Leveling: Differential Leveling, Elevations, and Percent Grades

NCEES FE Civil spec: Surveying, E. Leveling (e.g., differential, elevations, percent grades). FE Reference Handbook: No dedicated leveling formulas; Civil Engineering chapter, Transportation (grades on vertical curves).

What the FE Civil Spec Covers

Section E covers how elevations are carried from a benchmark to new points with a level and rod, and how elevation differences become grades. Expect a short set of level notes with one or more turning points, a question on the arithmetic check, curvature and refraction for long sights, and percent grade between two points. Grades then feed vertical curve problems in Transportation.

Key Concepts

Benchmark (BM): a point of known elevation. Turning point (TP): a temporary point used to move the level forward.
Backsight (BS): rod reading on a point of known elevation (a plus sight). Foresight (FS): rod reading on a point whose elevation you want (a minus sight).
Height of instrument (HI): elevation of the line of sight.
Curvature and refraction: over long sights the level line drops away from the curved earth; refraction bends the sight back down slightly. Balanced sight distances cancel both.
Percent grade: rise over horizontal run times 100, positive uphill in the direction of stationing.

Core Formulas

H I = Elev_{known} + B S Elev_{new} = H I - F S

Arithmetic check for a set of notes:

\sum B S - \sum F S = Elev_{end} - Elev_{start}

Combined curvature and refraction correction (sight length K in km or M in thousands of feet):

h_{cr} = 0.0675 K^{2} (m) h_{cr} = 0.0206 M^{2} (ft)

Percent grade between points 1 and 2:

g (%) = \frac{Elev _{2} - Elev _{1}}{horizontal distance} \times 100

Where It Is in the FE Reference Handbook

The handbook has no dedicated leveling formulas, so HI = Elev + BS and Elev = HI minus FS must be memorized. Grades are used in the Civil Engineering chapter, Transportation section, with the vertical curve formulas. Practice with a handbook search strategy so you do not waste time looking for a formula that is not there.

Worked Example

Run levels from BM 1 (elevation 245.32 ft) to point B through two turning points. Readings in order: BS 5.67 on BM 1, FS 3.21 on TP1, BS 4.88 on TP1, FS 7.45 on TP2, BS 2.15 on TP2, FS 6.02 on B. Then find the grade from B to point C, which is 260.0 ft away horizontally at elevation 247.86 ft.

Station	BS	HI	FS	Elevation
BM 1	5.67	250.99		245.32
TP1	4.88	252.66	3.21	247.78
TP2	2.15	247.36	7.45	245.21
B			6.02	241.34

Step 1, first setup. HI = 245.32 + 5.67 = 250.99; TP1 = 250.99 minus 3.21 = 247.78 ft.

Step 2, carry forward. HI = 247.78 + 4.88 = 252.66; TP2 = 252.66 minus 7.45 = 245.21. HI = 245.21 + 2.15 = 247.36; B = 247.36 minus 6.02 = 241.34 ft.

Step 3, arithmetic check.

\sum B S - \sum F S = 12.70 - 16.68 = - 3.98 = 241.34 - 245.32 ✓

Step 4, grade from B to C.

g = \frac{247.86 - 241.34}{260.0} \times 100 = \frac{6.52}{260.0} \times 100 = + 2.51%

Exam Strategy

Lay the readings out in a BS, HI, FS, Elevation table exactly as in the example. Even when the question only gives two or three readings, the table makes it obvious which reading is added and which is subtracted.

Some questions describe a reading on an overhead point, such as the underside of a bridge beam, with the rod held upside down. Treat that rod reading as negative: the point is above the line of sight, so its elevation is HI plus the reading.

Curvature and refraction only matter for long sights. At 500 ft the combined correction is 0.0206 times 0.5 squared, or about 0.005 ft, which is negligible for most FE questions; at 1 mile (5.28 thousand ft) it grows to about 0.57 ft. Apply it only when the question signals a long sight or unbalanced distances.

Common Mistakes

Subtracting a backsight or adding a foresight. BS is always added to get HI; FS is always subtracted.
Reusing the old HI after the level moves. Each setup has its own HI.
Dividing by slope distance instead of horizontal distance when computing grade.
Dropping the sign of the grade. A downhill grade in the direction of travel is negative.
Squaring the sight length in the wrong units in the curvature and refraction formula (km or thousands of feet).

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A benchmark has elevation 100.0 m. A leveling setup reads a backsight of 1.5 m on the benchmark and a foresight of 2.2 m on a new point. What is the new point's elevation? See the worked solution or practice the Surveying deck
From benchmark BM (elevation 120.00 m): BS 1.25 m, FS 2.10 m to TP1; then BS 1.60 m on TP1, FS 0.95 m to point X. The elevation of X is most nearly: See the worked solution or practice the Surveying deck
A road rises 2 m over a horizontal distance of 100 m. What is its grade? See the worked solution or practice the Surveying deck
A crest vertical curve 300 m long joins a grade of +2% to a grade of −4%. The external distance from the point of vertical intersection down to the curve is most nearly: See the worked solution or practice the Surveying deck

Related Calculators and Examples

Frequently Asked Questions

What is the arithmetic check in a level loop?

The sum of all backsights minus the sum of all foresights must equal the ending elevation minus the starting elevation. If it does not, there is an arithmetic error in the notes.

Why should backsight and foresight distances be balanced?

When the distances from the level to the backsight and foresight rods are about equal, errors from earth curvature, refraction and a slightly tilted line of sight are the same on both readings and cancel.

How do I convert percent grade to an angle?

Percent grade is rise over run times 100. The angle is the inverse tangent of the grade as a decimal, so a 5% grade is about 2.86 degrees. Road grades are almost always given in percent.

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