VERTICAL CURVES

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CHAPTER 3

• PART II
• VERTICAL CURVES
• HORIZONTAL SIGHT DISTANCE

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Vertical Alignment

• Specifies the elevation of points along a roadway
• Provides a transition between two grades
• Sag curves and crest curves
• Equal-tangent curves - half the curve length
  positioned before the PVI half after

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Notation

• Curve point naming is similar to horizontal
  curves, with addition of V for vertical
• PVC Point of Vertical Curvature
• PVI Point of Vertical Intersection
  (of initial and final tangents)
• PVT Point of Vertical Tangency
• Curve positioning and length usually referenced
  in stations
• Stations represent 1000 m or 100 ft
• e.g., 1258.5 ft ? 12 58.5 (i.e., 12 stations
  58.5 ft)

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Notation

• G1 is initial roadway grade
• Also referred to as initial tangent grade
• G2 is final roadway (tangent) grade
• A is the absolute value of the difference in
  grades (generally expressed in percent)
• A G2 G1
• L is the length of the vertical curve measured in
  a horizontal plane (not along curve center line,
  like horizontal curves)

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Fundamentals

• Parabolic curves are generally used for design
• Parabolic function ? y ax 2 bx c y
  roadway elevation x distance from PVC c
  elevation of PVC
• Also usually design for equal-length tangents
• i.e., half of curve length is before PVI and half
  after

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First Derivative

• First derivative gives slope
• At PVC, x 0, so , by
  definition
• G1 is initial slope (in ft/ft or m/m) as
  previously defined

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Second Derivative

• Second derivative gives rate of change of slope
• However, the average rate of change of slope, by
  observation, can also be written as
• Giving,

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Offsets
Offsets are vertical distances from initial
tangent to the curve
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Offset Formulas

• For an equal tangent parabola,
• Y offset (in m or ft) at any distance, x, from
  the PVC
• A and L are as previously defined
• It follows from the figure that,

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K Values

• The rate of change of grade at successive points
  on the curve is a constant amount for equal
  increments of horizontal distance, and
• Equals the algebraic difference between
  intersecting tangent grades divided by the length
  of curve, or A/L in percent per ft (m)
• The reciprocal L/A is the horizontal distance
  required to effect a 1 change in gradient and
  is, therefore, a measure of curvature
• The quantity L/A is termed K

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K Values

• The K-value can be used directly to compute the
  high/low points for crest/sag vertical curves
  (provided the high/low point is not at a curve
  end) by,
• xhl K ? G1
• Where x distance from the PVC to the high/low
  point
• Additionally, K-values have important
  applications in the design of vertical curves,
  which we will see shortly

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Vertical Curves

• Controlling factor sight distance
• Stopping sight distance should be provided as a
  minimum
• Rate of change of grade should be kept within
  tolerable limits
• Drainage of sag curves is important
  consideration, grades not less than 0.5 needed
  for drainage to outer edge of roadway

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Vertical Alignment Relationships
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Example Problem Vertical Curve

• A vertical curve crosses a 4 diameter pipe at
  right angles. Pipe at sta 11085 with centerline
  elevation of 1091.60. PVI at sta 11000
  elevation 1098.4. Equal tangent curve, 600
  long with initial and final grades of 1.2 and
  -1.08. Using offsets determine the depth below
  the surface of the curve the top of the pipe and
  determine the station of the highest point of the
  curve.

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Solution
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Solution Continued
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Stopping Sight Distance Crest Curves

• Two different factors are important for crest
  curves
• The drivers eye height in vehicle, H1
• Height of a roadway obstruction object, H2

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SSD Curve Design

• It is necessary, when designing vertical curves,
  to provide adequate stopping-sight distance
  (SSD)
• Because curve construction is expensive, we want
  to minimize curve length, subject to adequate SSD

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SSD and Curve Design

• SSD formulation was given in Chapter 2, i.e., ds
  d dr (Eq. 2.50)
• It is repeated in Chapter 3 as Eq. 3.12

Table 3.1 gives SSD values in 5mph increments
based on this equation and a11.2ft/s2 and tr
2.5s
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Minimum Curve Length

• By using the properties of a parabola for an
  equal tangent curve, it can be shown that the
  minimum length of curve, Lm, for a required SSD
  is

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Minimum Curve Length

• For the sight distance required to provide
  adequate SSD, current AASHTO design standards use
  the following specifications
• H1 (drivers eye height) 3.5 ft (1080 mm)
• H2 (object height) 2.0 ft (600 mm)

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Minimum Curve Length

• Substituting these values into previous two
  equations yields

Since using these equations can be cumbersome,
tables have been developed, utilizing KL/A
(discussed earlier)
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Example 3.5

• A highway is being designed to AASHTO guidelines
  with a 70-mph design speed and, at one section,
  an equal tangent vertical curve must be designed
  to connect grades of 1.0 and 2.0. Determine
  the minimum length of vertical curve necessary to
  meet SSD requirements.

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3.5 Solution
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K-values for adequate SSD
Design Controls for Crest Vertical Curves Based
on SSD
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Example 3.6

• Solve Example Problem 5 using the K-values in
  Table 3.2.

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Sag Vertical Curves

• Four criteria for establishing length of sag
  curves
• Headlight sight distance
• Passenger comfort
• Drainage control
• General appearance

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Headlight Sight Distance

• At night, the portion of highway that is visible
  to the driver is dependent on the position of the
  headlights and the direction of the light beam
• Headlights are assumed to be 2 ft (600 mm) and
  1-degree upward divergence of the light beam from
  the longitudinal axis of the vehicle
• Equations 3-19 through 3-23 describe the required
  sight distance for sag curves

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Sag Vertical Curve Length

• The most controlling factor is headlight sight
  distance
• If for economic reasons such lengths cannot be
  provided, fixed source lighting should be
  provided to assist the driver.

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Min Sag Curve Length

• Like crest curves, we need expressions for
  determining the minimum length of crest curve
  required for adequate SSD

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Minimum Curve Length
For the sight distance required to provide
adequate SSD, current AASHTO design standards use
the following specifications H (headlight
height) 2.0 ft (600 mm) ? (headlight angle) 1
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Minimum Sag Curve Length
Substituting the recommended values for beta and
H gives
If not sure which equation to use, assume SSD lt L
first (for either sag or crest curves)
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K Values for Adequate SSD
Design Controls for Sag Vertical Curves Based on
SSD
Table 3.3
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Passing Sight Distance Crest Vertical Curve
Design

• Only a factor for vertical curves
• A consideration for two-lane highways
• Sag curves have unobstructed sight distance
• Assume driver eye height and height of object on
  roadway surface both 3.5

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Stopping Sight Distance Horizontal Curve Design

• Adequate sight distance must be provided in the
  design of horizontal curves
• Cost of right of way or the cost of moving
  earthen materials often restrict design options
• When such obstructions exist, stopping sight
  distance is checked and measured along the
  horizontal curve from the center of the traveled
  lane

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Sight Distance Relationships
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Sight Distance Example

• Horizontal curve with 2000 radius 12lanes
  60mph design speed. Determine the distance that
  must be cleared from the inside edge of the
  inside lane to provide sufficient stopping sight
  distance.

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Sight Distance Example Continued
SSD is determined from Table 3.1 for 60mph
design speed