Lecture 4 Part 1 Horizontal Curves

The Design of Highway Alignments

Norman W. Garrick Lecture 4 Part 1

Street and Highway Design

Blue Ridge Parkway

Trinity College Lower Long Walk

Storrs Heights An un-engineered alignment

A Driveway in Willington, CT Likely engineered alignment

Geometric Design of Highways The engineering aspects of alignment design is usually referred to as geometric design

Highway alignment is in reality a three-dimensional problem

Design & construction is difficult in 3-D so highway design is typically treated as three 2-D problems: Horizontal alignment, vertical alignment, cross-section

This often create a dysfunctional situation when the designer forgets that the three dimensions must work together as one alignment – the Blue Ridge Parkway and the Trinity Lower Long Walk shows how the three dimensions can be coordinated to good overall effect

Storrs Heights and the Willington driveway illustrate a more naturalistic alignment

Austin, TX

Near Cincinnati, OH

Components of Highway Design

Plan View

Profile View

Horizontal Alignment

Vertical Alignment

Horizontal Alignment

Today we focus on •  Components of the horizontal alignment •  Properties of a simple circular curve •  Properties of a spiral curve

Horizontal Alignment

Tangents Curves

Tangents & Curves

Tangent

Curve

Tangent to Circular Curve

Tangent to Spiral Curve to Circular 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 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

Definition used in highway design

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

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

(note: D is not Δ )

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 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]

No Spiral

With 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

As with circular curve the central angle is also important for spiral Recall for circular curve

Δc = Lc D / 100

But for spiral

Δs = Ls D / 200

Central (or Deflection) Angle of Euler Spiral

The total deflection angle for a spiral/circular curve system is

Δ = Δc + 2 Δs

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

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

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

i)  k for the spiral, ii)  ii) length of each spiral leg, iii)  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