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Spring 2015

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Horizontal Alignment Geometric Elements of Horizontal Curves Superelevation Design Transition or Spiral Curves Sight Distance

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PC PT Circular Curve Tangent Point of Curvature Point of Tangency

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SC ST Circular Curve Tangent Tangent to Spiral Spiral to Tangent Spiral TS Spiral to Curve CS Curve to Spiral

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Design Elements of Horizontal Curves Deflection Angle Also known as Δ

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Design Elements of Horizontal Curves Larger D = smaller Radius

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Design Elements of Horizontal Curves E=External Distance M=Length of Middle Ordinate

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Design Elements of Horizontal Curves LC=Length of Long Cord

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Basic Formulas Where, e = superelevation f = side friction factor V = vehicle speed (mph) R = radius of curve (ft) Basic Formula that governs vehicle operation on a curve:

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Basic Formulas Where, e = superelevation f = side friction factor V = vehicle speed (mph) R = radius of curve (ft) Minimum radius:

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Minimum Radius with Limiting Values of “e” and “f”

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Desirable superelevation: for R > R min Where, V= design speed in ft/s or m/s g = gravity (9.81 m/s 2 or 32.2 ft/s 2 ) R = radius in ft or m Various 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.

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Method 1: ◦ Superelevation and side friction are directly proportional to the inverse of the radius (straight relationship between 1/R=0 and 1/R =1/R min ) 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 f max ◦ Superelevation is introduced only after the maximum side friction is used

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Method 3: ◦ 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 e max ◦ No side friction is provided on flat curves ◦ May result in negative side friction Method 4: ◦ Same approach as Method 3, but use average running speed rather than design speed ◦ Uses speeds lower than design speed ◦ Eliminate problems with negative side friction Method 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 3 ◦ Represents a practical distribution for superelevation over the range of curvature ◦ This is the method used for computing values shown in Exhibits 3-25 to 3-29

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e = 0 e max Reciprocal of Radius Side Friction Factor Five Methods f max M2 M1 M3 M5 M4 1/R f

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Important considerations: ◦ Governed by four factors: Climate conditions Terrain (flat, rolling, mountainous) Type of area (rural vs urban) Frequency of slow-moving vehicles ◦ Design should be consistent with driver expectancy ◦ Max 8% for snow/ice conditions ◦ Max 12% low volume roads ◦ Recurrent congestion: suggest lower than 6%

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In overall sense, the method of rotation about the centerline (Method 1) is usually the most adaptable Method 2 is usually used when drainage is a critical component in the design In the end, an infinite number of profile arrangements are possible; they depend on drainage, aesthetic, topography among others

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Median width Pivot points Example where pivot points are important Bad design Good design 15 ft to 60 ft

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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 superelevation ◦ The tangent runout: The length needed to accomplish a change in outside-lane cross slope rate to zero (flat)

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Tangent Runout

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Superelevation Runoff

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Transition Design Control

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= relative gradient in previous overhead

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Values for n 1 and b w in equation

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See Exhibit 3-32 for values of L t and L r

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Location: 1/3 on curve Location: 2/3 on tangent

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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 curves The 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

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Need to verify for maximum and minimum lengths

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Superelevation runoff should be accomplished on the entire length of the spiral curve transition Equation for tangent runout when Spirals are used:

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The sight distance is measured from the centerline of the inside lane Need to measure the middle-ordinate values (defined as M) Values of M are given in Exhibit 3-53 Note: Now M is defined as HSO or Horizontal sightline offset.

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Included for your benefit

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e = 0 e max (for the design speed) Reciprocal of Radius Side Friction Factor Selection of f design and e design (Method 5) f max (for the design speed) f design 1/R f

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e = 0 e max Reciprocal of Radius Side Friction Factor Selection of f design and e design f max f design R f = V 2 /(gf max ) R o = V 2 /(ge max ) R 0 : f = 0, e = e max R min = V 2 /[g(f max + e max )] 1/R f

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e = 0 e max (for the design speed) Reciprocal of Radius Side Friction Factor Selection of f design and e design f max (for the design speed) f design f design = α(1/R)+β(1/R) 2 α = f max R min [1-{R min /(R 0 -R min )}] β = f max R min 3 /(R 0 -R min ) 1/R f

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Superelevation Design for High Speed Rural and Urban Highways

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Example: Design Speed: 100 km/h f max = e max = 0.06 Question? What should be the design friction factor and design superelevation for a curve with a radius of 600 m?

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1. Compute R f, R 0, and R min : R f = V 2 /(gf max ) = / (9.81 x 0.128) = 615 m R 0 = V 2 /(ge max ) = / (9.81 x 0.06) = 1311 m R min = V 2 /[g(f max + e max )] = / [9.81( )] R min = 418 m

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e = 0 e max = 0.06 Side Friction Factor Selection of f design and e design (example) f max = f design 1 / / / 418 1/R f

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2. Compute α and β: α = x 418 x [1 – 418 / (1311 – 418) ] = m β = x / (1311 – 418) = m 2 3. Compute f design and e design : First, estimate the right-hand side of equation for designing superelevation e + f = V 2 /(gR) = / (9.81 x 600) = Then, f design = / / = e design = – = (< e max = 0.06)

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e = 0 e max = 0.06 Side Friction Factor f max = f design 1 / / / / Selection of f design and e design (example) 1/R f

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Selection of f design and e design (example) R=600 ft

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