Complex Vertical Curves - Theory & Concepts

Complex Vertical Curves

Learning Objectives

Understand the function of parabolic vertical curves in highway and railway profiles.
Differentiate between crest and sag vertical curves.
Master the design of symmetrical vertical curves, including elevation offsets and high/low point location.
Understand the geometry, applications, and computations for unsymmetrical vertical curves.
Determine safe vertical curve lengths based on stopping and passing sight distance criteria.

Design and analysis of complex vertical curves for highways and railways. 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

Classifications 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

Symmetrical Parabolic Vertical Curve

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

Variables for Symmetrical Curves

$L$ : Length of the vertical curve (measured horizontally).
$g_1$ : Initial grade or tangent (in percent or decimal).
$g_2$ : Final grade or tangent (in percent or decimal).
$A$ : Algebraic difference in grades ( $A = g_2 - g_1$ ).
$x$ : Horizontal distance from the PVC.
$y$ : Vertical offset from the initial tangent to the curve at distance $x$ .
$E$ : External distance, which is the vertical offset from the PVI to the curve.

Equation of the Parabola

Parabolic Vertical Offset

Calculates the vertical offset from the tangent line to any point on a symmetrical vertical curve.

y = \frac{A}{200 L} x^2

Variables

Symbol	Description	Unit
$y$	Vertical offset from the initial tangent	m or ft
$A$	Algebraic difference in grades	%
$L$	Length of the vertical curve	m or ft
$x$	Horizontal distance from the PVC	m or ft

Assuming $A$ is in percent and $L$ , $x$ are in meters or feet. If $A$ is a decimal, remove the 200.

Curve Elevation Equation

Calculates the elevation at any point x from the PVC by taking the elevation of the tangent line at x and adding (for sag) or subtracting (for crest) the offset y.

\text{Elev}_x = \text{Elev}_{PVC} \pm g_1 x \pm y

Variables

Symbol	Description	Unit
$\text{Elev}_x$	Elevation on the curve at distance x	m or ft
$\text{Elev}_{PVC}$	Elevation of the PVC	m or ft
$g_1$	Initial grade	decimal
$x$	Horizontal distance from the PVC	m or ft
$y$	Vertical offset	m or ft

External Distance

External Distance

Calculates the maximum offset occurring at the PVI (where x = L/2).

E = \frac{A L}{800}

Variables

Symbol	Description	Unit
$E$	External distance (vertical offset at PVI)	m or ft
$A$	Algebraic difference in grades	%
$L$	Length of the vertical curve	m or ft

High or Low Point

High/Low Point Location

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).

Horizontal Distance to High/Low Point

Calculates the horizontal distance from the PVC to the highest or lowest point.

x_m = \frac{g_1 L}{g_1 - g_2}

Variables

Symbol	Description	Unit
$x_m$	Horizontal distance to the high/low point	m or ft
$g_1$	Initial grade	%
$g_2$	Final grade	%
$L$	Length of the vertical curve	m or ft

Unsymmetrical Parabolic Vertical Curves

Unsymmetrical Parabolic Vertical Curve

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 ( $L_1$ ) is not equal to the horizontal distance from the PVI to the PVT ( $L_2$ ).

Applications of Unsymmetrical Curves

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 $L = L_1 + L_2$ . The common point where the two parabolas meet is directly below (or above) the PVI.

External Distance for Unsymmetrical Curves

Calculates the vertical offset at the PVI for an unsymmetrical curve.

E = \frac{L_1 L_2}{200(L_1 + L_2)} (g_1 - g_2)

Variables

Symbol	Description	Unit
$E$	External distance at PVI	m or ft
$L_1$	Length from PVC to PVI	m or ft
$L_2$	Length from PVI to PVT	m or ft
$g_1$	Initial grade	%
$g_2$	Final grade	%

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 ( $x_m$ ) is determined by checking which side the point falls on.

