15.1 Staking Horizontal Alignment and Curves

15.1 Staking Horizontal Alignment and Curves

Quick Answer: Do not occupy the point of intersection (PI) to set curve hubs. Set the point of curvature (PC) and point of tangency (PT) on the tangents, then occupy the PC (or a control point) to set a point on curve (POC). Deflection from the forward tangent is δ = ℓ / (2R) (radians) and the chord is c = 2 R sin(δ). Worked: PC 18+40.00, R = 750 ft, Δ = 28°00′ → L = 366.52 ft, T = 187.00 ft, PT 22+06.52. The POC at 20+00 has ℓ = 160.00 ft, δ = 6.112° (6°06′42″), c = 159.70 ft.

Domain V item C of the 2022 Civil Engineering Surveying (CES) test plan from the California Board for Professional Engineers, Land Surveyors, and Geologists (BPELSG) is locating and setting points along an alignment, including horizontal curves. Chapter 10 computed radius, tangent, length, and deflection as office geometry. This section is field setting: where the tripod stands, what you backsight, what distance you measure, and how you offset from centerline (CL) to edge of pavement (EOP) or face of curb. Independent OpenExamPrep teaching here is the stakeout procedure, not a second copy of Chapter 10's element sheet.

A CES stem that says set, stake, hub, tack, occupy, or offset to curb is Domain V even when the arithmetic looks like Domain III.E. Stationing is still chainage along the finished centerline (100 ft = 1 station). You do not station through the PI.

Why the PI is usually not occupied

The PI is where the back tangent and forward tangent meet. It is not on the arc and it is not a centerline hub between PC and PT. Construction almost never leaves a clear, safe, on-grade instrument point there. The PI may sit in a cut slope, a fill prism, a building, a canal, live traffic, or across a fence you cannot enter. You may use the PI as a backsight or as a computed coordinate. You set the PC and PT.

Occupying the PI to turn the intersection angle Δ is a reconnaissance or tangent-check move. It is not how you set 50-ft POCs on the arc. Deflection layout for points on the curve starts from the PC (or the PT, working backward), or from control coordinates.

Setting the PC and the PT

PC station = PI station − T

PT station = PC station + L

T = R tan(Δ/2) and L = R Δ π / 180. Keep the calculator in degree mode for tan(Δ/2). Convert Δ to radians only inside L = R Δ_rad. California highway and street work uses the arc definition D = 5729.58 / R; Chapter 10 covered why railroad chord-definition D is a different constant.

Field methods:

1. Measure along the tangent. From a set PI (or from a known station on the back tangent), measure T to the PC. From the PI along the forward tangent, measure T to the PT. Do not measure T along the curve to reach the PT.
2. Stake by coordinates. Compute PC and PT northing/easting from the alignment and occupy nearby project control with a total station.

Worked elements for the field set (new numbers, not the Chapter 10 R = 800 ft example):

• R = 750.00 ft, Δ = 28°00′, Δ/2 = 14°00′
• T = 750 tan 14° = 187.00 ft (186.996 ft)
• L = 750 × 28 × π / 180 = 366.52 ft
• Arc D = 5729.58 / 750 = 7.639°
• Long chord PC to PT: LC = 2 × 750 × sin 14° = 362.88 ft

PC given at 18+40.00:

• PI station = 18+40.00 + 1+87.00 = 20+27.00
• PT station = 18+40.00 + 3+66.52 = 22+06.52

Checks: PI − PC = T. PT − PC = L. PT − PI = L − T = 179.52 ft, which is not T. Writing PT = PI + T as a station is the same bust as in Chapter 10.2, now with a hub in the wrong place: that path leaves the PI along the forward tangent and misses the PT.

Hub procedure: drive a hub at the computed point, set a tack for the exact CL, and place a guard stake (lath) about 1 ft off the hub with the station facing the hub. The instrument occupies the tack, not the lath.

Setting a POC by deflection

Occupy the PC. Backsight the PI or another point on the back tangent. Transit to the forward tangent. Turn δ toward the inside of the curve. Measure c from the PC to the rod.

δ = ℓ / (2R) radians

δ = (ℓ / L) × (Δ / 2) (same angular unit)

δ° = ℓ D / 200 with arc-definition D and ℓ in feet

c = 2 R sin(δ)

δ is half the central angle to the POC. Using δ = ℓ / R sights twice the deflection and throws the rod off the curve.

Worked POC at station 20+00

ℓ = 20+00 − 18+40 = 160.00 ft (206.52 ft of arc still remain to the PT).

• δ = 160 / (2 × 750) = 0.106667 rad = 6.112° = 6°06′42″
• Check: (160 / 366.52) × 14° = 6.112°
• Check: 160 × 7.639 / 200 = 6.112°
• c = 2 × 750 × sin(6.112°) = 159.70 ft

The chord is 0.30 ft shorter than the 160-ft arc. Taping 160.00 ft on the chord overshoots the curve. Construction usually wants even stations, not even arcs from the PC. The first even station after 18+40.00 is 19+00 (ℓ = 60.00 ft).

From 19+00 to 20+00 the incremental deflection is D/2 = 3.820° per 100 ft of arc. That increment is constant in arc, not in chord. The last 6.52 ft to the PT add only 0.249°.

Offsets to EOP and face of curb

Typical-section offsets on a circular curve are radial: perpendicular to the local tangent, along the radius. Constant width w produces a concentric curve of radius R − w (inside) or R + w (outside).

Worked typical: 12.00 ft CL to EOP, plus 2.00 ft gutter to face of curb (14.00 ft CL to face). Curve to the right. Right-side EOP is inside (radius 738.00 ft). Left-side EOP is outside (radius 762.00 ft). Do not apply the 12.00 ft offset along the chord from the PC, along the back tangent, or along stationing as if the curve were a straight. Apply it at the POC along the radius.

A curb return or ramp with changing width is not concentric; then you compute the offset from the typical section at that station (Chapter 14 hinge/catch) and still swing it perpendicular to the local CL tangent.

Coordinate layout from control

Place the PC at N 5,000.00, E 2,000.00, forward tangent due north, curve right (center east of the PC). The 20+00 CL POC is the same point as the deflection shot:

• CL: N 5,158.79, E 2,017.00 (159.70 ft at 6.112° east of north)
• Right EOP 12.00 ft inside: N 5,156.25, E 2,028.73 (12.00 ft from the CL POC along the radius)

A total station on a random control hub can stake those coordinates without occupying the PC. CES still expects you to recognize that the deflection/chord pair and the coordinate pair are the same POC.

Exam traps

1. Occupying the PI to set POCs as if the PI were on the arc.
2. Taping the arc length as the chord.
3. Using δ = ℓ / R (central angle, twice the deflection).
4. Offsetting 12 ft along the chord or tangent instead of radially.
5. PT = PI + T as a station.
6. Degree/radian mix in sin(δ).

Tab Chapter 10's T, L, and δ formulas in your bound references. On exam day you recover the elements, then you set the point.