Vertical Curves - Theory & Concepts

Vertical Curves - Theory & Concepts

Learning Objectives

Understand the function and geometry of parabolic vertical curves.
Distinguish between crest and sag vertical curves and their primary design controls.
Calculate the Rate of Vertical Curvature (K-Value).
Compute stationing and elevation for points on vertical curves including high/low points.
Apply sight distance requirements to determine the minimum length of vertical curves.
Understand and calculate parameters for unsymmetrical vertical curves.

Design and analysis of parabolic vertical curves, including crest and sag curves for highway and railway profiles.

Overview of Vertical Curves

Vertical curves provide a smooth transition between two different grades (slopes) in the vertical alignment of a route. Unlike horizontal curves, which are circular, vertical curves are designed as symmetrical or unsymmetrical parabolas because a parabola provides a constant rate of change of grade, ideal for vehicular dynamics.

Types of Vertical Curves

Crest and Sag Curves

Crest Vertical Curve: Forms a convex profile (hill). It transitions from a positive grade to a negative grade, a positive to a flatter positive grade, or a negative to a steeper negative grade. The primary design control is Stopping Sight Distance (SSD).
Sag Vertical Curve: Forms a concave profile (valley). It transitions from a negative grade to a positive grade, a negative to a flatter negative grade, or a positive to a steeper positive grade. Design controls include headlight sight distance, passenger comfort, and drainage.

K

K-Value

A critical parameter in vertical curve design is the $K$ -value, which represents the horizontal distance required to effect a $1\%$ change in grade. It directly measures the "flatness" or "sharpness" of the curve. Highway design manuals (like AASHTO) provide minimum $K$ -values based on design speeds and sight distance requirements.

K-Value Formula

Calculates the horizontal distance required to effect a 1% change in grade.

K = \frac{L}{A}

Variables

Symbol	Description	Unit
$K$	Rate of vertical curvature	-
$L$	Length of the vertical curve	m or ft
$A$	Algebraic difference in grades, \|g_1 - g_2\|	%

Elements of a Vertical Curve

Key Points and Terminology

PVC (Point of Vertical Curvature): The beginning of the vertical curve. Also called BVC.
PVT (Point of Vertical Tangency): The end of the vertical curve. Also called EVC.
PVI (Point of Vertical Intersection): The intersection of the initial grade ( $g_1$ ) and the final grade ( $g_2$ ).
$g_1, g_2$ : The grades of the intersecting tangents, expressed in percent (e.g., $+3\%$ or $-2\%$ ).
$L$ : The length of the vertical curve, measured horizontally.
$A$ : The algebraic difference in grades ( $A = g_2 - g_1$ ).
$r$ : The rate of change of grade ( $r = A / L$ ).

Stationing and Elevation Computations

Stationing Computations

Compute stationing of the Point of Vertical Curvature (PVC) and Point of Vertical Tangency (PVT).

\text{Sta. PVC} = \text{Sta. PVI} - \frac{L}{2}

\text{Sta. PVT} = \text{Sta. PVI} + \frac{L}{2}

Variables

Symbol	Description	Unit
$\text{Sta. PVC}$	Station of Point of Vertical Curvature	-
$\text{Sta. PVT}$	Station of Point of Vertical Tangency	-
$\text{Sta. PVI}$	Station of Point of Vertical Intersection	-
$L$	Length of the curve	m or ft

Elevation Computations

Compute elevations of the Point of Vertical Curvature (PVC) and Point of Vertical Tangency (PVT).

\text{Elev. PVC} = \text{Elev. PVI} - g_1 \left(\frac{L}{2}\right)

\text{Elev. PVT} = \text{Elev. PVI} + g_2 \left(\frac{L}{2}\right)

Variables

Symbol	Description	Unit
$\text{Elev. PVC}$	Elevation of Point of Vertical Curvature	-
$\text{Elev. PVT}$	Elevation of Point of Vertical Tangency	-
$\text{Elev. PVI}$	Elevation of Point of Vertical Intersection	-
$g_1$	Initial grade in decimal format	-
$g_2$	Final grade in decimal format	-
$L$	Length of the curve	m or ft

Properties of a Parabolic Vertical Curve

Parabolic Geometry

The curve is typically symmetrical; the horizontal distance from PVC to PVI is $L/2$ , and from PVI to PVT is $L/2$ .
The vertical offset from a tangent to the parabola is proportional to the square of the horizontal distance from the point of tangency (PVC or PVT).
The midpoint of the curve ( $C$ ) lies exactly halfway vertically between the PVI and the midpoint of the chord connecting PVC and PVT.

