HORIZONTAL ALIGNMENT Spring 2015

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

Simple Curve Circular Curve Tangent PC PT Point of Tangency Point of Curvature

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

Design Elements of Horizontal Curves Deflection Angle Also known as Δ Deflection Angle

Design Elements of Horizontal Curves Larger D = smaller Radius

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

Design Elements of Horizontal Curves LC=Length of Long Cord

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

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

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

Superelevation Design Desirable superelevation: for R > Rmin Where, V= design speed in ft/s or m/s g = gravity (9.81 m/s2 or 32.2 ft/s2) 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.

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 fmax Superelevation is introduced only after the maximum side friction is used

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 emax 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

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

Design of Horizontal Alignment 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%

Method 1 Centerline

Method 2 Inside Edge

Method 3 Outside Edge

Method 4 Straight Cross Slope

Which Method? 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

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

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

Transition Design Control Tangent Runout

Transition Design Control Superelevation Runoff

Transition Design Control

Transition Design Control http://techalive.mtu.edu/modules/module0003/Superelevation.htm

Minimum Length of Superelevation Runoff

Minimum Length of Superelevation Runoff  = relative gradient in previous overhead

Minimum Length of Superelevation Runoff Values for n1 and bw in equation

Minimum Length of Tangent Runout See Exhibit 3-32 for values of Lt and Lr

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

Superelevation Runoff

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 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

Transition Curves -Spirals Need to verify for maximum and minimum lengths

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

Sight distance on Horizontal Curve 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.

Included for your benefit Example Application Included for your benefit

Selection of fdesign and edesign (Method 5) fmax (for the design speed) Side Friction Factor e = 0 fdesign emax (for the design speed) Reciprocal of Radius 1/R

Selection of fdesign and edesign Rf = V2/(gfmax) Rmin = V2/[g(fmax + emax)] fmax Side Friction Factor e = 0 fdesign emax Ro = V2/(gemax) Reciprocal of Radius 1/R R0: f = 0, e = emax

Selection of fdesign and edesign fdesign = α(1/R)+β(1/R)2 fmax (for the design speed) Side Friction Factor α = fmaxRmin[1-{Rmin/(R0-Rmin)}] e = 0 fdesign β = fmaxRmin3/(R0-Rmin) emax (for the design speed) Reciprocal of Radius 1/R

Superelevation Design for High Speed Rural and Urban Highways

Example: Design Speed: 100 km/h fmax = 0.128 emax = 0.06 Question? What should be the design friction factor and design superelevation for a curve with a radius of 600 m?

1. Compute Rf, R0, and Rmin: Rf = V2/(gfmax) = 27.782 / (9.81 x 0.128) = 615 m R0 = V2/(gemax) = 27.782 / (9.81 x 0.06) = 1311 m Rmin = V2/[g(fmax + emax)] = 27.782 / [9.81(0.128+0.06)] Rmin = 418 m

Selection of fdesign and edesign (example) fmax = 0.128 Side Friction Factor e = 0 fdesign emax = 0.06 1 / 1311 1 / 615 1 / 418 1/R

2. Compute α and β: α = 0.128 x 418 x [1 – 418 / (1311 – 418) ] = 28.45 m β = 0.128 x 4183 / (1311 – 418) = 10502 m2 3. Compute fdesign and edesign : First, estimate the right-hand side of equation for designing superelevation e + f = V2/(gR) = 27.782 / (9.81 x 600) = 0.131 Then, fdesign = 28.45 / 600 + 10502 / 6002 = 0.076 edesign = 0.131 – 0.076 = 0.055 (< emax = 0.06)

Selection of fdesign and edesign (example) fmax = 0.128 Side Friction Factor e = 0 fdesign emax = 0.06 0.076 1 / 1311 1 / 615 1 / 418 1/R 1 / 600

Selection of fdesign and edesign (example) R=600 ft