15.2 Staking Vertical Alignment and Grades

15.2 Staking Vertical Alignment and Grades

Quick Answer: Fill the vertical curve at even stations with elev = elev_PVC + g1 x + (A / (2L)) x². Worked crest: g1 = +1.60%, g2 = −2.40%, L = 500 ft, PVC 31+50 at 412.80 ft. High point 414.40 ft at 33+50. PVT 410.80 ft at 36+50. A blue top is a hub whose top is the design elevation. On a tangent, drop the parabola and use elev = elev0 + g Δx (37+50 = 408.40 ft).

Domain V item C includes setting points on the vertical alignment. Chapter 11 derived the parabola and the high point. This section is field grades: what you write on the lath, what a blue top is, how you fill even stations, and how you interpolate on a tangent that is not on the curve. Independent OpenExamPrep layout teaching uses the Chapter 11 formula with new numbers so the stake list is not a copy of the 520-ft crest.

Blue tops versus grade stakes

A blue top is a hub driven so the top of the hub is at the design elevation of the surface you are controlling (subgrade, aggregate base, or finished pavement — read the typical section). The top is marked with blue keel. The grader or paver works to that top. If the controlled surface is aggregate base 0.50 ft below finished pavement, the blue top is 0.50 ft below finished grade, not at pavement.

A grade stake (lath) is not at grade. It stands beside the point and tells cut or fill to grade from a reference mark on the stake, plus station and offset. Chapter 14 covered marking codes. Here the number you compute is the proposed elevation, then cut = existing − proposed when existing is higher, fill = proposed − existing when proposed is higher.

Do not mix roles. Painting existing ground blue is not a blue top. A hub driven to an arbitrary convenient height with "C 1.20" on a lath is a grade stake, not a blue top. CES will name hubs, blue tops, or cuts; compute the elevation first, then the mark.

The stake formula (Chapter 11 geometry, field use)

Equal-tangent parabolic vertical curve. A = g2 − g1. Measure x horizontally from the PVC (point of vertical curvature) to the stake station.

elev = elev_PVC + g1 x + (A / (2L)) x²

Use one unit system: decimal grades (ft/ft) with x and L in feet, or percent grades with x and L in stations. Mixing percent A into a decimal A/(2L) with L in feet makes offsets 100 times too large — a full-height bust on every hub.

PVI elevation is not on the curve. elev_PVI = elev_PVC + g1 (L/2) sits on the incoming tangent. The curve at that station is the PVI elevation plus E = A L / 8. Blue-topping 416.80 when the curve is 414.30 builds 2.50 ft of extra fill.

Worked stake list: new crest, 50-ft stations

Given: g1 = +1.60% = +0.016 ft/ft, g2 = −2.40% = −0.024 ft/ft, L = 500 ft, PVC station 31+50.00, PVC elevation 412.80 ft.

A = −0.024 − 0.016 = −0.040 ft/ft (−4.0%). A < 0 → crest, offsets below the tangents.

PVI at x = 250 ft, station 34+00: elev_PVI = 412.80 + 0.016 × 250 = 416.80 ft

PVT at x = 500 ft, station 36+50: elev_PVT = 416.80 + (−0.024) × 250 = 410.80 ft

Parabola check at x = L: 412.80 + 0.016 × 500 + (−0.040 / 1,000) × 500² = 412.80 + 8.00 − 10.00 = 410.80 ft. Matches.

A / (2L) = −0.040 / 1,000 = −0.00004

elev = 412.80 + 0.016 x − 0.00004 x²

High point: x_ext = −g1 L / A = −(0.016)(500) / (−0.040) = 200 ft, station 33+50, elevation 414.40 ft. That is 50 ft before the PVI. |g1| = 1.60% is flatter than |g2| = 2.40%, so the summit sits before the PVI, toward the PVC. Reporting the PVI as the blue-top high point is the Chapter 11 trap in the field.

Midpoint offset E = A L / 8 = (−0.040)(500) / 8 = −2.50 ft. Curve at 34+00 = 416.80 − 2.50 = 414.30 ft.

Fill every 50-ft station from PVC to PVT. Do not interpolate linearly from 412.80 to 410.80 — that is a chord, not the parabola.

Worked row at 32+50 (x = 100): y = (−4.0)(100)² / (200 × 500) = −40,000 / 100,000 = −0.40 ft; elev = 412.80 + 1.60 − 0.40 = 414.00 ft.

If existing ground at 34+00 CL is 413.10 ft and proposed is 414.30 ft, the lath reads 1.20 ft fill, not cut. Swapping existing and proposed flips the earthwork sign — the usual stake-face miss.

Interpolating grade on a tangent

Before the PVC and after the PVT the profile is a straight grade. Skip A and x².

elev = elev_known + g × Δx with g in ft/ft and Δx in feet.

Incoming g1 = +1.60% behind the PVC:

• 31+20 (30 ft before PVC): 412.80 − 0.016 × 30 = 412.32 ft
• 30+50 (100 ft before PVC): 412.80 − 1.60 = 411.20 ft

Outgoing g2 = −2.40% ahead of the PVT:

• 37+00 (50 ft past PVT): 410.80 − 1.20 = 409.60 ft
• 37+25 (75 ft past PVT): 410.80 − 0.024 × 75 = 410.80 − 1.80 = 409.00 ft
• 37+50 (100 ft past PVT): 410.80 − 2.40 = 408.40 ft

Averaging PVC and PVT (411.80 ft) is the chord midpoint, not a station on the parabola and not a tangent grade. Using 411.80 as a 34+00 blue top is 2.50 ft below the real curve (414.30) and 5.00 ft below the PVI tangent — three different wrong numbers from one bad average.

If a stem labels PVC/PVI/PVT, use the parabola between PVC and PVT. If it labels two elevations on a straight grade line with no vertical curve, use the tangent proportion from Chapter 11.3.

Field notes column order

A useful stake list is station | offset | x | curve (or tangent) elev | existing | cut/fill | mark. Compute all even stations before you occupy the first hub. Recheck the PVT against the outgoing tangent. Then occupy, set blue tops or mark lath, and move. Recalculating A at every rod shot under Prometric time pressure is how mixed-layout items get abandoned.

Exam traps

1. Blue-topping the PVI tangent (416.80) instead of the curve (414.30 at 34+00).
2. Linear interpolation PVC to PVT.
3. Percent A stuffed into a decimal A/(2L).
4. High point assumed at the PVI (here it is 33+50, not 34+00).
5. Using the parabola off the curve on the incoming or outgoing tangent.
6. Cut/fill sign from swapping existing and proposed.

Tab elev = elev_PVC + g1 x + (A/(2L)) x² and x_ext = −g1 L / A in the bound references. The field job is filling the table and marking the stake.