Vertical Curves - Theory & Concepts

Design and analysis of parabolic vertical curves, including crest and sag curves for highway and railway profiles.

Overview of Vertical Curves

Vertical curves provide a smooth transition between two different grades (slopes) in the vertical alignment of a route. Unlike horizontal curves, which are circular, vertical curves are designed as symmetrical or unsymmetrical parabolas because a parabola provides a constant rate of change of grade, ideal for vehicular dynamics.

Types of Vertical Curves

Crest and Sag Curves

  • Crest Vertical Curve: Forms a convex profile (hill). It transitions from a positive grade to a negative grade, a positive to a flatter positive grade, or a negative to a steeper negative grade. The primary design control is Stopping Sight Distance (SSD).
  • Sag Vertical Curve: Forms a concave profile (valley). It transitions from a negative grade to a positive grade, a negative to a flatter negative grade, or a positive to a steeper positive grade. Design controls include headlight sight distance, passenger comfort, and drainage.

Rate of Vertical Curvature (KK-Value)

K-Value

A critical parameter in vertical curve design is the KK-value, which represents the horizontal distance required to effect a 1%1\% change in grade. It directly measures the "flatness" or "sharpness" of the curve. Highway design manuals (like AASHTO) provide minimum KK-values based on design speeds and sight distance requirements.
$$ K = \\frac{L}{A} $$
Where:
  • LL = Length of the vertical curve
  • AA = Algebraic difference in grades (∣g1−g2∣|g_1 - g_2| in percent)

Elements of a Vertical Curve

Key Points and Terminology

  • PVC (Point of Vertical Curvature): The beginning of the vertical curve. Also called BVC.
  • PVT (Point of Vertical Tangency): The end of the vertical curve. Also called EVC.
  • PVI (Point of Vertical Intersection): The intersection of the initial grade (g1g_1) and the final grade (g2g_2).
  • g1,g2g_1, g_2: The grades of the intersecting tangents, expressed in percent (e.g., +3%+3\% or −2%-2\%).
  • LL: The length of the vertical curve, measured horizontally.
  • AA: The algebraic difference in grades (A=g2−g1A = g_2 - g_1).
  • rr: The rate of change of grade (r=A/Lr = A / L).

Stationing and Elevation Computations

$$ \\text{Sta. PVC} = \\text{Sta. PVI} - \\frac{L}{2} \\quad \\text{and} \\quad \\text{Sta. PVT} = \\text{Sta. PVI} + \\frac{L}{2} $$
$$ \\text{Elev. PVC} = \\text{Elev. PVI} - g_1 \\left(\\frac{L}{2}\\right) \\quad \\text{and} \\quad \\text{Elev. PVT} = \\text{Elev. PVI} + g_2 \\left(\\frac{L}{2}\\right) $$
Where:
  • g1,g2=g_1, g_2 = Grades in decimal format
  • L=L = Length of the curve

Properties of a Parabolic Vertical Curve

Parabolic Geometry

  • The curve is typically symmetrical; the horizontal distance from PVC to PVI is L/2L/2, and from PVI to PVT is L/2L/2.
  • The vertical offset from a tangent to the parabola is proportional to the square of the horizontal distance from the point of tangency (PVC or PVT).
  • The midpoint of the curve (CC) lies exactly halfway vertically between the PVI and the midpoint of the chord connecting PVC and PVT.

Fundamental Equations

$$ r = \\frac{g_2 - g_1}{L} $$
Where:
  • r=r = Rate of change of grade (% per unit distance)
  • g1=g_1 = Initial grade (%)
  • g2=g_2 = Final grade (%)
  • L=L = Length of curve (stations or meters)
$$ Y_x = Y_{PVC} + g_1 x + \\frac{1}{2} r x^2 $$
Where:
  • Yx=Y_x = Elevation on the curve at distance x from PVC
  • YPVC=Y_{PVC} = Elevation of the PVC
  • x=x = Horizontal distance from the PVC
  • g1=g_1 = Initial grade (decimal format or %/station)
  • r=r = Rate of change of grade

Important

In the elevation equation, ensure consistent units. If g1g_1 is in percent, xx must be in 100-m100\text{-m} or 100-ft100\text{-ft} stations. If g1g_1 is a decimal (e.g., 0.030.03), xx can be in meters or feet.

