Horizontal Curves

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Geometric Design of Highways:

Geometric Design of Highways Highway Alignment is a three-dimensional problem Design & Construction would be difficult in 3-D so highway design is split into three 2-D problems Horizontal alignment, vertical alignment, cross-section

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Austin, TX

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Near Cincinnati, OH

Components of Highway Design:

Components of Highway Design Plan View Profile View Horizontal Alignment Vertical Alignment

Horizontal Alignment:

Horizontal Alignment Today’s Class: Components of the horizontal alignment Properties of a simple circular curve

Horizontal Alignment:

Horizontal Alignment Tangents Curves

Tangents & Curves:

Tangents & Curves Tangent Curve Tangent to Circular Curve Tangent to Spiral Curve to Circular Curve

Layout of a Simple Horizontal Curve:

Layout of a Simple Horizontal Curve R = Radius of Circular Curve BC = Beginning of Curve (or PC = Point of Curvature) EC = End of Curve (or PT = Point of Tangency) PI = Point of Intersection T = Tangent Length (T = PI – BC = EC - PI) L = Length of Curvature (L = EC – BC) M = Middle Ordinate E = External Distance C = Chord Length Δ = Deflection Angle

Properties of Circular Curves:

Properties of Circular Curves Degree of Curvature Traditionally, the “steepness” of the curvature is defined by either the radius (R) or the degree of curvature (D) In highway work we use the ARC definition Degree of curvature = angle subtended by an arc of length 100 feet

Degree of Curvature Equation for D:

Degree of Curvature Equation for D Degree of curvature = angle subtended by an arc of length 100 feet By simple ratio: D/360 = 100/2*Pi*R Therefore R = 5730 / D (Degree of curvature is not used with metric units because D is defined in terms of feet.)

Length of Curve:

Length of Curve By simple ratio: D/ Δ = ? D/ Δ = 100/L L = 100 Δ / D Therefore L = 100 Δ / D Or (from R = 5730 / D, substitute for D = 5730/R) L = Δ R / 57.30 (D is not Δ .)

Properties of Circular Curves:

Properties of Circular Curves Other Formulas… Tangent: T = R tan( Δ/2) Chord: C = 2R sin(Δ/2) Mid Ordinate: M = R – R cos(Δ/2) External Distance: E = R sec(Δ/2) - R

Spiral Curve:

Spiral Curve A transition curve is sometimes used in horizontal alignment design It is used to provide a gradual transition between tangent sections and circular curve sections. Different types of transition curve may be used but the most common is the Euler Spiral. Properties of Euler Spiral (reference: Surveying: Principles and Applications, Kavanagh and Bird, Prentice Hall ]

Characteristics of Euler Spiral:

Degree of Curvature of a spiral at any point is proportional to its length at that point The spiral curve is defined by ‘ k ’ the rate of increase in degree of curvature per station (100 ft) In other words, k = 100 D/ Ls   Characteristics of Euler Spiral

Characteristics of Euler Spiral:

Degree of Curvature of a spiral at any point is proportional to its length at that point The spiral curve is defined by ‘ k ’ the rate of increase in degree of curvature per station (100 ft) In other words, k = 100 D/ Ls   Characteristics of Euler Spiral

Central (or Deflection) Angle of Euler Spiral:

As with circular curve the central angle is also important for spiral Recall for circular curve Δ c = L c D / 100 But for spiral Δ s = L s D / 200   Central (or Deflection) Angle of Euler Spiral The total deflection angle for a spiral/circular curve system is Δ = Δ c + 2 Δ s

Length of Euler Spiral:

Length of Euler Spiral Note: The total length of curve (circular plus spirals) is longer than the original circular curve by one spiral leg

Example Calculation – Spiral and Circular Curve:

Example Calculation – Spiral and Circular Curve The central angle for a curve is 24 degrees - the radius of the circular curve selected for the location is 1000 ft.   Determine the length of the curve (with no spiral) L = 100 Δ / D or L = Δ R / 57.30 = 24*1000/57.30 = 418.8 ft R = 5730 / D >> D = 5.73 degree

Example Calculation – Spiral and Circular Curve:

Example Calculation – Spiral and Circular Curve The central angle for a curve is 24 degrees - the radius of the circular curve selected for the location is 1000 ft.   If a spiral with central angle of 4 degrees is selected for use, determine the k for the spiral, ii) length of each spiral leg, iii) total length of curve Δs = 4 degrees Δs = Ls D / 200 >> 4 = Ls * 5.73/200 >> Ls = 139.6 ft k = 100 D/ Ls = 100 * 5.73/ 139.76 = 4.1 degree/100 feet Total Length of curve = length with no spiral + Ls = 418.8+139.76 = 558.4 feet

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