What is a Vertical Curve?
A vertical curve is a parabolic curve used in road and railway design to provide a smooth transition between two grades. There are two main types of vertical curves:
- Crest Curve: Forms a hill or high point where the roadway changes from an uphill grade to a downhill grade.
- Sag Curve: Forms a valley or low point where the roadway changes from a downhill grade to an uphill grade.
Key Vertical Curve Parameters
- PVI (Point of Vertical Intersection): The theoretical point where the two tangent grades would intersect if there was no curve.
- PVC (Point of Vertical Curve): The beginning point of the vertical curve.
- PVT (Point of Vertical Tangent): The ending point of the vertical curve.
- g₁: Initial grade (in percent).
- g₂: Final grade (in percent).
- L: Length of the vertical curve.
- K-Value: The horizontal distance required for a 1% change in grade (L/A, where A is the absolute difference between grades).
Vertical Curve Formulas
For a parabolic vertical curve, the elevation at any point along the curve can be calculated using:
E = E₁ + g₁x + (g₂ - g₁)x²/(2L)
Where:
- E = Elevation at distance x from PVC
- E₁ = Elevation at PVC
- g₁ = Initial grade (decimal)
- g₂ = Final grade (decimal)
- x = Distance from PVC
- L = Length of vertical curve
High/Low Point Calculation
For crest and sag curves, the high or low point can be calculated using:
x = -g₁L/(g₂ - g₁)
This point only exists on the curve if 0 ≤ x ≤ L.
K-Value and Design Standards
K-values are used to ensure adequate sight distance for drivers. Minimum K-values are specified in design standards based on:
- Crest Curves: Stopping sight distance
- Sag Curves: Headlight sight distance and driver comfort
Higher design speeds require larger K-values to ensure safety.
Applications in Engineering
Vertical curve calculations are essential for:
- Highway and road design
- Railway alignment design
- Drainage system planning
- Earthwork quantity calculations
- Construction stakeout and layout