The Design of Modern Steel Bridges - TEDI

196 The Design of Modern Steel Bridges
ends. If the weight of the cable is uniform along its length, the deflected shape
of the cable will be a catenary. The analysis of the catenary shape is much
more complicated than that of a parabola, which would have been the deflected
shape of the cable if its weight were uniformly distributed on the horizontal
projected length of the cable. The error caused by this approximation on the
length, sag or tension of the cable is found to be insignificant for practical
implications in even the extreme real-life cases. The relationship between the
sag and the tension in the cable can be obtained by equating the bending
moment at mid-length of the cable to zero and is given by
where
f ¼ wL2
8H
¼ gC2
8s
f ¼ vertical sag
w ¼ weight of cable per unit horizontal length
L ¼ horizontal span between cable anchorages
H ¼ horizontal component of the cable tension
g ¼ weight per unit volume of the cable material
C ¼ inclined distance between cable anchorages
s ¼ tensile stress in the cable.
ð7:1Þ
It may be noted that the vertical sag of the cable is inversely proportional to
the tension in it.
Due to the sag the curved length of the cable is larger than the chord length by
8f 2
3L sec3 y
where y is the angle of the cable with the horizontal.
Due to the tension s the cable will have stretched elastically by an amount
Cs/E; hence the length of the unstressed cable (i.e. lying on the ground)
should be
C þ
3C sec2 Cs
ð7:2Þ
y E
The self-weight of the cable reduces the cable tension in the bottom half of
the cable and increases the cable tension in the top half; the tension in the cable
will be maximum at its top end and minimum at the bottom; the maximum
tension is given by
where
8f 2
H 1 þ h
( )
2
1=2
4f
þ
L L
H ¼ horizontal component of the cable tension at the ends and
h ¼ vertical distance between the cable anchorages.