the minimum length for crest curves. First, the one of the equations is used, and if the results are not consistent with the initial assumption, then the other equation is used.

Similar sets of equation are used for sag curves, but there is no limitation on the height of the obstacle on the road.

Although, if there is a sag curve and there is an overpass beyond which there is an object on the driver's direction, the height of the object should be known and the equation to determine the length is different too. Also, charts and graphs are available for different combinations of speed, gradient difference, curve length, and K-values, which can be used in design. Here is a link to Texas DOT's guidelines for vertical curves which has provided these graphs. These equations along with a picture of a vertical curve is also in a file attached to this page.