Turning Point on L1 Side

Calculates the turning point if it is on the side of L1 (measured from the PVC).

x_m = \frac{g_1 L_1^2}{200 E}

Variables

Symbol	Description	Unit
$x_m$	Distance from PVC to turning point	m or ft
$g_1$	Initial grade	%
$L_1$	Length from PVC to PVI	m or ft
$E$	External distance at PVI	m or ft

Turning Point on L2 Side

Calculates the turning point if it is on the side of L2 (measured from the PVT backwards).

x_m = \frac{g_2 L_2^2}{200 E}

Variables

Symbol	Description	Unit
$x_m$	Distance from PVT backwards to turning point	m or ft
$g_2$	Final grade	%
$L_2$	Length from PVI to PVT	m or ft
$E$	External distance at PVI	m or ft

To find the elevation at any point $x$ on an unsymmetrical curve, you must first determine whether $x$ is on the $L_1$ side or the $L_2$ side.

Elevation on L1 Side

Calculates offset and elevation for points where x < L1 (measured from PVC).

y = \frac{x^2}{L_1^2} E

\text{Elev}_x = \text{Elev}_{PVC} + g_1 x \pm y

Variables

Symbol	Description	Unit
$y$	Vertical offset on L1 side	m or ft
$x$	Distance from PVC	m or ft
$L_1$	Length from PVC to PVI	m or ft
$E$	External distance at PVI	m or ft
$\text{Elev}_x$	Elevation on curve	m or ft
$\text{Elev}_{PVC}$	Elevation of PVC	m or ft
$g_1$	Initial grade	decimal

Elevation on L2 Side

Calculates offset and elevation for points where x < L2 (measured from PVT backwards).

y = \frac{x^2}{L_2^2} E

\text{Elev}_x = \text{Elev}_{PVT} - g_2 x \pm y

Variables

Symbol	Description	Unit
$y$	Vertical offset on L2 side	m or ft
$x$	Distance backwards from PVT	m or ft
$L_2$	Length from PVI to PVT	m or ft
$E$	External distance at PVI	m or ft
$\text{Elev}_x$	Elevation on curve	m or ft
$\text{Elev}_{PVT}$	Elevation of PVT	m or ft
$g_2$	Final grade	decimal

PVI in Unsymmetrical Curves

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 ( $h_1$ ) can see an object of height ( $h_2$ ) on the road ahead at a distance equal to the safe Stopping Sight Distance ( $S$ ).

Crest Curve Length (SSD, S < L)

If sight distance is less than the curve length.

L = \frac{A S^2}{200 (\sqrt{h_1} + \sqrt{h_2})^2}

Variables

Symbol	Description	Unit
$L$	Length of the vertical curve	m or ft
$A$	Algebraic difference in grades	%
$S$	Stopping Sight Distance	m or ft
$h_1$	Driver eye height	m or ft
$h_2$	Object height	m or ft

Crest Curve Length (SSD, S > L)

If sight distance is greater than the curve length.

L = 2S - \frac{200 (\sqrt{h_1} + \sqrt{h_2})^2}{A}

Variables

Symbol	Description	Unit
$L$	Length of the vertical curve	m or ft
$A$	Algebraic difference in grades	%
$S$	Stopping Sight Distance	m or ft
$h_1$	Driver eye height	m or ft
$h_2$	Object height	m or ft

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 ( $h_2$ ) is typically considered equal to the passenger car eye height ( $1.08 \text{ m}$ ), as the opposing vehicle is the target.

Crest Curve Length (PSD, S < L)

Approximate metric standard for curve length given passing sight distance.

L = \frac{A S^2}{280}

Variables

Symbol	Description	Unit
$L$	Length of the vertical curve	m or ft
$A$	Algebraic difference in grades	%
$S$	Passing Sight Distance	m or ft

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 ( $S$ ).

Sag Curve Length (Headlight, S < L)

Minimum curve length for sag curve based on headlight sight distance.

L = \frac{A S^2}{200 (h + S \tan \beta)}

Variables

Symbol	Description	Unit
$L$	Length of the vertical curve	m
$A$	Algebraic difference in grades	%
$S$	Stopping Sight Distance	m
$h$	Headlight height, typically 0.60 m	m
$\beta$	Upward divergence angle, typically 1 degree	deg

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 $y = (A / 200L) x^2$ .
The high or low point of a symmetrical vertical curve is found at $x_m = (g_1 L) / (g_1 - g_2)$ from the PVC.
Unsymmetrical curves consist of two different parabolas meeting at the PVI, used when strict spatial constraints exist. The external distance $E$ and parabolic offsets are calculated distinctly for the $L_1$ and $L_2$ 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.

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