WikiEngineer :: Transportation

Geometric Design - Spiral Curve Design

Spiral curves (aka transition or easement curves) are generally used to provide a gradual transition in curvature from a straight section of road to a curved section (or from tangents to circular curves). Figure 1 shows the placement of spiral curves in relation to circular curves. Figure 2 shows the components of a spiral curve. Spiral curves are necessary on high-speed roads from the standpoint of comfortable operation and gradually bringing about the full superelevation ? of the curves. A spiral should be utilized with a circular curve with a superelevation ? of 3% or greater. Spiral curves were developed for locomotives and are still used widely today in the industry.

Figure 1: The Placement of a Spiral Curve

Figure 2: Components of a Spiral Curve^[2]

Definitions ^[1]:

SCS PI = Point of intersection of main tangents.
TS = Point of change from tangent to spiral curve.
SC = Point of change from spiral curve to circular curve.
CS = Point of change from circular curve to spiral curve.
ST = Point of change from spiral curve to tangent.
LC = Long chord.
LT = Long tangent.
ST = Short tangent.
PC = Point of curvature for the adjoining circular curve.
PT = Point of tangency for the adjoining circular curve.
T_s = Tangent distance from TS to SCS PI or ST to SCS PI.
E_s = External distance from the SCS PI to the center of the circular curve.
R_c = Radius of the adjoining circular curve.
D_c = Degree of curve of the adjoining circular curve, based on a 100 foot arc (English units only).
D = Degree of curve of the spiral at any point, based on a 100 foot arc (English units only).
l = Spiral arc from the TS to any point on the spiral (l = l_s at the SC).
l_s = Total length of spiral curve from TS to SC.
L = Length of the adjoining circular curve.
θ_s = Central (or spiral) angle of arc l_s.
∆ = Total central angle of the circular curve from TS to ST.
∆_c = Central angle of circular curve of length L extending from SC to CS.
p = Offset from the initial tangent.
k = Abscissa of the distance between the shifted PC and TS.
Y_c = Tangent offset at the SC.
X_c = Tangent distance at the SC.
x and y = coordinates of any point on the spiral from the TS.

Equations to use:

D_c = {{18000\over \pi}\Bigg/ R_c}

D_c = {200 \times {\theta_s\over l_s}}

l_s = {200 \times {\theta_s\over D_c}}

\theta_s = {{l_s \times D_c}\over 200}

\Delta = {{180\times L}\over {\pi \times R_c}}

\theta_s\hbox{ (decimal degrees) }= {180\over \pi}\times \theta_s\hbox{ (radians)}

X_c = {\left(l_s\over 100\right)\times \left(100-0.0030462\theta^{\raise1pt 2}_s \right)}

Y_c = {\left(l_s\over 100\right)\times \left(0.58178\theta_s-0.000012659\theta^{\raise1pt 3}_s \right)}

p = {Y_c - R_c\times \left(1.0-\cos \theta_s \right)}

A = {{20000\times \theta_s}\over l^{\raise1pt 2}_s}

k = {{l_s\over 2}-0.000127A^2 \times \left(l_s\over 100\right)^5}

T_s = {\left(R_c + p\right)\times tan {\Delta\over 2} + k}

E_s = {\left(R_c+p\right)\times arcsec {\Delta\over 2} + p}

LT = {X_c - \left(Y_c\times \cot \theta_s\right)}

ST = {Y_c\over {\sin \theta_s}}

LC = {l_s-0.00034A^2\times \left({l_s\over 100}\right)^5}

\Delta_c = {\Delta-2\times \theta_s}

Definitions:

References

Iowa Department of Transportation
- http://www.iowadot.gov/design/dmanual/manual.html
Hickerson, T.F., Route Location and Design. 2nd ed. (New York: McGraw-Hill, Inc., 1964), pg 168.

Spiral Curves

Geometric Design - Spiral Curve Design

Definitions [1]:

Equations to use:

Definitions:

References

Definitions ^[1]: