GEFS (FOUNDAMENTALS OF SURVEYING)
CBLAMSIS UNIVERSITY OF THE CORDILLERAS VERTICAL CURVES
VERTICAL ALIGNMENT
In road construction, to avoid sudden change in direction when two tangents with slopes
greater than or less than zero intersects, it is customary to introduce a vertical curve at
every such point where the angle is large enough to warrant it. The vertical curve that
may be used to connect tangents to gradually change slope from positive to negative
(summit curve) or from negative to positive (sag curve) are compound curves and
parabolic curves. Generally parabolic curves are being used because parabola effects
the transition rather better theoretically than the circle, but its selection for the purpose is
due principally to its greater simplicity of application.
Types of Parabolic Curves
1. Symmetrical Parabolic Curves
2. Unsymmetrical Parabolic Curves
Elements of a parabolic curve
L = length of curve, the horizontal projection of the parabolic curve from PC to PT.
H = vertical tangent offset from the vertex, V, to a point below it (summit curve) or
above it (sag curve) on the curve.
G1 = grade (slope) of the back tangent in %
g1 = grade (slope) of the back tangent in decimal
G2 = grade (slope) of the forward tangent
g2 = grade of the forward tangent in decimal
A = change in grade from PC to PT in %
Summit curves
V
PT
H
PT
PC V
H
L
PC
L
Sag curves
L
PC
L
H
PC PT V
H
V PT
, GEFS (FOUNDAMENTALS OF SURVEYING)
CBLAMSIS UNIVERSITY OF THE CORDILLERAS VERTICAL CURVES
Symmetrical Parabolic Curves
Symmetrical parabolic curve are curves with their equation as the equation of a parabola
either opening upward or downward. They are called symmetrical parabolic curve
because the length of curve to the left and to the right of the vertex, V, are equal and
equal to one-half of the total length of curve, L.
The figure shown below is a profile. The vertical axis represents the elevation at any
stations and horizontal axis represent the stations. For example, sta. PC is known, sta. V
equals sta. PC plus L/2 and sta. PT equals sta. PC plus L. Also, if elev. PC is known,
Elev. V equals elev. PC plus Y.
Elevations
Elev. V V
H
PT
Y
Elev. PC
PC
L/2 L/2
L
Sta. PC Sta. V Sta. PT
Stations
Relationships of the elements of a symmetrical parabolic curve
C
g1(L/2)
Figure X
(g1 – g2)(L/2)
V
H - g2(L/2)
PT
H
PC B
L/2 L/2
L
CBLAMSIS UNIVERSITY OF THE CORDILLERAS VERTICAL CURVES
VERTICAL ALIGNMENT
In road construction, to avoid sudden change in direction when two tangents with slopes
greater than or less than zero intersects, it is customary to introduce a vertical curve at
every such point where the angle is large enough to warrant it. The vertical curve that
may be used to connect tangents to gradually change slope from positive to negative
(summit curve) or from negative to positive (sag curve) are compound curves and
parabolic curves. Generally parabolic curves are being used because parabola effects
the transition rather better theoretically than the circle, but its selection for the purpose is
due principally to its greater simplicity of application.
Types of Parabolic Curves
1. Symmetrical Parabolic Curves
2. Unsymmetrical Parabolic Curves
Elements of a parabolic curve
L = length of curve, the horizontal projection of the parabolic curve from PC to PT.
H = vertical tangent offset from the vertex, V, to a point below it (summit curve) or
above it (sag curve) on the curve.
G1 = grade (slope) of the back tangent in %
g1 = grade (slope) of the back tangent in decimal
G2 = grade (slope) of the forward tangent
g2 = grade of the forward tangent in decimal
A = change in grade from PC to PT in %
Summit curves
V
PT
H
PT
PC V
H
L
PC
L
Sag curves
L
PC
L
H
PC PT V
H
V PT
, GEFS (FOUNDAMENTALS OF SURVEYING)
CBLAMSIS UNIVERSITY OF THE CORDILLERAS VERTICAL CURVES
Symmetrical Parabolic Curves
Symmetrical parabolic curve are curves with their equation as the equation of a parabola
either opening upward or downward. They are called symmetrical parabolic curve
because the length of curve to the left and to the right of the vertex, V, are equal and
equal to one-half of the total length of curve, L.
The figure shown below is a profile. The vertical axis represents the elevation at any
stations and horizontal axis represent the stations. For example, sta. PC is known, sta. V
equals sta. PC plus L/2 and sta. PT equals sta. PC plus L. Also, if elev. PC is known,
Elev. V equals elev. PC plus Y.
Elevations
Elev. V V
H
PT
Y
Elev. PC
PC
L/2 L/2
L
Sta. PC Sta. V Sta. PT
Stations
Relationships of the elements of a symmetrical parabolic curve
C
g1(L/2)
Figure X
(g1 – g2)(L/2)
V
H - g2(L/2)
PT
H
PC B
L/2 L/2
L
