Bridge geometry - Computational Analysis and Design of Bridge Structures - page 557

Center

r = 1/ e

Y

P e

L

Center

r = 1/ c

p

Center

L

r = 1/ s y

θ

P s

X

x

0

Figure 18.10 Spiral curve and its local coordinate system.

Therefore, local coordinates x and y at any given curve length are the inte-
gration forms of Equation 18.4:

l l

c − c

L

c − c

L ∫ ∫

e s 2 e s 2

x = cos c l + l dl y , = sin c l + l dl (18.5)

s s

2 2

0 0

Given a curve length ordinate l, l , point on a spiral and curve properties can
be computed by Equations 18.1, 18.3, and 18.5. When computing coordi-
nates by Equation 18.5, Simpson's Rule can be used as a generic numerical
integration method.
As a spiral is a part of compound plane curve as usual, local coordi-
nates and tangent at any point on a spiral as shown in Figure 18.10 have
to be transformed to global coordinate system by a simple rotation and a
translation.

18.3.5 vertical parabola calculation
Parabolas used in roadway vertical curves are for making a grade transition.
Similar to spiral, the grade of a vertical parabola is proportional to hori-
zontal curve length, or the grade change to horizontal length is constant:

dy
dx

g − g

e s

g x

( ) =

= g + x (18.6)

s

X