Vertical Curve Length Calculator

Inputs and Outputs of the Vertical Curve Length Calculator

Inputs

• Approach Grade g₁ (%): Upward grades are positive, downward grades negative (e.g., +2.0, -3.5).
• Departure Grade g₂ (%): Grade after the point of vertical intersection, using the same sign convention.
• Design Sight Distance S (ft): Typically stopping sight distance or passing sight distance, depending on the design objective.
• Curve Type: “Crest” or “Sag” vertical curve.

Outputs

• Algebraic grade difference A (%): A = |g₂ - g₁|.
• Minimum curve length L (ft): Satisfies the sight-distance requirement for the chosen curve type.
• K value (ft/%): K = L / A, useful for comparing to design-speed guidelines.
• Sometimes an indicator of whether L ≥ S or L < S, which determines which AASHTO equation controls.

Formula: Vertical Curve Design Equations (AASHTO-Based)

The main geometric parameter is the algebraic difference between the approach and departure grades, expressed as a percent:

A = |g₂ - g₁|

AASHTO provides slightly different forms of the sight-distance equations depending on whether the required sight distance S is shorter or longer than the vertical curve length L. The calculator checks both regimes and selects the appropriate formula automatically.

A representative crest vertical curve case with S ≤ L can be written using MathML as:

where:

• S = sight distance (ft)
• h₁ = driver eye height (ft)
• h₂ = object or headlight target height (ft)
• A = algebraic grade difference (%), using absolute value

For crest vertical curves, AASHTO typically assumes h₁ = 3.5 ft and h₂ = 2.0 ft for stopping sight distance. For sag vertical curves at night, the “object” is the illuminated point on the pavement from headlight beams, often represented with a headlight height of about 2.0 ft and a standard upward beam angle. The calculator uses standard values appropriate for highway vertical curve design.

When the required sight distance is longer than the curve length (S > L), alternative AASHTO equations apply. In that regime the relationship between L, S, and the height terms changes, and forms such as L = 2S - 200(h₁ + h₂)/A are used. The tool evaluates the correct form based on the inputs so that you do not need to select the case manually.

Interpreting the K Value for Crest and Sag Vertical Curves

Once the minimum curve length L is computed, the K value is defined as:

K = L / A

K has units of ft/% and is widely used in design manuals because it provides a simple way to compare different combinations of grades and lengths. Larger K values correspond to “flatter” vertical curves that generally give better sight distance and ride comfort for a given design speed.

AASHTO and many highway agencies publish minimum recommended K values for a range of design speeds. The table below summarizes typical values for stopping-sight-distance-based design of two-lane and multilane rural highways; always check the latest edition of the AASHTO Green Book and your agency’s standards.

Compare the computed K from this calculator to guideline values:

• If K (computed) < K (recommended), the crest or sag vertical curve is too sharp for the chosen design speed; consider lengthening the curve or reducing the grade break.
• If K (computed) ≥ K (recommended), the highway vertical curve meets or exceeds the minimum for sight distance.

Worked Example: Crest Vertical Curve Length

Assume a two-lane rural highway crest vertical curve with:

• Approach grade g₁ = +2%
• Departure grade g₂ = -2%
• Stopping sight distance S = 400 ft

Step 1 – Compute algebraic grade difference:

A = |g₂ - g₁| = |-2 - (+2)| = 4%

Step 2 – Use the appropriate AASHTO crest-curve equation for S ≤ L (the calculator checks the correct case). Using standard eye and object heights, the resulting minimum vertical curve length for this configuration is approximately:

L ≈ 171 ft

Step 3 – Compute K:

K = L / A ≈ 171 / 4 ≈ 43 ft/%

Step 4 – Compare to recommended K. For example, at a design speed of 50 mph, the table above shows K_{min, crest} ≈ 64. Since 43 < 64, this crest vertical curve would not meet stopping sight distance requirements for 50 mph and should be lengthened or redesigned.

How to Use the Vertical Curve Length Calculator

1. Enter the approach grade g₁ and departure grade g₂ as percentages, with upgrades positive and downgrades negative.
2. Enter the design sight distance S in feet, typically based on the chosen design speed and whether you are using stopping or passing sight distance.
3. Select the curve type as “Crest” or “Sag” depending on your roadway profile.
4. Run the calculation to obtain L, A, and the resulting K value.
5. Compare the computed K to the recommended K for your design speed from the AASHTO Green Book or your agency’s design manual.

Assumptions, Limitations, and Good Practice

This calculator focuses on sight-distance-based minimum vertical curve length and K value for crest and sag curves. It does not substitute for a full geometric design or independent engineering judgment. Key assumptions and limitations include:

• Source: Equations are based on standard formulations consistent with the AASHTO Green Book; verify against the current edition and any local modifications.
• Units: Grades are in percent (%), sight distance S and length L are in feet (ft).
• Heights: Typical driver eye height of about 3.5 ft and object/headlight heights of about 2.0 ft are assumed for stopping sight distance calculations.
• Range of validity: Intended for typical highway and street design speeds and grade differences. Very small A values (near-zero grade breaks) can produce very large K values and may not be critical for sight distance.
• Curve type selection: For crest vertical curves, sight distance is governed by line of sight over the crest. For sag vertical curves, nighttime headlight sight distance typically governs.
• Other criteria: Comfort (rate of change of acceleration), drainage, aesthetics, and agency-specific constraints on minimum and maximum grades are not checked here and must be evaluated separately.
• Design vs. checking: Use this tool as a quick check or for preliminary highway vertical curve design. Final designs should be checked in detail using agency-approved software and procedures.

By pairing the curve length and K value from this calculator with design-speed-based guidelines and local standards, you can quickly screen alternative profiles and refine your vertical curve design before more detailed analysis.