Problem:
A current roadway is climbing a hill at an angle of +3.0%. The roadway starts at station 100+00 and elevation of 1000 m. At station 110+00, there is an at-grade railroad crossing that goes over the sloped road. Since designers are concerned for the safety of drivers crossing the tracks, it has been proposed to cut a level tunnel through the hill to pass beneath the railroad tracks and come out on the opposite side. A vertical crest curve would connect the existing roadway to the proposed tunnel with a grade of (-0.5)%. The prospective curve would start at station 100+00 and have a length of 2000 meters. Engineers have stated that there must be at least 10 meters of separation between the railroad tracks and the road to build a safe tunnel. Assume an equal tangent curve. With the current design, is this criteria met?
Solution:
Fundamentals of Transportation/Vertical Curves 163
One way to solve this problem would be to compute the elevation of the curve at station 110+00 and then see if it is at least 10 meters from the tangent.
Another way would be to use the Offset Formula. Since A, L, and x are all known, this problem can be easily solved. Set x to 1000 meters to represent station 110+00.
The design DOES NOT meet the criteria.
Example 3: Stopping Sight Distance
Problem:
A current roadway has a design speed of 100 km/hr, a coefficient of friction of 0.1, and carries drivers with perception-reaction times of 2.5 seconds. The drivers use cars that allows their eyes to be 1 meter above the road. Because of ample roadkill in the area, the road has been designed for carcasses that are 0.5 meters in height. All curves along that road have been designed accordingly.
The local government, seeing the potential of tourism in the area and the boost to the local economy, wants to increase the speed limit to 110 km/hr to attract summer drivers. Residents along the route claim that this is a horrible idea, as a particular curve called "Dead Man's Hill" would earn its name because of sight distance problems. "Dead Man's Hill" is a crest curve that is roughly 600 meters in length. It starts with a grade of +1.0% and ends with (-1.0)%. There has never been an accident on "Dead Man's Hill" as of yet, but residents truly believe one will come about in the near future.
A local politician who knows little to nothing about engineering (but thinks he does) states that the 600-meter length is a long distance and more than sufficient to handle the transition of eager big-city drivers. Still, the residents push back, saying that 600 meters is not nearly the distance required for the speed. The politician begins a lengthy campaign to "Bring Tourism to Town", saying that the residents are trying to stop "progress". As an engineer, determine if these residents are indeed making a valid point or if they are simply trying to stop progress?
Solution:
Using sight distance formulas from other sections, it is found that 100 km/hr has an SSD of 465 meters and 110 km/hr has an SSD of 555 meters, given the criteria stated above. Since both 465 meters and 555 meters are less than the 600-meter curve length, the correct formula to use would be:
Since the 1055-meter minimum curve length is greater than the current 600-meter length on "Dead Man's Hill", this curve would not meet the sight distance requirements for 110 km/hr.
This seems like a very large gap. The question becomes, was the curve even good enough at 100 km/hr? Using the same formula, the result is:
740 meters for a minimum curve length is far greater than the existing 600-meter curve. Therefore, the residents are correct in saying that "Dead Man's Hill" is a disaster waiting to happen. As a result, the politician, unable to hold public confidence by his "progress" comment, was forced to resign.
Fundamentals of Transportation/Vertical Curves 164
Thought Question
Problem
Sag curves have sight distance requirements because of nighttime sight distance constraints. The headlights on cars have a limited angle at which they can shine with bright enough intensity to see objects far off in the distance. If the government were to allow a wider angle of light to be cast out on standard car headlights, would this successfully provide more stopping sight distance?
Solution
Yes, of course. For a single car traveling on a road with many sag curves, the design speed could be increased since more road could be seen. However, when additional cars were added to that same road, problems would begin to appear. With a greater angle of light being cast from headlights, drivers in opposing lanes would be severely blinded, forcing them to slow down to avoid causing an accident. Just think of the last time somebody drove by with their 'brights' on and blinded you. This problem could cause more accidents and force people to slow down, thus producing a net loss overall.
Sample Problem
Problem (Solution)
Additional Questions
• Homework
• Additional Questions
Variables
• - Curve Length
• - Elevation of designated point, such as PVC, PVT, etc.
• - Grade
• - Absolute difference of grade percentages for a certain curve, in percent
• - Elevation of curve
• - Offset between grade tangent from PVC and curve elevation for a specific station
• - Height of driver's eye above roadway surface
• - Height of object above roadway surface
• - Height of headlight
• - Inclined angle of headlight beam, in degrees
• - Sight Distance in question
• - Rate of curvature
• - Minimum Curve Length
Fundamentals of Transportation/Vertical Curves 165
Key Terms
• PVC: Point of Vertical Curve
• PVI: Point of Vertical Intersection
• PVT: Point of Vertical Tangent
• Crest Curve: A curve with a negative grade change (like on a hill)
• Sag Curve: A curve with a positive grade change (like in a valley)
References
[1] http://street.umn.edu/flash_curve_crest.html [2] http://street.umn.edu/flash_curve_sag.html