11.3 Line and Grade from Plans and Profiles

11.3 Line and Grade from Plans and Profiles

Quick Answer: Line is the horizontal location (station and offset) on the plan. Grade is the proposed elevation on the profile. Rate of grade = Δelev / Δstation. Worked tangent: station 10+00 at 102.40 ft and 14+50 at 98.20 ft → 450 ft of run, −4.20 ft of fall, grade −0.933%, elevation at 12+00 = 100.53 ft.

The Domain III professional activity is determine line and grade (plans and profiles). Item F already gave you vertical-curve math; this section is the sheet-reading half: proposed profile versus existing ground, interpolating a tangent grade, vertical clearance, and utility inverts. Setting hubs and marking stakes is Chapter 14–15 — only a preview here.

Plan versus Profile

A civil plan-profile sheet is two coordinated drawings of the same alignment.

Station 10+00 means 1,000 ft along the alignment from the zero station, not 10 ft and not 10.00 stations until you convert. 14+50 is 1,450 ft. Difference = 450 ft.

Offset is perpendicular distance from the alignment: 12 ft Lt is 12 ft left of centerline looking up-station. Grade for a curb, ditch, or pipe may be a fixed vertical offset from centerline profile (a typical section). If the stem gives only centerline profile, do not invent a crown unless the typical section is in the problem.

Profile Grade versus Existing Ground

Two lines on a profile are easy to swap under time pressure:

• Proposed profile grade — the design surface (finished pavement, channel invert, pipe invert — read the label).
• Existing ground — the topo surface from the survey.

At one station:

• Cut when existing ground is above proposed grade: cut = existing − proposed.
• Fill when existing ground is below proposed grade: fill = proposed − existing.

If existing at 12+00 is 101.10 ft and proposed is 100.53 ft, you have 0.57 ft of cut, not fill. Calling every positive difference “cut” without checking which line is higher is the trap.

The proposed line may be a vertical curve from §11.1–11.2 or a tangent grade. If the profile dimension labels two points on a straight grade line, use the tangent interpolation below. If it labels PVC/PVI/PVT, use the parabola — linear interpolation between PVC and PVT is wrong.

Vertical Clearance and Utility Inverts

Vertical clearance is the difference in elevation between two objects at (approximately) the same station: proposed soffit versus existing ground, proposed pavement versus a utility, one pipe versus another.

An invert is the inside bottom of a pipe or channel. Profiles often show invert of existing storm and proposed finished grade at a crossing. Subtract, and name which is higher.

Example: proposed grade 88.40 ft, existing storm invert 86.15 ft. Cover from invert to proposed grade = 88.40 − 86.15 = 2.25 ft. Whether 2.25 ft meets a local cover specification is a project rule; CES asks you to read both elevations and subtract. Do not add them, and do not treat the invert as a finished-grade stake unless the stem says the design surface is the invert.

Proposed versus existing structure corners use the same subtraction on the profile or on a cross-section (Chapter 12). The interference question is “do these two elevations leave a gap?” not “what is K?”

Rate of Grade Between Two Profile Points

On a tangent (straight) grade between two profile points:

g = (elev₂ − elev₁) / (sta₂ − sta₁)

Convert stations to feet (or both to stations). g decimal (ft/ft) × 100 = percent.

Worked: Point A: 10+00, elev 102.40 ft. Point B: 14+50, elev 98.20 ft.

Δstation = 1,450 − 1,000 = 450 ft
Δelev = 98.20 − 102.40 = −4.20 ft
g = −4.20 / 450 = −0.009333 ft/ft = −0.933% (falling).

Keep the exact ratio −4.20/450 for interpolation. Rounding to −0.93% too early, then multiplying, can shift the hundredths. −0.933% is the clean percent statement; the elevation arithmetic below uses the fraction.

Traps on this pair:

• Using 400 ft (10+00 to 14+00, dropping the +50) gives −1.05%.
• Using 1,000 ft as if 10+00 to 14+50 were 10 stations of something else.
• Dropping the sign and calling it a +0.933% climb.

Interpolating a Grade Between Stations

elev(x) = elev₁ + g × (x − sta₁)
or, without forming g first:
elev(x) = elev₁ + (elev₂ − elev₁) × (Δx / total run).

Elevation at 12+00 (200 ft past 10+00):

drop = 4.20 × (200 / 450) = 840 / 450 = 1.8667 ft
elev = 102.40 − 1.8667 = 100.533 ft → 100.53 ft

Check from the other end: 14+50 is 250 ft ahead of 12+00.
rise from B = 4.20 × (250 / 450) = 2.333 ft.
98.20 + 2.33 = 100.53 ft.

Do not average 102.40 and 98.20. That average is 100.30 ft, and it belongs at the midpoint station 12+25, not at 12+00. That is the interpolation trap the exam likes.

Other even stations on the same tangent (for reading a profile between ticks):

11+00: 102.40 − 4.20 × (100/450) = 102.40 − 0.933 = 101.47 ft.
13+00: 102.40 − 4.20 × (300/450) = 102.40 − 2.80 = 99.60 ft exactly.

If the profile between 10+00 and 14+50 were a vertical curve instead of a tangent, you would need PVC elevation, g1, A, and L — §11.1–11.2. Two labeled elevations alone define a straight grade, not a parabola.

Construction Staking Preview (Only)

Once line and grade are known, a field crew can occupy 12+00, 0.0 ft offset and set a mark at elevation 100.53 ft. Compared with existing ground 101.10 ft, the stake face would read a 0.57 ft cut to proposed grade.

That is the information on the stake. How the lath is marked (cut/fill, station, offset codes), where hinge and catch points sit on a cross-section, and how you occupy a horizontal or vertical curve in the field are Chapters 14 and 15. On this section, stop at: read station and offset from the plan, read or compute elevation from the profile, compare to existing or to a utility if asked.

Vertical-curve staking uses the intermediate elevations from §11.2 the same way: the stake at 22+00 on the worked crest is a grade of 522.25 ft, not the PVI’s 524.00 ft. Mixing PVI tangent elevation into a blue-top is a field bust with the same numbers as the §11.2 exam trap.

Exam Traps

1. Station arithmetic — 14+50 − 10+00 is 450 ft, not 4.50 ft and not 400 ft.
2. Averaging endpoint elevations for a non-midpoint station — 100.30 ft is 12+25, not 12+00.
3. Existing and proposed swapped — cut versus fill flips.
4. Invert added to grade instead of subtracted for clearance.
5. Parabola treated as a straight grade between PVC and PVT, or a straight grade treated as a parabola without g1, A, and L.