Horizontal curve elements and stationing on the PE Civil Transportation exam

A horizontal curve problem on the Transportation exam gives you a radius and an intersection angle, or the survey data to find them, and asks for a station, a length, an offset or a coordinate. The geometry is in the Handbook's Horizontal Design section: the simple-curve elements, both definitions of degree of curvature, deflection angles, compound and reverse curves, and latitudes and departures. What it doesn't print is how stationing runs. You bring that.

What you actually need to own

• The elements from R and Δ. T = R tan(Δ/2), L = RΔπ/180, long chord C = 2R sin(Δ/2), E = R[sec(Δ/2) − 1], M = R[1 − cos(Δ/2)]. T runs from the PC or PT to the PI, E from the PI to the arc midpoint, M from the arc midpoint to the long-chord midpoint. Since E/M = sec(Δ/2), E is always the longer.
• Two definitions of degree of curvature. Arc: the angle subtended by 100 ft of arc, D_a = 5,729.6/R, so L = 100Δ/D_a. Chord: the angle subtended by a 100-ft chord, R = 50/sin(D_c/2). For the same D the chord-definition radius is larger, more so as the curve sharpens. Either way, D depends on R alone.
• Stationing. Stations run along the centerline: up the back tangent to the PC, then around the arc to the PT: PC = PI − T, PT = PC + L. At a station equation (BK = AH), measure within each stationing and add the pieces; never subtract across the equation.
• Points on the curve. The arc from the PC to a point is l₁ = Rδ₁π/180, the chord to it is C₁ = 2R sin(δ₁/2), and the deflection from the back tangent is δ₁/2. A measured chord must become an arc before it goes on a station. The long chord leaves the PC deflected Δ/2 toward the inside.
• Coordinates. Departure = distance × sin(bearing), latitude = distance × cos(bearing). Working from the PI back to the PC reverses the back-tangent bearing. The radius at the PC is perpendicular to the back tangent, turned toward the inside of the curve.
• Compound, reverse and spiral layouts. A two-centered compound curve is solved as a triangle, with the law of sines, from the two radii and the two central angles. A reverse curve between parallel tangents shares the offset: p = (R1 + R2)(1 − cos Δ). On a spiral–curve–spiral, each spiral takes its own angle out of Δ and adds its own length to the stationing.
• The Green Book judgment calls. On a highway main line, the flatter arc of a compound curve should have a radius no more than 1.5 times the sharper one (2:1 is the turning-roadway preference). That comes from Green Book Chapter 3, not the Handbook.

Where people lose points

• Stationing the PT as PI + T. That runs along the tangents. The PT is PC + L.
• Adding the long chord, or a measured chord, to the PC station in place of the arc it subtends.
• Treating the PI as the midcurve station. The PI is off the road. Midcurve is PC + L/2.
• Using the PC's plus as the arc to the first full station. That runs backward; use the next full station minus the PC.
• Mixing up the definitions. In one problem, a 22° chord-definition curve has R = 262.0 ft, while the arc formula gives 260.4 ft. The two are close enough that the wrong one looks right.
• An inaccessible PI. Δ is the exterior angle of the traverse triangle, so it equals the sum of the two measured interior angles. In the law of sines, pair each side with the angle opposite it.
• Bearing slips: a reversed direction, sine and cosine swapped, the radial turned outward.
• Solving a compound curve as one simple curve over the full Δ, or putting a reverse curve's whole offset on one radius.

How to study it

Take one curve: R, Δ and a PI station. Compute the five elements, station the PC and PT, list the deflection to every full station, and find the PC's coordinates from the PI. Redo it under the chord definition, then with the PI in a pond so you have to traverse around it. The PC and PT you station here anchor superelevation runoff, and the same stationing discipline carries into vertical curves.

Part of the Horizontal Design area of the PE Civil Transportation exam. → Start practicing free — the whole practice bank, free with an account.