Bridge geometry - Computational Analysis and Design of Bridge Structures - page 556

360′ 385′ 410′ 435′

600′

330′

310′

295′

285′

275′

270′

265′

260′

255′

250′

245′

Figure 18.9 Perspective view mainline and cross sections.

18.3.4 spiral calculation
Spirals used in roadway plane curves are for making a curvature transition.
A spiral for this purpose is simply defined as a curve whose curvature is
proportional to curve length, or the curvature change-to-curve length ratio
is constant:

c − c

e s

c l

( ) =

c + l (18.1)

s

L

where:
c denotes curvature
subscripts s and e denote starting and ending of spiral, respectively
L is the total length of spiral
l is the curve length ordinate

From the definition of curvature, reciprocal of curve radius, the differential
of sweeping angle θ is

(18.2)

d θ = ( ) c l dl

Integrating Equation 18.2 with the substitution of 18.1 and considering
zero initial sweeping angle, sweeping angle at any given curve length ordi-
nate can be obtained as

c −

2

c
L

e s 2

θ ( l = c l + l (18.3)

s

Taking a local coordinate system as shown in Figure 18.10, the differentials
of ordinates x and y can be written as

c − c

L

c − c

L

e s 2 e s 2

dx = cos c l + l dl dy ; = sin c l + l dl (18.4)

s s

2 2