Highway Curves (Kavanagh)

BC (Beginning of Curve)

The point where alignment changes from straight tangent to circular curve. Also called: • PC (Point of Curve) • TC (Tangent to Curve) Located distance T from the PI along the back tangent. — Kavanagh Ch. 11, p. 381

EC (End of Curve)

The point where alignment changes from circular curve back to tangent. Also called: • PT (Point of Tangency) • CT (Curve to Tangent) Located distance T from PI; station = BC + L. — Kavanagh Ch. 11, p. 381

PI (Point of Intersection)

The point where the back tangent and forward tangent would intersect if extended. The PI is NOT on the centerline—the curve bypasses it. Δ = deflection angle measured at PI. — Kavanagh Ch. 11, p. 381

Subtangent Distance (T)

T = R × tan(Δ/2) Distance from PI to BC (or PI to EC) along tangent line. Curve is symmetrical, so both subtangents equal T. — Kavanagh Ch. 11, Eq. 11.1, p. 382

Long Chord (C)

C = 2R × sin(Δ/2) Straight-line distance from BC to EC. Always shorter than arc length L. — Kavanagh Ch. 11, Eq. 11.2, p. 382

Arc Length (L)

L = 2πR × (Δ/360) or L = 100 × (Δ/D) when using degree of curve Actual distance along the curve from BC to EC. — Kavanagh Ch. 11, Eq. 11.5 & 11.7, p. 383-384

Middle Ordinate (M)

M = R × (1 - cos(Δ/2)) Perpendicular distance from midpoint of long chord to midpoint of arc. — Kavanagh Ch. 11, Eq. 11.3, p. 383

External Distance (E)

E = R × (sec(Δ/2) - 1) or E = R × (1/cos(Δ/2) - 1) Distance from PI to midpoint of curve. — Kavanagh Ch. 11, Eq. 11.4, p. 383

Degree of Curve (D) - Highway

Central angle subtended by 100 ft of ARC. D = 5,729.58 / R Larger D = sharper curve (smaller radius). Smaller D = flatter curve (larger radius). — Kavanagh Ch. 11, Eq. 11.6, p. 383

Degree of Curve (D) - Railway

Central angle subtended by 100 ft of CHORD. Railway definition differs from highway definition. Both relate sharpness to a 100-ft reference. — Kavanagh Ch. 11, p. 383

Calculating BC and EC Stations

BC station = PI station - T EC station = BC station + L COMMON MISTAKE: Do NOT add T to PI for EC! Stationing follows the centerline (the curve), not through the PI. — Kavanagh Ch. 11, p. 385

Total Deflection Angle (BC to EC)

Total deflection = Δ/2 The deflection angle from BC tangent to EC equals half the central angle. Used to verify curve stakeout. — Kavanagh Ch. 11, p. 382

Deflection Angle Formula

Deflection = (arc/L) × (Δ/2) For any station: Deflection = (arc from BC to station / total L) × (Δ/2) Deflections are CUMULATIVE from BC. — Kavanagh Ch. 11, p. 389

Subchord Calculation

C = 2R × sin(deflection angle) Used to calculate actual chord distances for field layout. Chords are always slightly shorter than arcs. — Kavanagh Ch. 11, Eq. 11.8, p. 390

Vertical Curve Type

Parabola (not circular) General equation: y = ax² + bx + c Where: • a = (g₂ - g₁)/(2L) • b = g₁ • c = elevation at BVC • x = distance from BVC — Kavanagh Ch. 11, p. 403, 409

BVC, PVI, EVC

BVC = Beginning of Vertical Curve PVI = Point of Vertical Intersection (where grades meet) EVC = End of Vertical Curve PVI is at the midpoint: BVC + L/2 = PVI = EVC - L/2 — Kavanagh Ch. 11, p. 405

Algebraic Difference in Grades (A)

A = g₂ - g₁ Where grades are in percent. Example: g₁ = -3.2%, g₂ = +1.8% A = 1.8 - (-3.2) = 5.0 — Kavanagh Ch. 11, p. 407

High/Low Point Location

x = -g₁ × (L/A) or x = -g₁ × K Where x is distance from BVC to high/low point. Result is always positive. — Kavanagh Ch. 11, Eq. 11.15, p. 406

K-Value (Rate of Vertical Curvature)

K = L / A Horizontal distance (ft or m) required for 1% change in slope. Used in design tables for sight distance requirements. L = K × A (minimum curve length) — Kavanagh Ch. 11, Eq. 11.17, p. 409

Tangent Offset (d) at PVI

d = A × L / 800 or d = (midchord elevation - PVI elevation) / 2 Offsets are proportional to x² (square of distance). — Kavanagh Ch. 11, Eq. 11.20, p. 410

Crest vs. Sag Curves

Crest: Summit curve (hill top) • Subtract tangent offsets from tangent elevation • Design controlled by sight distance over crest Sag: Valley curve (dip) • Add tangent offsets to tangent elevation • Design controlled by headlight range — Kavanagh Ch. 11, p. 405, 410

Spiral Curve Purpose

1. Overcome abrupt change from tangent to circular curve 2. Transition from normal crown to full superelevation Radius changes uniformly from ∞ (tangent) to R (circular curve). — Kavanagh Ch. 11, p. 411

Spiral Curve Points

TS = Tangent to Spiral (radius = ∞) SC = Spiral to Curve (radius = R) CS = Curve to Spiral ST = Spiral to Tangent Spiral length used for superelevation transition. — Kavanagh Ch. 11, p. 411

Curve Layout Field Check

1. Set up theodolite at BC, zero on PI 2. Turn off Δ/2 toward EC 3. Line of sight should fall on EC stake 4. If not, recheck T computation and field measurement Then proceed with deflection staking. — Kavanagh Ch. 11, p. 391

Moving Up on the Curve

When line of sight is blocked: 1. Move instrument to last established station 2. Zero horizontal circle 3. Backsight BC with telescope inverted 4. Transit telescope to normal position 5. Continue with original deflection list — Kavanagh Ch. 11, p. 392