1. Horizontal Alignment – Circular Curves
CTC 440

2. Objectives Know the nomenclature of a horizontal curve
Know how to solve curve problems
Know how to solve reverse/compound curve problems

3. Simple Horizontal Curve
Circular arc tangent to two straight (linear) sections of a route

5. 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

6. 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)

7. Degree of Curvature Highway agencies –arc definition
Railroad agencies –chord definition

8. Arc Definition-Derivision
Dc/100’ of arc is proportional to 360 degrees/2*PI*r
Dc=18,000/PI*r

9. 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

10. 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)/ metric

11. From: Highway Engineering, 6th Ed. 1996,
Paul Wright, ISBN

12. Example Problem Δ=30 deg E=100’ minimum to avoid a building
Choose an even degree of curvature to meet the criteria

13. Example Problem
Solve for R knowing E and Deflection Angle (R= ’ 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= ft which is > 100’ ok)

14. 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

15. 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

16. Compound Curves

17. Compound Curves Avoid if possible for most road alignments
Used for ramps (RS<=0.5*RL)
Used for intersection radii (3-centered compound curves)

18. Use of Compound Curves

19. Use of compound curves: intersections

20. 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

21. Reverse Compound Curves

22. Reverse Compound Curves
Avoid if possible for most road alignments
Used for design of auxiliary lanes (see AASHTO)

23. Use of RCC: Auxiliary Lanes
Source: AASHTO, Figure IX-72, Page 784

24. 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

25. 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

26. 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

27. Worksheet: Identifying Finding Tangents and PI’s

28. 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

29. Next lecture
Spiral Curves