# Sag Vertical Curves

1. A 1400-ft–long sag vertical curve (equal tangent) has a PVC at station 115 + 00 and elevation 750 ft. The initial grade is −3.5% and the final grade is +6.5%. Determine the elevation and stationing of the low point, PVI, and PVT.

2. An equal-tangent sag vertical curve is designed with the PVC at station 109 + 00 and elevation 950 ft, the PVI at station 110 + 77 and elevation 947.34 ft, and the low point at station 110 + 50. Determine the design speed of the curve.

3. An equal-tangent vertical curve connects a −2% and a +3% grade. The low point of the curve is at elevation 297.88 ft. If the PVI is at elevation 295 ft, what is the design speed of the curve?

3. An equal-tangent sag vertical curve is designed for 45 mi/h. The low point is 237 ft from the PVC at station 112 + 37 and the final offset at the PVT is 19.355 ft. If the PVC is at station 110 + 00, what is the elevation difference between the PVT and a point on the curve at station 111 + 00?

4. An equal-tangent vertical curve connects an initial grade of 3% and a final grade of +1% and is designed for 60 mi/h. The PVI is at station 250+50 and elevation 732 ft. What is the station and elevation of the lowest point on the curve?

5. An overpass is being built over the PVI of an existing equal-tangent sag curve. The sag curve has a 70-mi/h design speed and G1 = −5%, G2 = +3%. Determine the minimum necessary clearance height of the overpass and the resultant elevation of the bottom of the overpass over the PVI. (Ignore the cross-sectional width of the overpass.)

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