Solution: Vertical Curves Set 1
Problem (1)
PVI station and location are 74+26.50 and 772.30 ft. What are the endpoints' stations and elevations of a 750.00 ft long curve with incoming and outgoing grades of -4.00% and +2.00%?
For problems (2) through (4), compute the following items
(a) Curve table at full station interval. Include curve point grades.
(b) Stations and elevations of the curve's highest and lowest points?
Problem (2)
PVI station and location are 56+50.00 and 1325.00 ft. Incoming and outgoing grades are +5.0% and -3.5%. A 500.00 ft long vertical curve will be used.
(a) Curve Table
| Station | di, sta | Elevi, ft | gi, % |
| 59+00 EVC | 5.0000 | 1316.250 | -3.5 |
| 58+00 | 4.0000 | 1318.900 | -1.8 |
| 57+00 | 3.0000 | 1319.850 | -0.1 |
| 56+00 | 2.0000 | 1319.100 | +1.6 |
| 55+00 | 1.0000 | 1316.650 | +3.3 |
| 54+00 BVC | 0.0000 | 1312.500 | +5.0 |
Math checks in red.
(b) Highest and lowest points
Because this is a crest curve, the lowest point will occur at either the BVC or EVC.
In this case lowest point is at BVC: 54+00.00 @ 1312.50 ft
Highest:
Problem (3)
A 425.00 ft long curve ends at station and elevation 88+00.00 and 1220.00. Its initial and final grades are -4.00% and +2.75%.
Only the answers are given, computations left to the reader.
(a) Curve table
| Station | di, sta | Elevi, ft | gi, % |
| 92+25 EVC | 4.2500 | 1217.344 | +2.75 |
| 92+00 | 4.0000 | 1216.706 | +2.35 |
| 91+00 | 3.0000 | 1216.147 | +0.76 |
| 90+00 | 2.0000 | 1215.176 | -0.82 |
| 89+00 | 1.0000 | 1216.794 | -2.41 |
| 88+00 BVC | 0.0000 | 1220.000 | -4.00 |
Math checks in red.
(b) Highest and lowest points
Sag curve so either BVC or EVC is highest point
Highest: BVC, 88+00.00 @ 1220.000 ft
Lowest: 90+51.852 @ 1214.963 ft
Problem (4)
A curve begins at station 32+00.00 and elevation 912.00 ft; it ends at station 38+00.00 and elevation 915.00 ft. The PVI elevation is 918.00 ft.
Only the answers are given, computations left to the reader.
(a) Curve table
| Station | di, sta | Elevi, ft | gi, % |
| 38+00 EVC | 6.0000 | 915.000 | -1.00 |
| 37+00 | 5.0000 | 915.750 | -0.50 |
| 36+00 | 4.0000 | 916.000 | 0.00 |
| 35+00 | 3.0000 | 915.750 | +0.50 |
| 34+00 | 2.0000 | 915.000 | +1.00 |
| 33+00 | 1.0000 | 913.750 | +1.50 |
| 32+00 BVC | 0.0000 | 912.000 | +2.00 |
Math checks in red.
(b) Highest and lowest points
This is crest curve so the lowest point will be at BVC or EVC.
Lowest point: BVC, 32+00 @ 912.000 ft
Highest point occurs at grade = 0.00%: 36+00 @ 916.000 ft
Problem (5)
A 400.00 ft vertical curve begins at station 29+00.00 and elevation 954.00 ft. Its initial and final grades are -1.00% and -2.25%.
Part (a)
What are the station and elevation of the curve's midpoint?
The midpoint is not the PVI. It has the same station as the PVI, but is located on the curve.
Alternate solution
Part (b)
What are the stations and elevations of the curve's highest and lowest points?
Because both grades are negative, the highest point is the BVC, lowest is EVC.
Highest: 29+00.00 @ 954.00
Lowest:
