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 L-Length of Curve 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

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

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

Worksheet: Identifying Finding 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

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