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