A vertical curve calculator is used in road design and construction to determine the smooth transition between two road grades. It helps engineers design safe and comfortable roadways by ensuring gradual changes in elevation.
Examples of vertical curves include:
- Crest curve: Where two upward slopes meet, or an upward slope transitions to a downward slope.
- Sag curve: Where two downward slopes meet, or a downward slope transitions to an upward slope.
These calculators typically consider factors such as:
- Starting and ending grades
- Length of the curve
- Station and elevation of the Point of Vertical Intersection (PVI)
Vertical Curve Calculator
- PVC Station: 10+00
- PVC Elevation: 100.00 feet
- G₁: 2%
- G₂: -3%
- L: 300 feet
| Station | x (ft) | Tangent Elevation (ft) | Curve Elevation (ft) | E (ft) |
|---|---|---|---|---|
| 10+00 | 0 | 100.00 | 100.00 | 0.00 |
| 10+50 | 50 | 101.00 | 100.96 | -0.04 |
| 11+00 | 100 | 102.00 | 101.67 | -0.33 |
| 11+50 | 150 | 103.00 | 102.13 | -0.87 |
| 12+00 | 200 | 104.00 | 102.33 | -1.67 |
| 12+50 | 250 | 105.00 | 102.29 | -2.71 |
| 13+00 | 300 | 106.00 | 102.00 | -4.00 |
Vertical Curve Formula
The fundamental parabolic equation for a vertical curve is:
y = ax² + bx + c
Where:
- y: Elevation at any point on the curve
- x: Horizontal distance from the beginning of the curve
- a, b, c: Constants determined by the curve’s characteristics
For practical applications, road designers often use a simplified form:
E = (G₁ – G₂) * x² / (200 * L)
Where:
- E: Elevation difference between the curve and the tangent at any point
- G₁: Initial grade (%)
- G₂: Final grade (%)
- x: Distance from the beginning of the curve
- L: Total length of the curve
How to Calculate PVI Vertical Curve
Calculating the Point of Vertical Intersection (PVI) is crucial for vertical curve design. Follow these steps:
- Determine the initial and final grades (G₁ and G₂)
- Identify the stations and elevations of the Point of Vertical Curvature (PVC) and Point of Vertical Tangency (PVT)
- Calculate the horizontal distance between PVC and PVT
- Find the PVI station by adding half the curve length to the PVC station
- Calculate PVI elevation using the tangent grades: PVI Elevation = PVC Elevation + (G₁ * L/2)
Where L is the total curve length.
How do you calculate the K value for vertical curves?
The K value represents the horizontal distance required for a 1% change in grade.
To calculate K:
K = L / A
Where:
- L: Length of the vertical curve
- A: Algebraic difference between the initial and final grades (|G₁ – G₂|)
Example:
For a curve with L = 300 feet, G₁ = 2%, and G₂ = -3%:
A = |2 – (-3)| = 5%
K = 300 / 5 = 60
This K value of 60 means the curve provides 60 feet of horizontal distance for each 1% change in grade.
How to find the elevation along a vertical curve?
To find the elevation at any point along the vertical curve:
- Calculate the station of the point of interest
- Determine the distance (x) from the beginning of the curve
- Use the parabolic equation: Elevation = PVC Elevation + G₁ * x + E
- Where E is calculated using the formula from the “Vertical Curve Formula” section.
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