Fundamentals of Transportation/Vertical Curves

Vertical Curves are the second of the two important transition elements in geometric design for highways, the first being Horizontal Curves. A vertical curve provides a transition between two sloped roadways, allowing a vehicle to negotiate the elevation rate change at a gradual rate rather than a sharp cut. The design of the curve is dependent on the intended design speed for the roadway, as well as other factors including drainage, slope, acceptable rate of change, and friction. These curves are parabolic and are assigned stationing based on a horizontal axis.

Fundamental Curve Properties

Parabolic Formulation

Two types of vertical curves exist: (1) Sag Curves and (2) Crest Curves. Sag curves are used where the change in grade is positive, such as valleys, while crest curves are used when the change in grade is negative, such as hills. Both types of curves have three defined points: PVC (Point of Vertical Curve), PVI (Point of Vertical Intersection), and PVT (Point of Vertical Tangency). PVC is the start point of the curve while the PVT is the end point. The elevation at either of these points can be computed as $e_{P V C}$ and $e_{P V T}$ for PVC and PVT respectively. The roadway grade that approaches the PVC is defined as $g_{1}$ and the roadway grade that leaves the PVT is defined as $g_{2}$ . These grades are generally described as being in units of (m/m) or (ft/ft), depending on unit type chosen.

Both types of curves are in parabolic form. Parabolic functions have been found suitable for this case because they provide a constant rate of change of slope and imply equal curve tangents, which will be discussed shortly. The general form of the parabolic equation is defined below, where $y$ is the elevation for the parabola.

$y = a x^{2} + b x + c$

At x = 0, which refers to the position along the curve that corresponds to the PVC, the elevation equals the elevation of the PVC. Thus, the value of $c$ equals $e_{P V C}$ . Similarly, the slope of the curve at x = 0 equals the incoming slope at the PVC, or $g_{1}$ . Thus, the value of $b$ equals $g_{1}$ . When looking at the second derivative, which equals the rate of slope change, a value for $a$ can be determined.

$a = \frac{g_{2} - g_{1}}{2 L}$

Thus, the parabolic formula for a vertical curve can be illustrated.

$y = e_{P V C} + g_{1} x + \frac{(g_{2} - g_{1}) x^{2}}{2 L}$

Where:

$e_{p v c}$ : elevation of the PVC
$g_{1}$ : Initial Roadway Grade (m/m)
$g_{2}$ : Final Roadway Grade (m/m)
$L$ : Length of Curve (m)

Most vertical curves are designed to be Equal Tangent Curves. For an Equal Tangent Curve, the horizontal length between the PVC and PVI equals the horizontal length between the PVI and the PVT. These curves are generally easier to design.

Offset

Some additional properties of vertical curves exist. Offsets, which are vertical distances from the initial tangent to the curve, play a significant role in vertical curve design. The formula for determining offset is listed below.

$Y = \frac{A x^{2}}{200 L}$

Where:

$A$ : The absolute difference between $g_{2}$ and $g_{1}$ , multiplied by 100 to translate to a percentage
$L$ : Curve Length
$x$ : Horizontal distance from PVC along curve

Stopping Sight Distance

Sight distance is dependent on the type of curve used and the design speed. For crest curves, sight distance is limited by the curve itself, as the curve is the obstruction. For sag curves, sight distance is generally only limited by headlight range. AASHTO has several tables for sag and crest curves that recommend rates of curvature, $K$ , given a design speed or stopping sight distance. These rates of curvature can then be multiplied by the absolute slope change percentage, $A$ to find the recommended curve length, $L_{m}$ .

$L_{m} = K A$

Without the aid of tables, curve length can still be calculated. Formulas have been derived to determine the minimum curve length for required sight distance for an equal tangent curve, depending on whether the curve is a sag or a crest. Sight distance can be computed from formulas in other sections (See Sight Distance).

Crest Vertical Curves

The correct equation is dependent on the design speed. If the sight distance is found to be less than the curve length, the first formula below is used, whereas the second is used for sight distances that are greater than the curve length. Generally, this requires computation of both to see which is true if curve length cannot be estimated beforehand.

