horizontal alignment – circular...
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Horizontal Alignment – Circular Curves
CTC 440
Objectives
� Know the nomenclature of a horizontal curve
� Know how to solve curve problems
� Know how to solve reverse/compound
curve problems
Simple Horizontal Curve
� Circular arc tangent to two straight (linear) sections of a route
Circular Curves
� PI-pt of intersection
� PC-pt of curvature
� PT-pt of tangency
� R-radius of the circular arc
� Back tangent
� Forward (ahead) tangent
Circular Curves
� T-distance from the PC or PT to the PI
� ∆-Deflection Angle. Also the central angle of the curve (LT or RT)
� Dc -Degree of Curvature. The angle
subtended at the center of the circle by a 100’ arc on the circle (English units)
Degree of Curvature
� Highway agencies –arc definition
� Railroad agencies –chord definition
Arc Definition-Derivision
� Dc/100’ of arc is proportional to 360 degrees/2*PI*r
Dc=18,000/PI*r
Circular Curves
� E –External Distance � Distance from the PI to the midpoint of the circular arc measured
along the bisector of the central angle
� L-Length of Curve
� M-Middle Ordinate � Distance from the midpoint of the long chord (between PC & PT)
and the midpoint of the circular arc measured along the bisector of
the central angle
Basic Equations
� T=R*tan(1/2*∆)
� E=R(1/cos(∆/2)-1)
� M=R(1-cos(∆/2))
� R=18,000/(Π*Dc)
� L=(100*∆)/Dc
� L=(Π*R*∆)/180-------metric
From: Highway Engineering, 6th Ed. 1996, Paul Wright, ISBN 0-471-00315-8
Example Problem
� ∆=30 deg
� E=100’ minimum to avoid a building
� Choose an even degree of curvature to meet the criteria
Example Problem
� Solve for R knowing E and Deflection Angle (R=2834.77’ minimum)
� Solve for degree of curvature (2.02 deg
and round off to an even curvature (2 degrees)
� Check R (R=2865 ft)
� Calc E (E=101.07 ft which is > 100’ ok)
Practical Steps in Laying Out a Horizontal Alignment
� POB - pt of beginning
� POE - pt of ending
� POB, PI’s and POE’s are laid out
� Circular curves (radii) are established
� Alignment is stationed
� XX+XX.XX (english) – a station is 100’
� XX+XXX.XXX (metric) – a station is one km
Compound Curves
� Formed by two simple curves having one common tangent and one common
point of tangency
� Both curves have their centers on the same side of the tangent
� PCC-Point of Compound Curvature
Compound Curves
� Avoid if possible for most road alignments
� Used for ramps (RS<=0.5*RL)
� Used for intersection radii (3-centered
compound curves)
Use of Compound Curves
Use of compound
curves: intersections
Reverse Compound Curves
� Formed by two simple curves having one common tangent and one common
point of tangency
� The curves have their centers on the opposite side of the tangent
� PRC-Point of Reverse Curvature
Reverse Compound Curves
� Avoid if possible for most road alignments
� Used for design of auxiliary lanes (see
AASHTO)
Use of RCC: Auxiliary Lanes
Source: AASHTO, Figure IX-72, Page 784
Example: Taper Design C-3
� R=90m
� L=35.4m
� What is width?
� L=2Rsin∆ and w=2R(1-cos ∆)
� Solve for ∆ (first equation) and solve for
w (2nd equation)
� W-3.515m=11.5 ft
In General
� Horizontal alignments should be as directional as possible, but consistent
with topography
� Poor horizontal alignments look bad, decrease capacity, and cost money/time
Considerations
� Keep the number of curves down to a minimum
� Meet the design criteria
� Alignment should be consistent
� Avoid curves on high fills
� Avoid compound & reverse curves
� Correlate horizontal/vertical alignments
Lab Worksheet Find Tangents and PI’s
Deflection Angles-Practice
Back Tangent Azimuth=25 deg-59 sec
Forward (or Ahead) Tangent Azimuth=14 deg-10 sec
Answer: 11 deg 00’ 49”
Back Tangent Bearing=N 22 deg E
Forward Tangent Bearing=S 44 deg E
Answer: 114 deg
Back Tangent Azimuth=345 deg
Forward Tangent Azimuth=22 deg
Answer: 370 deg
Next lecture
� Spiral Curves
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