VERTICAL CURVES
• Curves are needed to provide smooth transitions between straight segments (tangents)
of grade lines for highways and railroads.
• Because these curves exist in vertical planes, they are called vertical curves.
• The function of each curve is to provide a gradual change in grade from the initial (back)
tangent to the grade of the second (forward) tangent.
• Because parabolas provide a constant rate of change of grade, they are ideal and almost
always applied for vertical alignments used by vehicular traffic.
758 VERTICAL CURVES
Ground profile
Tangent 2
Tangent 1
Figure 25.1
Grade line and
ground profile of a
proposed highway
section.
B
A
Tangent 3
Curve a
Curve b
Figure 1.atGrade
line and
ground
profile
proposed
highway
section.
Elevations
selected
points
(e.g.,
fullofora half
stations
in the
English system
of stationing, or 20, 30, or 40 m in the metric system) along vertical parabolic
curves are usually computed by the tangent-offset method. It is simple, straightforward, conveniently performed with calculators and computers, and self-checking.
After the elevations of curve points have been computed, they are staked in the
field to guide construction operations so the route can be built according to plan.
■ 25.2 GENERAL EQUATION OF A VERTICAL
PARABOLIC CURVE
The general mathematical expression of a parabola, with respect to an XY rectangular coordinate system, is given by
YP = a + bXP + cX2P
(25.1)
FACTORS TO BE TAKEN INTO ACCOUNT IN THE
DESIGN:
(1) providing a good fit with the existing ground profile, thereby minimizing the depths of
cuts and fills,
(2) balancing the volume of cut material against fill,
(3) maintaining adequate drainage,
(4) not exceeding maximum specified grades, and
(5) meeting fixed elevations such as intersections with other roads.
• In addition, the curves must be designed to
(a) fit the grade lines they connect,
(b) have lengths sufficient to meet specifications covering a maximum rate of change of
grade (which affects the comfort of vehicle occupants), and
(c) provide sufficient sight distance for safe vehicle operation
Elements of Vertical Curve
PC = point of curvature, also known as BVC (beginning of vertical
curve)
PT = point of tangency, also known as EVC (end of vertical curve)
PI = point of intersection of the tangents, also called PVI (point of
vertical intersection)
L = length of parabolic curve, it is the projection of the curve onto
a horizontal surface which corresponds to the plan distance.
S1 = horizontal distance from PC to the highest (lowest) point of
the summit (sag) curve
S2 = horizontal distance from PT to the highest (lowest) point of
the summit (sag) curve
h1 = vertical distance between PC and the highest (lowest) point of
the summit (sag) curve
h2 = vertical distance between PT and the highest (lowest) point of
the summit (sag) curve
g1 = grade (in percent) of back tangent (tangent through PC)
g2 = grade (in percent) of forward tangent (tangent through PT)
A = change in grade from PC to PT
a = vertical distance between PC and PI
b = vertical distance between PT and PI
H = vertical distance between PI and the curve
SYMMETRICAL
SYMMETRICAL PARABOLIC CURVE
SAMPLE PROBLEM
A grade of -4.2% grade intersects a grade of +3.0% at Station 11 + 488.00 of elevations 20.80
meters. These two center gradelines are to be connected by a 260 meter vertical parabolic
curve.
1.
At what station is the cross-drainage pipes be situated?
2.
If the overall outside dimensions of the reinforced concrete pipe to be installed is 95 cm,
and the top of the culvert is 30 cm below the subgrade, what will be the invert elevation at
the center?
UNSYMMETRICAL PARABOLIC CURVE
SAMPLE PROBLEM
• In a certain road construction undertaken by the DPWH it was decided to connect a
forward tangent of 3% and a back tangent of -5% by a 200 meter symmetrical parabolic
curve. It was later discovered that the grade intersection at station 10 +100; whose
elevation is 100m fall on a rocky section with the exposed boulder at elevation 102.67m.
To avoid rock excavation, the project engineer decided to adjust the vertical parabolic
curve in such a way that the curve will just clear the rock without altering the position of
PC and the grade of the tangents. Determine the stationing and the elevation of the new
PT.