6.1 Transportation: Geometric Design

Horizontal Curves

Geometric design lays out the highway's centerline so a vehicle at the design speed can travel safely and comfortably. The Fundamentals of Engineering (FE) Civil exam tests this from the NCEES FE Reference Handbook — your job is to locate and apply the right formula, not memorize it.

A simple horizontal curve is a circular arc connecting two tangents. Its sharpness is given by the degree of curve (D). Using the arc definition (the FE/U.S. highway convention, where D is the central angle subtending a 100-ft arc):

• R = 5729.58 / D (R in feet, D in degrees)

Key geometric elements (all in the Handbook):

Here Δ is the deflection (intersection) angle between the two tangents, PC is point of curvature, PI is point of intersection, and PT is point of tangency.

Superelevation and Side Friction

On a curve, the superelevation (e) (cross-slope, ft/ft) plus the side friction factor (f) must supply the centripetal acceleration. The governing relationship (U.S. customary units) is:

• e + f = V² / (15·R)

where V = design speed in mph and R = radius in feet. The 15 bundles gravity and the mph-to-ft/s conversion. The most common trap is mixing units — V must be mph and R must be feet for the 15 to be valid.

Worked example. A curve has design speed V = 60 mph, e_max = 0.08, and f = 0.12 (from the AASHTO/Handbook table at 60 mph). Find the minimum radius.

• e + f = V²/(15R) → R = V²/[15(e+f)]
• R = 60² / [15·(0.08 + 0.12)] = 3600 / (15·0.20) = 3600 / 3.0 = 1200 ft

So any radius ≥ 1200 ft is safe at 60 mph for those e and f values. Sharper (smaller R) would exceed the available friction. Designers round up to a standard radius.

Vertical Curves and Sight Distance

Vertical curves are equal-tangent parabolas joining two grades, g1 and g2 (in percent). A crest curve goes from a steeper up- to a flatter/down-grade (A = g2 − g1 < 0); a sag curve is the reverse. The algebraic grade difference is A = |g2 − g1| (percent).

The curve is characterized by the rate of vertical curvature K:

• K = L / A (length L in feet per 1% change in grade)

Larger K means a longer, flatter, safer curve. The elevation along a parabola from the BVC (begin vertical curve) is:

• y = g1·x + (A/200L)·x² (x in ft from BVC, elevations in ft)

The high/low point occurs where the slope is zero: x_turning = −g1·L/A (i.e., g1·L/(g1−g2)).

Stopping sight distance (SSD) is the distance to perceive, react, and brake to a stop:

• SSD = 1.47·V·t + 1.075·V²/a

with V in mph, perception–reaction time t = 2.5 s (AASHTO standard), and deceleration a = 11.2 ft/s². The first term is reaction distance, the second braking distance.

Worked SSD example (V = 50 mph):

• Reaction: 1.47·50·2.5 = 183.75 ft
• Braking: 1.075·50²/11.2 = 1.075·2500/11.2 = 239.96 ft
• SSD ≈ 424 ft

Crest curve length is then set so the driver's eye (3.5 ft) can see an object (2.0 ft) over the crest for at least SSD. Always confirm whether the case is S < L or S > L in the Handbook formula.

Design Speed, Stations, and Curve Length

Design speed is the selected speed used to determine the geometric features of a roadway; every controlling element — minimum radius, superelevation, sight distance, and vertical-curve length — is derived from it. Higher design speed demands flatter curves (larger R and K), longer sight distances, and more generous superelevation runoff.

Highway stationing measures distance along the alignment in 100-ft stations: station 12+50 is 1250 ft from the origin. Curve points are located by station: PT station = PC station + L (curve length). A common exam step is computing curve length, then chaining stations.

Worked curve-length example. A curve has R = 1432 ft (D = 4°) and deflection angle Δ = 30°. Its length is L = R·Δ·(π/180) = 1432·30·0.01745 = 749.6 ft, or equivalently L = 100·Δ/D = 100·30/4 = 750 ft. If the PC is at station 20+00, the PT is at 20+00 + 7+50 = station 27+50.

Crest vs sag controls differ: crest curves are governed by sight distance over the hump (object visibility); sag curves are governed by headlight sight distance at night, comfort (rider acceleration), and drainage. For both, minimum length L_min = K·A once the controlling K (from the design-speed table in the Handbook) is read. Mixing a crest K with a sag problem is a classic trap — confirm which table applies before plugging in.