1. Geometric Design (II)

2. Learning Objectives To calculate minimum radius of horizontal curve
To understand design concepts for transition curves and compute min length
To understand the role of SSD in horizontal and vertical design
To define and apply grade considerations
To develop vertical curves
(Chapter 6.1 ~ 6.4)

3. Horizontal Curve Minimum Curve Radius
Curve requiring the most centripetal force for the given speed
Given emax, umax, Vdesign R
Calculating the minimum radius for a horizontal curve is based on three factors: the design speed, the superelevation, and the side-friction factor.
The minimum radius serves not only as a constraint on the geometric design of the roadway, but also as a starting point from which a more refined curve design can be produced.
Any increase in the radius of the curve beyond this minimum radius will allow you to reduce the side-friction factor, the superelevation rate, or both.

4. Horizontal Curve Properties
Point of
Curvature
Tangency
Based on circular curve
R: radius of curve
D: degree of curve
: central angle
T: length of tangent
L: length of curve
LC: long chord
M: middle ordinate dist
E: external dist
D = 36000/(2R), the angle subtended by a 100 ft arc
: central angle, also the deflection angle between the tangents
T = R·tan(Δ/2)
L = 100 (Δ/D)
LC = 2R·sin(Δ/2)
M = R·[1 – cos(Δ/2)]
E = R·[sec(Δ/2)-1]

5. Horizontal Design Iterations
Design baseline
Curve radius above the minimum
Superelevation and side-friction factor not exceeding the maximum values
Design is revised to consider:
cost, environmental impacts, sight distances, aesthetic consequences, etc. 

6. Horizontal Curve Sight Distance
Sight line is a chord of the circular curve
Sight Distance is curve length measured along centerline of inside lane
Once you have a radius that seems to connect the two previously disjointed sections of roadway safely and comfortably, you need to make sure that you have provided an adequate stopping sight distance throughout your horizontal curve. 
Sight distance can be the controlling aspect of horizontal curve design where obstructions are present near the inside of the curve. To determine the actual sight distance that you have provided, you need to consider that the driver can only see the portion of the roadway ahead that is not hidden by the obstruction. In addition, at the instant the driver is in a position to see a hazard in the roadway ahead, there should be a length of roadway between the vehicle and the hazard that is greater than or equal to the stopping sight distance. 

7. Horizontal Curve Sight Distance
Figure 6-10
Once you have a radius that seems to connect the two previously disjointed sections of roadway safely and comfortably, you need to make sure that you have provided an adequate stopping sight distance throughout your horizontal curve. 
Sight distance can be the controlling aspect of horizontal curve design where obstructions are present near the inside of the curve. To determine the actual sight distance that you have provided, you need to consider that the driver can only see the portion of the roadway ahead that is not hidden by the obstruction. In addition, at the instant the driver is in a position to see a hazard in the roadway ahead, there should be a length of roadway between the vehicle and the hazard that is greater than or equal to the stopping sight distance. 

8. Transition Curves
Gradually changing the curvature from tangents to circular curves
Without Transition Curves
With Transition Curves

9. Transition Curves
Gradually changing the curvature from tangents to circular curves
Use a spiral curve
L: min length of spiral (ft)
V: speed (mph)
R: curve radius (ft)
C: rate of increase of centrifugal accel (ft/sec3), 1~3
Often, horizontal curves are more comfortable and more aesthetically pleasing if the change in roadway cross-section and curvature is effected over a short transitional segment. 
The gradual change in curvature is produced by using a spiral curve. The radius of the spiral curve starts at infinity and is gradually reduced to the radius of the circular curve that you designed originally.  Adding the spiral curve causes the centripetal acceleration to build up gradually, which is more comfortable for vehicle occupants.  

10. Transitional Curves
Gradually changing the cross-section of the roadway from normal to superelevated (Figure 6-9)
Often, horizontal curves are more comfortable and more aesthetically pleasing if the change in roadway cross-section and curvature is effected over a short transitional segment. 
The gradual change in curvature is produced by using a spiral curve. The radius of the spiral curve starts at infinity and is gradually reduced to the radius of the circular curve that you designed originally.  Adding the spiral curve causes the centripetal acceleration to build up gradually, which is more comfortable for vehicle occupants.  
Keep water drainage in mind while considering all of the available cross-section options

11. Vertical Alignment
Reduced Speed
Increased Speed

12. Vertical Alignment Grade
measure of inclination or slope, rise over the run
Cars: negotiate 4-5% grades without significant speed reduction
Trucks: significant speed changes
5% increase on short descending grades
7% decrease on short ascending grades
10% grade means that the elevation of the roadway increases by 10 ft for every 100 ft of horizontal distance

13. Grade Considerations
Maximum grade – depends on terrain type, road functional class, and design speed
Rural Arterials
Terrain
60mph
70mph
Level 3%
Rolling 4%
Mountainous 6% 5%

14. Grade Considerations Critical length of grade
Maximum length which a loaded truck can travel without unreasonable speed reduction
Based on accident involvement rates with 10mph speed reduction as threshold

15. Grade Considerations General Design Speed Reduction
For a given speed and speed reduction, the steeper the grade, the shorter the critical length of grade

16. Vertical Curves To provide transition between two grades Consider
Drainage (rainfall)
Driver safety (SSD)
Driver comfort
Use parabolic curves
Crest vs Sag curves

17. Vertical Curves

18. Vertical Curves Given  Develop the actual shape of the vertical curve
G1, G2: initial & final grades in percent
L: curve length (horizontal distance)
 Develop the actual shape of the vertical curve
PVI
point of vertical intersection
PVI: point where the two tangents intersect
VPC and VPT are the points along the roadway where the vertical curve begins and ends. 
PVC is generally designated as the origin for the curve and is located on the approaching roadway segment.
PVT serves as the end of the vertical curve and is located at the point where the vertical curve connects with the departing roadway segment.
point of vertical curvature
point of vertical tangency
G1%
G2%

19. Vertical Curves
Define curve so that PVI is at a horizontal distance of L/2 from PVC and PVT
Provides constant rate of change of grade: A
X=0, Ep=EPVC
X=L, Ep=EPVT=
X=L/2 =
Xhighest elev.: dEp/dx -> 0
G1%
G2%

20. Example G1 = 2% G2 = -4% Design speed = 70 mph
Is this a crest or sag curve?
What is A?

21. Vertical Curves Major control for safe operation is sight distance
MSSD should be provided in all cases (use larger sight distance where economically and physically feasible)
For sag curves, also concerned with driver comfort (large accelerations due to both gravitational and centrifugal forces)

22. Crest Vertical Curves
Critical length of curve, L, is where sight line is tangent to the crest
Assume driver eye height (H1) = 3.5 ft and object height (H2) = 2.0 ft and S=MSSD

28. Sag VC - Design Criteria
Headlight sight distance
Rider comfort
Drainage control
Appearance