Fundamental Equations

Rate of Change of Grade

Calculates the rate of change of grade.

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

Variables

Symbol	Description	Unit
$r$	Rate of change of grade	% per unit distance
$g_1$	Initial grade	%
$g_2$	Final grade	%
$L$	Length of curve	stations or meters

Elevation on the Curve

Calculates the elevation on the curve at any horizontal distance x from PVC.

Y_x = Y_{PVC} + g_1 x + \frac{1}{2} r x^2

Variables

Symbol	Description	Unit
$Y_x$	Elevation on the curve at distance x from PVC	m or ft
$Y_{PVC}$	Elevation of the PVC	m or ft
$x$	Horizontal distance from the PVC	m or ft
$g_1$	Initial grade	decimal or %/station
$r$	Rate of change of grade	-

Consistent Units for Elevation

In the elevation equation, ensure consistent units. If $g_1$ is in percent, $x$ must be in $100\text{-m}$ or $100\text{-ft}$ stations. If $g_1$ is a decimal (e.g., $0.03$ ), $x$ can be in meters or feet.

Location of the Highest or Lowest Point

Turning Point

The highest point (on a crest curve) or lowest point (on a sag curve) occurs where the tangent is completely horizontal (grade is zero). This point exists only if the grades $g_1$ and $g_2$ have opposite signs.

High or Low Point Location

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

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

Variables

Symbol	Description	Unit
$x_m$	Horizontal distance from PVC to the high/low point	m or ft
$g_1$	Initial grade	%
$g_2$	Final grade	%
$L$	Total length of curve	m or ft
$r$	Rate of change of grade	-

Vertical Offsets

Vertical Offset

Calculates the vertical offset from the tangent to the curve.

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

Variables

Symbol	Description	Unit
$y$	Vertical offset from the tangent to the curve	m or ft
$x$	Horizontal distance from PVC (or PVT)	m or ft
$A$	Algebraic difference in grades	%
$L$	Total length of curve	m or ft

Sight Distance on Vertical Curves

Curve Length vs. Sight Distance

The minimum length of a vertical curve ( $L$ ) is controlled by the required sight distance ( $S$ ), which is typically the Stopping Sight Distance (SSD). The formulas depend on whether the sight distance is greater than or less than the curve length.

Crest Vertical Curves

Crest Curve Length (S < L)

Minimum length of crest vertical curve when sight distance is less than curve length.

L = \frac{A S^2}{100 (\sqrt{2h_1} + \sqrt{2h_2})^2}

Variables

Symbol	Description	Unit
$L$	Minimum curve length	m
$S$	Sight distance	m
$A$	Algebraic difference in grades	%
$h_1$	Eye height of the driver (typically 1.08 m)	m
$h_2$	Height of the object (typically 0.60 m)	m

Crest Curve Length (S > L)

Minimum length of crest vertical curve when sight distance is greater than curve length.

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

Variables

Symbol	Description	Unit
$L$	Minimum curve length	m
$S$	Sight distance	m
$A$	Algebraic difference in grades	%
$h_1$	Eye height of the driver (typically 1.08 m)	m
$h_2$	Height of the object (typically 0.60 m)	m

Sag Vertical Curves

Headlight Sight Distance

For sag curves, sight distance is typically governed by the distance illuminated by the vehicle's headlights at night.

Sag Curve Length (S < L)

Minimum length of sag vertical curve when sight distance is less than curve length.

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

Variables

Symbol	Description	Unit
$L$	Minimum curve length	m
$S$	Sight distance	m
$A$	Algebraic difference in grades	%
$h$	Height of headlights (typically 0.60 m)	m
$\beta$	Upward angle of the headlight beam (typically 1 degree)	deg

Sag Curve Length (S > L)

Minimum length of sag vertical curve when sight distance is greater than curve length.

