Calculation example – Elevation calculation and grade along a profile vertical curve

Determine the elevation and grade value of an unknown point K which lies along a profile vertical curve (convex in this example).

*Commonly in profile vertical curves, a parabola is applied as an approximation to circular curve, considering that for large radius values and target precision, these elements practically coincide.

Known values: Curvature radius H^~R (positive for concave, negative for convex vertical curve), grade in S₁ and grade out S₂ (positive for ascending , negative for descending slopes), elevation of start point A of vertical curve H_A 

Solution: Elevation of point K at a distance x from curve start point A y_K, grade value at point K S_K 

The vertical distance of point K from point A is calculated:

y_K = (S₁/100)*x + x²/(2*H)

The elevation of point K is calculated:

H_K = H_A + y_K

Grade S_K at point K calculation:

S_K = S₁ + (x/H)*100

Where:

S₁ = grade in value at start point A of vertical curve

x = horizontal distance of point K from point A

H = curvature radius at peak of vertical curve (where grade s = 0%)

H_A = elevation of start point A of vertical curve