Cable-stayed bridges - Computational Analysis and Design of Bridge Structures - page 346

A B B ′

P P

Sag

L = cable span Δ L

Figure 11.19 Sag of a horizontal cable (to show axial deformation under force P ).

Figure 11.19 illustrates the cable geometry in reality compared to what it is
modeled mathematically.
The axial stiffness of a cable is simply defined as the axial force required
for causing a unit axial deformation or the elongation along the axis of
two anchors. When a cable is straight, the total elongation is the deforma-
tion of the cable so that the axial force that causes such an elongation is
higher. For example, Δ L as shown in Figure 11.19 is the deformation of
cable AB if sag does not exist. When cable sags away from its axis, not
all of the elongation is due to deformation, yet it is due to the geometry
change, so the axial force required to cause the same amount of elongation
is lower. As shown in Figure 11.19, Δ L , the elongation of the sagged cable
AB is the sum of the cable deformation and shortage of cable geometry.
That is how a sagged cable behaves as if the material has a lower Young's
modulus.
One fact about the cable sag is that the higher the existing axial force is,
the smaller the sag and thus the stiffness of the cable is closer to a straight
cable.
In preliminary analyses, one cable is usually meshed into one element by
its two anchors. In a regular FEA package, the stiffness of such a cable is
calculated based on a straight line between two anchor points. Its stiffness
is, therefore, overcalculated. The Ernst formula (Ernst 1965), as shown in
Equation 11.10, is usually adopted to calculate the cable's equivalent stiff-
ness or Young's modulus based on a given cable stress.

E
H AE

E

eq =

(11.10)

2 3

1 + [( ω ) / ( 12 P )]

where:
E is the Young's modulus of a cable, in kN/m 2
ω is the unit weight of the cable, in kN/m
H is the cable span in horizontal direction, in m
A is the cable area, in m 2
P is the cable force, in kN

Cable forces will be redistributed in the next phases after they are initially
jacked, or, in another words, cable forces are never constant from stage to