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

Variables

Symbol	Description	Unit
$L$	Minimum curve length	m
$S$	Sight distance	m
$A$	Algebraic difference in grades	%
$h$	Height of headlights (typically 0.60 m)	m
$\beta$	Upward angle of the headlight beam (typically 1 degree)	deg

Riding Comfort on Sag Curves

Centrifugal Acceleration

When a vehicle travels through a sag vertical curve, centrifugal force acts downwards, combining with the vehicle's weight and pressing the passengers into their seats. To maintain riding comfort, the length of the curve must be sufficient to limit the rate of change of centrifugal acceleration. The standard comfort criterion is to limit the vertical acceleration to $0.3 \text{ m/s}^2$ .

Curve Length for Comfort

Minimum length of sag vertical curve based on comfort standards.

L = \frac{A V^2}{395}

Variables

Symbol	Description	Unit
$L$	Length of sag curve for comfort	m
$A$	Algebraic difference in grades	%
$V$	Design speed	km/h

Unsymmetrical Vertical Curves

Unequal Tangents

An unsymmetrical vertical curve consists of two adjacent parabolic arcs that share a common point of tangency (PTC) located directly below or above the PVI. This is used when a symmetrical curve cannot fit due to physical constraints. The horizontal lengths from the PVI to the PVC ( $L_1$ ) and from the PVI to the PVT ( $L_2$ ) are unequal.

Unsymmetrical Curve Rates of Change

Rates of change of grade for the two arcs of an unsymmetrical curve.

r_1 = \frac{g_{PTC} - g_1}{L_1}

r_2 = \frac{g_2 - g_{PTC}}{L_2}

Variables

Symbol	Description	Unit
$r_1$	Rate of change of grade for the first arc	-
$r_2$	Rate of change of grade for the second arc	-
$g_1$	Initial grade	-
$g_2$	Final grade	-
$g_{PTC}$	Grade of the common tangent at the point directly above/below the PVI	-
$L_1$	Horizontal length from the PVI to the PVC	-
$L_2$	Horizontal length from the PVI to the PVT	-

Key Takeaways

Vertical curves are parabolic to ensure a constant rate of change of grade ( $r$ ).
The $K$ -value ( $K = L/A$ ) represents the horizontal distance required to effect a $1\%$ change in grade.
Crest curves transition over hills; sag curves transition through valleys.
The curve elevation equation is a quadratic function of the distance from the PVC: $Y_x = Y_{PVC} + g_1 x + (r x^2)/2$ .
The high or low point occurs where the slope is zero, located at distance $x_m = -g_1 / r$ from the PVC.
Sag curves must be long enough to provide adequate headlight sight distance and maintain riding comfort by limiting centrifugal acceleration.

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Vertical Curves - Theory & Concepts

Learning Objectives

Overview of Vertical Curves

Types of Vertical Curves

Crest and Sag Curves

Rate of Vertical Curvature (KKK-Value)

K-Value

K-Value Formula

Elements of a Vertical Curve

Key Points and Terminology

Stationing and Elevation Computations

Stationing Computations

Elevation Computations

Properties of a Parabolic Vertical Curve

Parabolic Geometry

Fundamental Equations

Rate of Change of Grade

Elevation on the Curve

Consistent Units for Elevation

Location of the Highest or Lowest Point

Turning Point

High or Low Point Location

Vertical Offsets

Vertical Offset

Sight Distance on Vertical Curves

Curve Length vs. Sight Distance

Crest Vertical Curves

Crest Curve Length (S < L)

Crest Curve Length (S > L)

Sag Vertical Curves

Headlight Sight Distance

Sag Curve Length (S < L)

Sag Curve Length (S > L)

Riding Comfort on Sag Curves

Centrifugal Acceleration

Curve Length for Comfort

Unsymmetrical Vertical Curves

Unequal Tangents

Unsymmetrical Curve Rates of Change

Rate of Vertical Curvature ( $K$ -Value)