2 Horizontal Alignment Geometric Elements of Horizontal Curves Transition or Spiral CurvesSuperelevation DesignSight Distance
3 Simple Curve Circular Curve Tangent PC PT Point of Tangency Point of Curvature
4 Curve with Spiral Transition Circular CurveSpiralTangentCSSCSTTSCurve to SpiralSpiral to CurveSpiral to TangentTangent to Spiral
5 Design Elements of Horizontal Curves Deflection AngleAlso known as ΔDeflection Angle
6 Design Elements of Horizontal Curves Larger D = smaller Radius
7 Design Elements of Horizontal Curves E=External DistanceM=Length of Middle Ordinate
8 Design Elements of Horizontal Curves LC=Length of Long Cord
9 Basic FormulasBasic Formula that governs vehicle operation on a curve:Where,e = superelevationf = side friction factorV = vehicle speed (mph)R = radius of curve (ft)
10 Basic Formulas Minimum radius: Where, e = superelevation f = side friction factorV = vehicle speed (mph)R = radius of curve (ft)
11 Minimum Radius with Limiting Values of “e” and “f”
12 Superelevation Design Desirable superelevation:for R > RminWhere,V= design speed in ft/s or m/sg = gravity (9.81 m/s2 or 32.2 ft/s2)R = radius in ft or mVarious methods are available for determining the desirable superelevation, but the equation above offers a simple way to do it. The other methods are presented in the next few overheads.
13 Methods for Estimating Desirable Superelevation Superelevation and side friction are directly proportional to the inverse of the radius (straight relationship between 1/R=0 and 1/R =1/Rmin)Method 2:Side friction is such that a vehicle traveling at the design speed has all the acceleration sustained by side friction on curves up to those requiring fmaxSuperelevation is introduced only after the maximum side friction is used
14 Method 3: Method 4: Method 5: Superelevation is such that a vehicle traveling at the design speed has all the lateral acceleration sustained by superelevation on curves up to those required by emaxNo side friction is provided on flat curvesMay result in negative side frictionMethod 4:Same approach as Method 3, but use average running speed rather than design speedUses speeds lower than design speedEliminate problems with negative side frictionMethod 5:Superelevation and side friction are in a curvilinear relationship with the inverse of the radius of the curve, with values between those of methods 1 and 3Represents a practical distribution for superelevation over the range of curvatureThis is the method used for computing values shown in Exhibits 3-25 to 3-29
15 Five Methods fmax e = 0 emax f M2 M1 M5 M3 1/R M4 Side Friction Factor Reciprocal of Radius1/RM4
16 Design of Horizontal Alignment Important considerations:Governed by four factors:Climate conditionsTerrain (flat, rolling, mountainous)Type of area (rural vs urban)Frequency of slow-moving vehiclesDesign should be consistent with driver expectancyMax 8% for snow/ice conditionsMax 12% low volume roadsRecurrent congestion: suggest lower than 6%
21 Which Method?In overall sense, the method of rotation about the centerline (Method 1) is usually the most adaptableMethod 2 is usually used when drainage is a critical component in the designIn the end, an infinite number of profile arrangements are possible; they depend on drainage, aesthetic, topography among others
22 Example where pivot points are important Bad designPivot pointsGood designMedian width15 ft to 60 ft
23 Transition Design Control The superelevation transition consists of two components:The superelevation runoff: length needed to accomplish a change in outside-lane cross slope from zero (flat) to full superelevationThe tangent runout: The length needed to accomplish a change in outside-lane cross slope rate to zero (flat)
34 Transition Curves -Spirals All motor vehicles follow a transition path as it enters or leaves a circular horizontal curve (adjust for increases in lateral acceleration)Drivers can create their own path or highway engineers can use spiral transitional curvesThe radius of a spiral varies from infinity at the tangent end to the radius of the circular curve at the end that adjoins the curve
35 Transition Curves -Spirals Need to verify for maximum and minimum lengths
36 Transition Curves Superelevation runoff should be accomplished on the entire length of the spiral curve transitionEquation for tangent runout when Spirals are used:
37 Sight distance on Horizontal Curve The sight distance is measured from the centerline of the inside laneNeed to measure the middle-ordinate values (defined as M)Values of M are given in Exhibit 3-53Note: Now M is defined as HSO or Horizontal sightline offset.
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