Location of the Highest or Lowest Point

Turning Point

The highest point (on a crest curve) or lowest point (on a sag curve) occurs where the tangent is completely horizontal (grade is zero). This point exists only if the grades g1g_1 and g2g_2 have opposite signs.
$$ x_m = \\frac{-g_1 L}{g_2 - g_1} = \\frac{-g_1}{r} $$
Where:
  • xm=x_m = Horizontal distance from PVC to the high/low point
  • g1,g2=g_1, g_2 = Grades (%)
  • L=L = Total length of curve

Vertical Offsets

$$ y = \\frac{A x^2}{200 L} $$
Where:
  • y=y = Vertical offset from the tangent to the curve
  • x=x = Horizontal distance from PVC (or PVT)
  • A=A = Algebraic difference in grades (%)
  • L=L = Total length of curve

Sight Distance on Vertical Curves

Curve Length vs. Sight Distance

The minimum length of a vertical curve (LL) is controlled by the required sight distance (SS), which is typically the Stopping Sight Distance (SSD). The formulas depend on whether the sight distance is greater than or less than the curve length.

Crest Vertical Curves

$$ L = \\frac{A S^2}{100 (\\sqrt{2h_1} + \\sqrt{2h_2})^2} \\quad \\text{(When } S < L \\text{)} $$
$$ L = 2S - \\frac{200 (\\sqrt{h_1} + \\sqrt{h_2})^2}{A} \\quad \\text{(When } S > L \\text{)} $$
Where:
  • S=S = Sight distance (m)
  • A=A = Algebraic difference in grades (%)
  • h1=h_1 = Eye height of the driver (typically 1.08 m)
  • h2=h_2 = Height of the object (typically 0.60 m)

Sag Vertical Curves

Headlight Sight Distance

For sag curves, sight distance is typically governed by the distance illuminated by the vehicle's headlights at night.
$$ L = \\frac{A S^2}{200 (h + S \\tan \\beta)} \\quad \\text{(When } S < L \\text{)} $$
$$ L = 2S - \\frac{200 (h + S \\tan \\beta)}{A} \\quad \\text{(When } S > L \\text{)} $$
Where:
  • h=h = Height of headlights (typically 0.60 m)
  • β=\beta = Upward angle of the headlight beam (typically 1∘1^\circ)

Riding Comfort on Sag Curves

Centrifugal Acceleration

When a vehicle travels through a sag vertical curve, centrifugal force acts downwards, combining with the vehicle's weight and pressing the passengers into their seats. To maintain riding comfort, the length of the curve must be sufficient to limit the rate of change of centrifugal acceleration. The standard comfort criterion is to limit the vertical acceleration to 0.3 m/s20.3 \text{ m/s}^2.
$$ L = \\frac{A V^2}{395} $$
Where:
  • L=L = Length of sag curve for comfort (m)
  • A=A = Algebraic difference in grades (%)
  • V=V = Design speed (km/h)

Unsymmetrical Vertical Curves

Unequal Tangents

An unsymmetrical vertical curve consists of two adjacent parabolic arcs that share a common point of tangency (PTC) located directly below or above the PVI. This is used when a symmetrical curve cannot fit due to physical constraints. The horizontal lengths from the PVI to the PVC (L1L_1) and from the PVI to the PVT (L2L_2) are unequal.
$$ r_1 = \\frac{g_{PTC} - g_1}{L_1} \\quad \\text{and} \\quad r_2 = \\frac{g_2 - g_{PTC}}{L_2} $$
Where:
  • r1,r2=r_1, r_2 = Rates of change of grade for the two arcs
  • gPTC=g_{PTC} = The grade of the common tangent at the point directly above/below the PVI
Key Takeaways
  • Vertical curves are parabolic to ensure a constant rate of change of grade (rr).
  • The KK-value (K=L/AK = L/A) represents the horizontal distance required to effect a 1%1\% change in grade.
  • Crest curves transition over hills; sag curves transition through valleys.
  • The curve elevation equation is a quadratic function of the distance from the PVC: Yx=YPVC+g1x+(rx2)/2Y_x = Y_{PVC} + g_1 x + (r x^2)/2.
  • The high or low point occurs where the slope is zero, located at distance xm=−g1/rx_m = -g_1 / r from the PVC.
  • Sag curves must be long enough to provide adequate headlight sight distance and maintain riding comfort by limiting centrifugal acceleration.
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