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 Δ-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*Δ) E=R((1/cos(Δ/2))-1) M=R(1-cos(Δ/2)) R=18,000/(Π*Dc) L=(100*Δ)/Dc

L=(Π*R*Δ)/180-------metric

From: Highway Engineering, 6th Ed. 1996, Paul Wright, ISBN 0-471-00315-8

Example Problem

Δ=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=2RsinΔ and w=2R(1-cos Δ) Solve for Δ (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 WorksheetFind Tangents and PI’s

Deflection Angles-PracticeBack Tangent Azimuth=25 deg-59 secForward (or Ahead) Tangent Azimuth=14 deg-10 secAnswer: 11 deg 00’ 49”

Back Tangent Bearing=N 22 deg E Forward Tangent Bearing=S 44 deg EAnswer: 114 deg

Back Tangent Azimuth=345 deg Forward Tangent Azimuth=22 deg Answer: 370 deg

Next lecture

Spiral Curves