Geometric Design (II)

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)

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.

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]

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. 

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. 

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. 

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

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.  

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

Vertical Alignment Reduced Speed Increased Speed

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

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%

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

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

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

Vertical Curves

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%

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%

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

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)

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

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