Module 8: Vertical Alignment on Mohammad Movahedi

Vertical Curve Fundamentals

Mon, 04 May 2026 00:00:00 +0000

Vertical Curve Fundamentals

Vertical curves are used in highway design to provide a smooth transition between two intersecting grade tangents. Unlike horizontal curves, which are typically circular arcs, vertical curves are designed as parabolic arcs. A parabolic profile provides a constant rate of change of grade, which minimizes vertical acceleration and passenger discomfort while maintaining consistent steering control.

Curve Classifications

Vertical curves are classified based on the relative direction of the grade change:

Elevation on Vertical Curves

Mon, 04 May 2026 00:00:00 +0000

Elevation on Vertical Curves

Determining elevations along a parabolic vertical curve is a frequent task in road design and a heavily tested topic on the PE Civil Transportation exam. Geometric calculations utilize the properties of a vertical parabola, where the vertical offset from a tangent grade is proportional to the square of the horizontal distance from the tangent point.

The Parabolic Curve Equations

To calculate the elevation of any point on a vertical curve, we define a coordinate system with its origin at the PVC:

Stopping Sight Distance

Mon, 04 May 2026 00:00:00 +0000

Stopping Sight Distance

Stopping Sight Distance (SSD) is the minimum distance required for a driver traveling at a given speed to see a hazard on the roadway, react, and bring the vehicle to a complete stop before colliding with the object. SSD is a fundamental control for horizontal alignment, vertical alignment, and intersection design.

Components of Stopping Sight Distance

SSD is composed of two distinct components:

Perception-Reaction Distance: The distance traveled from the instant the driver sees a hazard to the instant they apply the brakes.
Braking Distance: The distance traveled while the vehicle is decelerating to a stop under braking.

|<-------------------------- Stopping Sight Distance (SSD) -------------------------->|
| |
|<--- Perception-Reaction Distance --->|<---------------- Braking Distance ---------->|
| (Vehicle travels at design speed V) | (Vehicle decelerates under braking) |
*--------------------------------------*----------------------------------------------*
Driver sees hazard Brakes applied Stop

Governing Equations

1. USCS (English) Units

The general formula for SSD in USCS units is:

Crest Vertical Curves

Mon, 04 May 2026 00:00:00 +0000

Crest Vertical Curves

Crest vertical curves are designed to connect an uphill grade to a downhill grade (or a steeper uphill grade to a flatter uphill grade, etc.), where the algebraic difference in grade is negative ($g_1 > g_2$). The primary design control for a crest vertical curve is ensuring adequate sight distance—specifically Stopping Sight Distance (SSD)—so that a driver can see a hazard over the crest of the hill.

Design Parameters for Sight Distance

The relationship between the length of the curve, the grades, and the sight distance is based on the height of the driver’s eye and the height of the object. AASHTO Green Book standard values for SSD are:

Sag Vertical Curves

Mon, 04 May 2026 00:00:00 +0000

Sag Vertical Curves

Sag vertical curves connect a downgrade to an upgrade (or a steep downgrade to a flatter downgrade, etc.), where the algebraic difference in grade is positive ($g_1 < g_2$). Unlike crest curves, where sight distance is restricted by the roadway surface itself, sight distance on sag curves during the day is generally unrestricted.

Consequently, the design of sag vertical curves is governed by four criteria:

Headlight Sight Distance (the primary safety control for nighttime driving)
Passenger Comfort (limiting vertical acceleration)
Drainage Control (preventing water accumulation in the low point)
General Appearance (preventing an abrupt or sagging look)

Headlight Sight Distance Criteria

During nighttime, the distance the driver can see is limited by the reach and angle of the vehicle’s headlights. AASHTO design standards assume:

Passing Sight Distance on Vertical Curves

Mon, 04 May 2026 00:00:00 +0000

Passing Sight Distance on Vertical Curves

Passing Sight Distance (PSD) is the minimum distance required for a driver on a two-lane highway to safely complete a passing maneuver around a slower vehicle without colliding with an opposing vehicle. Providing PSD on vertical curves is much more demanding than providing Stopping Sight Distance (SSD) due to the greater sight distance required.

Design Parameters for Passing Sight Distance

Unlike SSD, where the object on the road is small ($h_2 = 2.00\text{ ft}$), the object in a passing scenario is an oncoming vehicle. AASHTO Green Book standards assume:

Vertical Clearance

Mon, 04 May 2026 00:00:00 +0000

Vertical Clearance

Vertical clearance is the minimum vertical distance between the roadway surface and an overhead structure, such as an overpass bridge girder, sign truss, or pedestrian bridge. Ensuring adequate clearance is critical for preventing overhead collisions from commercial trucks, and ensuring that overhead structures do not block a driver’s line of sight (underpass sight distance).

Design Standards for Vertical Clearance

According to AASHTO and federal design standards:

Minimum Clearance: The standard minimum vertical clearance for highways and freeways is $16.0\text{ ft}$ ($4.9\text{ m}$). On local or collector roads, a minimum clearance of $14.0\text{ ft}$ to $15.0\text{ ft}$ may be permitted.
Overlay Allowance: It is standard practice to design for an additional $6\text{ inches}$ ($0.5\text{ ft}$) of clearance (making the design clearance $16.5\text{ ft}$) to accommodate future asphalt overlays without reducing the clearance below the legal limit.

Profile and Clearance Geometry

Clearance is evaluated by comparing the profile elevation of the roadway with the elevation of the bottom of the overhead structure’s girders. Because the roadway is often on a vertical curve (most critically a sag curve), the clearance varies across the width of the bridge.