Geometric Design
Horizontal Alignment
• Curves (circular)
• Tangents
• Transitions
Vertical Alignment
• Curves (parabolic)
• Tangents
IT222 Plan and Profile Drawing
Types of Horizontal Curves
•
•
•
•
Simple Curve
Compound Curves
Reverse Curves
Transition (Spiral) Curves
Curve Parameter Use
1.
2.
3.
What parameter is needed to find the beginning of a
curve relative to the PI?
What other parameters affect the value of the
parameter in question 1?
To prevent a subdivision street curve from being too
close to a lot line, the centerline cannot be less
than 65 ft from the PI. Parameter(s) to find R?
Degree of Curvature Arc Definition
„
„
Most frequently
used in highway
design
It is the central
angle formed by
two radii that
extend from the
center of a circle
to the ends of an
arc measuring
100 feet long
Degree of Curvature Arc Definition
„
Ratio between the degree of curvature (D)
and 360° is the same as the ratio between
100 feet of arc and the circumference (C)
of a circle having the same radius
D__ =
360°
100__
C
Degree of Curvature Arc Definition
„
Since the circumference of a circle = 2πR
D__ =
360°
„
R = 5729.578 / D
100__
2πR
Degree of Curvature Arc Definition
ƒ Smaller D means larger R;
ƒ Larger D means smaller R
ƒ For a 100’ arc:
ƒ D = 32 degrees, R = 179.05’
ƒ D = 8 degrees, R = 716.20’
Degree of Curvature –
Chord Definition
„
„
Most frequently used
in railway practice
Degree of curve is
the central angle
formed by two radii
drawn from the
center of the circle to
the ends of a chord
100 feet long
Degree of Curvature –
Chord Definition
„
„
The following trigonometric relationship
exists:
sin (D/2) = 50/R
Solving for R:
R = 50 / (sin(D/2))
Horizontal Alignment
„
Principle consideration
„
Length of radius or
degree of curvature,
based on
„
„
Design speed
Sight distance
„
As limited by headlights or
obstructions
Horizontal Alignment
„
The smaller the radius, the sharper the
curve
„
„
For high speed highways, curves must be flat
and have large radius
Typical radii
„
„
„
„
12,000 feet on interstate highways
1,000 feet on major arterials
500 feet on industrial access roads
150 feet on residential roads
Geometry:
Intersection or Delta Angle
„
Angle PI-PC-PT = Delta / 2 because:
„
Sum of the angles for the isosceles triangle
PC-PI-PT = 180
„
The angle at the PI = 180 - Delta
„
Therefore, the other two angles =
(180 – (180 - Delta))/2 = Delta/2
Simple Curves
PI
Sta.4+10.23
PI Sta = PC Sta + T
PT
St
a
3
.2
60
1+
a.
St
St
a.
=
5
PC +5
St 2.93
a
+
L
PT
PC
Begin Project Sta 0+00
Laying Out a Horizontal Curve
„
Surveyors stake out curves for construction
„
„
„
Stakes placed at PC, PT and full stations (every
100 feet)
For sharp curves, may need stakes at 25, 50 and
75-feet stations
Two Methods
„
„
Deflection Angle Method
Tangent Offset Method
Laying Out a Horizontal Curve
„
Deflection Angle Method
„
„
Curve distances assumed to start at PC
Deflection angle: angle between tangent
and chord
Laying Out a Horizontal Curve
PI
(V)
„
Deflection angle
between the
tangent and
chord is half the
PC
subtended arc (A)
δ/2
q
PT
(B)
δ
Laying Out a Horizontal Curve
PI
(V)
„
Angle
between 2
chords is half
the
subtended arc
r
PC
(A)
PT
(B)
D/2
D
Laying Out a Horizontal Curve
„
Staking out points on the curve
„
Need:
„
„
„
Deflection angles measured from the PC
(cumulative)
Lengths of the chords joining consecutive
stations
First deflection angle (δ1) and last deflection
angle (δ2) are usually < 1 station
„
„
C1 = 2R sin (δ1/2)
C2 = 2R sin (δ2/2)
PI
(V)
) I or Delta
δ1/2
r
q
TC/PC
(A)
s
t
p
δ1
D
D
D
D
δ2
CT/PT
(B)
Laying Out a HorizontalPICurve
(V)
l1
δ1
=
Δ
L
or I
=
l2
δ2
δ1/2
D/2
D/2
„
Deflection angles from the PC
„
„
„
„
„
„
VAp = δ1/2
p
PC
VAq = δ1/2 + D/2
(A)
VAr = δ1/2 + D/2 + D/2
VAs = δ1/2 + D/2 + D/2 + D/2
VAt = δ1/2 + D/2 + D/2 + D/2 + D/2
δ1
VAB = δ1/2 + D/2 + D/2 + D/2 + D/2 + δ2/2
q
r
D/2
s
D/2
t
δ2/2
D D D D
δ2
PT
(B)
Laying Out a Horizontal Curve
„
Entire horizontal curve can be laid out
from PC
„
„
„
Sighting deflection angle PI-PC-A = δ/2
Taping chord distance PC-A
PC and PT can be found by solving for T
and starting at PI
?
??
?
?
Degree of Curve vs Chord
Length
Degree of Curve
Arc Length (ft)
Chord (ft)
2o
100
99.995
3o
100
99.989
6o
100
99.954
10o
100
99.873
Nominal Chord Lengths –
Guidelines for curve layout
Da
Nominal Chord Length to Use
< 3o
Full Stations
3o – 7o
Half Stations
7o – 14o
Quarter Stations
HC Layout example
HC Layout example - detail
Field Notes
Layout Table
Profile from Surveying Text