$S < L : L_{m} = \frac{A S^{2}}{200 {(\sqrt{h_{1}} + \sqrt{h_{2}})}^{2}}$

$S > L : L_{m} = 2 S - \frac{200 {(\sqrt{h_{1}} + \sqrt{h_{2}})}^{2}}{A}$

Where:

$L_{m}$ : Minimum Curve Length (m)
$A$ : The absolute difference between $g_{2}$ and $g_{1}$ , multiplied by 100 to translate to a percentage
$S$ : Sight Distance (m)
$h_{1}$ : Height of driver's eye above roadway surface (m)
$h_{2}$ : Height of objective above roadway surface (m)

Sag Vertical Curves

Just like with crest curves, the correct equation is dependent on the design speed. If the sight distance is found to be less than the curve length, the first formula below is used, whereas the second is used for sight distances that are greater than the curve length. Generally, this requires computation of both to see which is true if curve length cannot be estimated beforehand.

$S < L : L_{m} = \frac{A S^{2}}{200 (H + S \tan β)}$

$S > L : L_{m} = 2 S - \frac{200 (H + S \tan β)}{A}$

Where:

$A$ : The absolute difference between $g_{2}$ and $g_{1}$ , multiplied by 100 to translate to a percentage
$S$ : Sight Distance (m)
$H$ : Height of headlight (m)
$β$ : Inclined angle of headlight beam, in degrees

To find the position of the low point on a SAG vertical curve: x is the horizontal distance between the PVC and Low Point

$x = \frac{G 1 L}{G 1 - G 2}$

$G 1$ : Grade Down (%)
$G 2$ : Grade Up (%)
$L$ : Length of Vertical Curve (station) ei. 600 ft =6

Passing Sight Distance

In addition to stopping sight distance, there may be instances where passing may be allowed on vertical curves. For sag curves, this is not an issue, as even at night, a vehicle in the opposing can be seen from quite a distance (with the aid of the vehicle's headlights). For crest curves, however, it is still necessary to take into account. Like with the stopping sight distance, two formulas are available to answer the minimum length question, depending on whether the passing sight distance is greater than or less than the curve length. These formulas use units that are in metric.

$S < L : L_{m} = \frac{A (P S D^{2})}{864}$

$S > L : L_{m} = 2 P S D - \frac{864}{A}$

Where:

$A$ : The absolute difference between $g_{2}$ and $g_{1}$ , multiplied by 100 to translate to a percentage
$P S D$ : Passing Sight Distance (m)
$L_{m}$ : Minimum curve length (m)

Demonstrations

Examples

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, which is 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

Variables

$L$ - Curve Length
$e$ - Elevation of designated point, such as PVC, PVT, etc.
$g$ - Grade
$A$ - Absolute difference of grade percentages for a certain curve, in percent
$y$ - Elevation of curve
$Y$ - Offset between grade tangent from PVC and curve elevation for a specific station
$h_{1}$ - Height of driver's eye above roadway surface
$h_{2}$ - Height of object above roadway surface
$H$ - Height of headlight
$β$ - Inclined angle of headlight beam, in degrees
$S$ - Sight Distance in question
$K$ - Rate of curvature
$L_{m}$ - Minimum Curve Length

Key Terms

PVC: Point of Vertical Curves
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)

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Fundamentals of Transportation/Vertical Curves

Contents

Fundamental Curve Properties

Parabolic Formulation

Offset

Stopping Sight Distance

Crest Vertical Curves

Sag Vertical Curves

Passing Sight Distance

Demonstrations

Examples

Example 1: Basic Curve Information

Example 2: Adjustment for Obstacles

Example 3: Stopping Sight Distance

Thought Question

Sample Problem

Additional Questions

Variables

Key Terms

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Fundamentals of Transportation/Vertical Curves

Fundamental Curve Properties

Parabolic Formulation

Offset

Stopping Sight Distance

Crest Vertical Curves

Sag Vertical Curves

Passing Sight Distance

Demonstrations

Examples

Example 1: Basic Curve Information

Example 2: Adjustment for Obstacles

Example 3: Stopping Sight Distance

Thought Question

Sample Problem

Additional Questions

Variables

Key Terms

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