Vertical Curve Alignment

Vertical alignment

Longitudinal centreline profile: a series of grades + vertical curves.

Grade

Rise/horizontal distance in % (1 in 20 = 5%).

Grade design factors

Percentage (cut/fill, drainage, speed, cost, overpasses) and length (vehicle speed).

Grades vs vehicles

Cars unaffected up to 6%; trucks slow from 4-5%.

Max grade

Set by low-power trucks' climbing ability.

Min grade

0.5% (absolute min 0.3%) for drainage / prevent aquaplaning.

Long steep grades

Cause speed variation → rear-end crashes, queuing, fuel use, emissions; may need auxiliary climbing lanes.

Why vertical curves

Smooth grade transition (comfort) + adequate sight distance (safety) + appearance.

Curve shape

Parabolic: crest (convex) or sag (concave).

Crest vs sag

Classify using SIGNED A = G2 − G1: A > 0 → SAG; A < 0 → CREST.

Curve length

Use the ABSOLUTE value when calculating length: L = K|A| (otherwise a crest gives a negative length).

K value

Length (m) per 1% grade change; defines parabola size.

K selection criteria

Sight distance (always), appearance (low embankments/flat terrain), comfort (general).

PVI / PVC / PVT

Grade intersection point / start of curve / end of curve.

Parabola coefficients

y = ax² + bx + c with: a = (G2 − G1)/(200L) [SIGNED grade change

Crest criteria

Governed by sight distance; appearance may govern for small grade changes.

Crest heights

h1 = 1.1 m (eye), h2 = 0.2 m (object).

SL procedure

Assume SL and redo.

Appearance check

Table 8.6 is a CREST-curve appearance check; do NOT apply it automatically to sags. Sag curves are checked with their own applicable appearance, comfort and headlight criteria.

Sag criteria

Appearance (short sags look like a kink), comfort, headlight sight distance (night, unlit roads).

Sag comfort limit

Normally limit vertical acceleration to 0.05g; 0.10g may be accepted on low-volume roads or at intersections.

Sag headlight parameters

Headlight height h = 0.65 m (min); beam angle q = 1°.

Sag overhead obstructions

Check overpasses etc.

SSD worst case

Use the greatest DOWNWARD grade (most negative a).

High/low point of curve

Vertex at x = −b/(2a), then substitute into y = ax² + bx + c; verify 0 ≤ x ≤ L. (Only exists when grades have opposite signs.)

PVC / PVT elevations

ElevPVC = ElevPVI − G1L/200; ElevPVT = ElevPVI + G2L/200.

Controlling sag K

Calculate BOTH comfort K and headlight-sight-distance K, then adopt the larger valid value.

Design process

1) Crest or sag (signed A)? 2) Calc K (crest: sight distance; sag: comfort + headlight, adopt larger) → L = K|A|; 3) parabola equation for (x, y) points; 4) check H/V coordination; 5) appearance check (Table 8.6 for crests).

Appendix reference

Vertical-curve formulae are in Appendix J of the 2026 guide (lecture's Appendix L is an older-edition reference).