48
STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS
Two additional équations, deduced from équations (2.55) for 0O = 0 and z = z0,
are respectively
f=^sinh(^
(2.59)
W
\Ix /
Fx
V.
2o = — cosn
w
Fx
w
(2.60)
FYom the overall equilibrium conditions, équations (2.51) and (2.52), it follows
that Fx = F cos ô and wl — F sin 6. The ratio of these latter équations leads to
tan 0 = sinh I — )
(2-61)
For a fixed value of zq and w and an initially fixed value of 0 = 0r, then the three
initial values Fx = Fxr, x — xr, and t = lT can be calculated from équations
(2.59), (2.60), and (2.61). For values of 0
0r, the corresponding values of
Fx,x,£. and v are calculated from équations (2.58)-(2.61). The restoring force
of a single cable is simply
q(y) = Fx = Fx(v)
(2.62)
One can now deduce that the restoring force for the pair of identical cables (the
single cable of Figure 2.20 and its mirror image across the plane v = 0) is given
by the superposition of the results just derived for the single cable, or
q(v) = Fx(y) - Fx(-v)
(2.63)
To facilitate the calculations and interprétations of the restoring forces, équations (2.58) through (2.61) can be cast in nondimensional form with the aid of
the following définitions:
= —;
WZO
—
Fxr
r xr =
W20
(2.64a)
e = L
2o
^0
(2.64b)
X
= —;
xr
zo
_
V
= —;
v = —
20
20
(2.64c)
With équations (2.64), équations (2.58)-(2.61) become, after some rearrangement,
=- = sinh- (tan£)
(2.65)