Complex Vertical Curves
Design and analysis of complex vertical curves for highways and railways.
Overview
Vertical curves provide a smooth transition between different grades (slopes) on a highway or railway profile. Unlike horizontal curves, which are typically circular or spiraled, vertical curves in highway design are almost exclusively parabolic. Parabolas are used because they provide a constant rate of change of grade, which ensures a smooth ride and simplifies elevation calculations.
Types of Vertical Curves
Vertical curves are classified primarily into two types based on their geometry:
- Crest Vertical Curves: Curves that connect an ascending grade to a descending grade, or where the change in grade is negative (convex upwards).
- Sag Vertical Curves: Curves that connect a descending grade to an ascending grade, or where the change in grade is positive (concave upwards).
Symmetrical Parabolic Vertical Curves
Definition
A symmetrical parabolic vertical curve is one where the horizontal distance from the Point of Vertical Curvature (PVC) to the Point of Vertical Intersection (PVI) is equal to the horizontal distance from the PVI to the Point of Vertical Tangency (PVT). This means the PVI is located exactly at the horizontal midpoint of the curve.
Key Elements and Variables
- : Length of the vertical curve (measured horizontally).
- : Initial grade or tangent (in percent or decimal).
- : Final grade or tangent (in percent or decimal).
- : Algebraic difference in grades ().
- : Horizontal distance from the PVC.
- : Vertical offset from the initial tangent to the curve at distance .
- : External distance, which is the vertical offset from the PVI to the curve.
Equation of the Parabola
The vertical offset () from the tangent line to any point on a symmetrical vertical curve is given by the general parabolic equation:
$$
y = \\frac{A}{200 L} x^2
$$(Assuming is in percent and , are in meters or feet. If is a decimal, remove the 200). The elevation at any point from the PVC is calculated by taking the elevation of the tangent line at and adding (for sag) or subtracting (for crest) the offset .
$$
\\text{Elev}_x = \\text{Elev}_{PVC} \\pm g_1 x \\pm y
$$External Distance
The maximum offset occurs at the PVI (where ). This offset is called the external distance ():
$$
E = \\frac{A L}{800}
$$High or Low Point
The highest point (on a crest curve) or lowest point (on a sag curve) does not necessarily occur at the center of the curve. It occurs where the tangent to the curve is perfectly horizontal (i.e., the grade is zero). The horizontal distance () from the PVC to the high/low point is:
$$
x_m = \\frac{g_1 L}{g_1 - g_2}
$$Unsymmetrical Parabolic Vertical Curves
Definition
An unsymmetrical parabolic vertical curve is essentially two different parabolas that meet tangentially at a common point, usually vertically aligned with the PVI. The horizontal distance from the PVC to the PVI () is not equal to the horizontal distance from the PVI to the PVT ().
Applications
Unsymmetrical curves are used when strict physical constraints dictate the curve geometry, such as needing to tie into an existing intersection, bridge deck, or clearing a specific overhead or underground obstacle where a symmetrical curve cannot fit.
Geometry of Unsymmetrical Curves
The total length of the curve is . The common point where the two parabolas meet is directly below (or above) the PVI. The vertical offset () at the PVI for an unsymmetrical curve is given by:
$$
E = \\frac{L_1 L_2}{200(L_1 + L_2)} (g_1 - g_2)
$$The turning point (highest or lowest point) of an unsymmetrical curve may fall on either side of the PVI depending on the steepness of the grades. The horizontal distance from the PVC to the turning point () is determined by checking which side the point falls on.
If the turning point is on the side of (i.e., measured from the PVC):
$$
x_m = \\frac{g_1 L_1^2}{200 E}
$$If the turning point is on the side of (i.e., measured from the PVT backwards):
$$
x_m = \\frac{g_2 L_2^2}{200 E}
$$To find the elevation at any point on an unsymmetrical curve, you must first determine whether is on the side or the side.
For (measured from PVC):
$$
y = \\frac{x^2}{L_1^2} E \\quad \\text{and} \\quad \\text{Elev}_x = \\text{Elev}_{PVC} + g_1 x \\pm y
$$For (measured from PVT backwards):
$$
y = \\frac{x^2}{L_2^2} E \\quad \\text{and} \\quad \\text{Elev}_x = \\text{Elev}_{PVT} - g_2 x \\pm y
$$Important
The location of the PVI in an unsymmetrical curve is not the turning point, nor is it the midpoint of the curve. It is simply the common point of tangency directly above or below where the two unequal tangents meet.
Sight Distance on Vertical Curves
A primary criterion for determining the length of a vertical curve is providing adequate sight distance for safety.
Crest Curves: Stopping Sight Distance (SSD)
Stopping Sight Distance (SSD)
On crest curves, the sightline is blocked by the roadway surface itself. The length of the curve must be sufficient so that a driver with an eye height () can see an object of height () on the road ahead at a distance equal to the safe Stopping Sight Distance ().
If (sight distance is less than the curve length):
$$
L = \\frac{A S^2}{200 (\\sqrt{h_1} + \\sqrt{h_2})^2}
$$If (sight distance is greater than the curve length):
$$
L = 2S - \\frac{200 (\\sqrt{h_1} + \\sqrt{h_2})^2}{A}
$$Crest Curves: Passing Sight Distance (PSD)
Passing Sight Distance (PSD)
For two-lane highways, it may be necessary to provide Passing Sight Distance (PSD), which is significantly longer than SSD. It ensures a driver has sufficient visibility to overtake a slower vehicle safely without colliding with an oncoming vehicle. In this case, the object height () is typically considered equal to the passenger car eye height (), as the opposing vehicle is the target.
If :
$$
L = \\frac{A S^2}{280} \\quad \\text{(approximate metric standard)}
$$Sag Curves: Headlight Sight Distance
Headlight Sight Distance
On sag curves, sight distance during the day is rarely an issue because the driver can see across the entire curve. However, at night, the limiting factor is how far the vehicle's headlights illuminate the road ahead. The length of the curve must ensure that the headlight beam covers the required Stopping Sight Distance ().
$$
L = \\frac{A S^2}{200 (h + S \\tan \\beta)} \\quad \\text{for } S < L
$$Where is the headlight height (typically ) and is the upward divergence angle of the light beam (typically ).
Key Takeaways
- Vertical curves in highway design are parabolic to provide a constant rate of change of grade.
- In symmetrical curves, the PVI is located at the horizontal midpoint. The vertical offset from the tangent follows .
- The high or low point of a symmetrical vertical curve is found at from the PVC.
- Unsymmetrical curves consist of two different parabolas meeting at the PVI, used when strict spatial constraints exist. The external distance and parabolic offsets are calculated distinctly for the and sides.
- The length of a crest vertical curve is primarily dictated by the required Stopping Sight Distance (SSD) or Passing Sight Distance (PSD), factoring in driver eye height and object height.
- The length of a sag vertical curve is primarily dictated by headlight sight distance for safe nighttime driving.