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3. Nomenclature & Parts

A single figure showing a spiraled horizontal curve with all its nomenclature, parts, and points labeled can be confusing. To make things a little more clear, we will use a series of figures emphasizing and explaining different attributes. Although ours will be a symmetric curve system, the same principles apply to differing entrance and exit spirals, as long as sufficient basic information is known.

Let's start with two tangents and a circular curve of radius Rc and length L connecting them, Figure E-6.

Figure E-6
Simple curve between tangents

To make room for spirals inclusion requires pushing the circular arc in from the tangents and shortening its length. Adding two spirals, each Ls long, reduces the circular arc length to Lc. This results in the TS-SC-CS-ST curve system, Figure E-7. The OBC and OEC are offset positions of the BC and EC. Radial lines from the OBC and OEC to the arc's center of curvature are parallel with perpendicular lines at the TS and ST respectively.

Ls: Spiral length
Lc: Reduced arc length

Rs: Spiral radius (varies)
Rc: Circular arc radius

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

OBC: Offset BC
OEC: Offset EC
Figure E-7
Spiraled horizontal curve

The radius of the entrance spiral, Rs, decreases uniformly from infinity at the TS to the circular arc radius, Rc, at the SC. The radius of the exit spiral increases uniformly from the arc radius at the CS to infinity at the ST.

The entrance spiral's degree of curvature, Ds, varies from 0°00'00" at the TS to the circular arc's, Dc, at the SC. Because it changes uniformly over the spiral's length, its rate is Equation E-1:

Equation E-1 k: Spiral's Degree of Curvature rate change
Dc: Circular arc D
Ls: Spiral length

 

Although reversed, the exit spiral in a symmetric system has the same k.

Figure E-8 shows the central angles (Δ) for the spirals and circular arc.

Δs: Spiral Central Angle
Δc: Circular Curve Central Angle
Figure E-8
Central Angles

Because the radial lines at the TS and OBC are parallel, the central angle of the entrance spiral is the same as the central arc angle from the OBC to the SC. The same is true for the exit spiral. The spiral angle is determined from Equation E-2.

Equation E-2

The original circular arc's central angle is reduced by the spirals', Equation E-3.

Equation E-3 Δc: Reduced arc central angle
Δ: Total central angle
Δs: Spiral central angle

 

And its shortened length is determined from Equation E-4 (modified from Chapter C. Horizontal Curves Equation C-2).

Equation E-4

Location of the TS from the PI along with tangent distances and offsets from the TS to the OBC and SC are shown in Figure E-9.


 
(a) (b)
LCs: Spiral long chord
A: Spiral deflection angle

Ts: Spiral tangent
Xo: Tangent distance to OBC
o: Tangent offset to OBC

 X: Tangent distance to SC
 Y: Tangent offset to SC
Figure E-9
Tangent Distances

 

 

 

 

The distances and offsets are computed using Equations E-5 through E-10

Equation E-5
Equation E-6
Equation E-7
Equation E-8
Equation E-9
Equation E-10

Because spirals are short and very flat, arc distances are often used to approximate chord distances. The is the case with Equations E-5 through E-7: the spiral length, Ls, is used for the spiral chord, LCs. In most cases, this assumption will be fine, however, we'll discuss a more exact, though more complex, spiral computing method also.