Bridge geometry - Computational Analysis and Design of Bridge Structures - page 553

the curved girders in horizontal view, and Figure 18.2 shows the elevation
view of these haunch girders.
18.3 curve calculatIons
Given the most commonly used curve types, such as straight lines, arcs,
spirals, and parabolas, calculations needed to obtain a point on curve are
simple and straightforward mathematically. Challenges, however, arise
from the engineering depiction of a curve in a way of easy representing
actual roadway curves in three-dimensional (3D) space and accurately
controlling geometries of any bridge component. To model a bridge in 3D
based on a spatial curve preset by roadway, defining an appropriate curve
model is fundamental. Procedures for sampling points along a 3D curve or
road surface can then be established.
18.3.1 Bridge mainline curve model
The bridge mainline, or the deck centerline, is the reference line of modeling
a bridge in 3D. Geometries and locations of most bridge components can be
derived or located by referring to the bridge mainline. Often the mainline
of a bridge can be the same as the centerline of the roadway. Vertically, it
is aligned on top of the deck. Figure 18.3 shows an example of a bridge
mainline. As a spatial curve shown in Figure 18.3, geometries of a bridge
mainline contain plane curves, or the horizontal curves, and vertical curves
(Wang and Fu 2013).
Following the practices of roadway design and route locations (Hickerson
1959), spatial curves can be described in horizontal (plane) and vertical
curves separately. A pure mathematical description of a roadway curve in
spatial is not practical at all in road engineering. Therefore, a bridge main-
line can be described separately by its (1) plane curves and (2) vertical curves.
Plane curves are compound curves in general, which may contain straight
lines, arcs, and spirals with smooth connections from one to another.
Figure 18.4 shows a plane curve as an example. Smooth connection from one
component to another means tangents at connecting points are continuous at
least. In most cases, as the example shown in Figure 18.4, smooth connection
can further mean that curvatures at connecting points are continuous. Under
such a restriction, a spiral segment is needed to connect a straight line and
an arc. The connecting of straight line and arc, especially those with small
radii, which causes the curvature changes from zero to a constant, is dis-
couraged. Having parameters of each connecting components including the
type and length of curve segment and radius of an arc or ending radius of a
spiral, plus the starting point and tangent defined, geometric properties, such
as location, tangent, and curvature at any given curve length ordinate, can