(^334) A Textbook of Engineering Mechanics
l 1 = Horizontal length between A and C, and
l 2 = Horizontal length between C and B.
Let us imagine the portion CB of the string to be extended to CB 1 , such that the new support B 1
is at the same level as that of A. Similarly, imagine the portion AC of the cable to be cut short to A 1 C,
such that the new support A 1 is at the same level as that of B. From the geometry of the figure, we find
that the cable ACB 1 has a span of 2l 1 and a central dip of (yc + d), whereas the cable A 1 CB has a span
of 2l 2 and a central dip of yc.
We have discussed in Art. 15.8 that the length of the cable,
8 2
3
Ll yc
l
=+
∴ Length of the cable ACB 1 ,
22
11 1
11
8( ) 8( )
22
32 6
Ll ydcclyd
ll
++
=+ =+
× ...(i)
Similarly, length of the cable A 1 CB,
22
22 2
22
88
22
32 6
Ll yyccl
ll
=+ =+
×
...(ii)
Now total length of the cable ACB,
2 2
12
12
12
1 8( ) 8
22
22 6 6
Ll lLL ydccy
ll
- ⎛⎞+
==+⎜⎟⎜⎟++
⎝⎠
2 2
12
12
2( ) 2
33
llydccy
ll
=+ + +
2 2
12
2( ) 2
33
l ydccy
ll
=+ + ...(Q l
1 + l 2 = l)
Note. While using the above relation for the length of the curve, first of all position of the
point C is to be located.
Example 15.7. A foot bridge is carried over a river of span 60 m. If the supports are 3 m
and 12 m higher than the lowest point of the cable, find the length of the cable AB
Solution. Given : Span (l) = 60 m ; Depth of the lowest point from the support B (yc) = 3 m
or difference between the levels of the supports (d) = 12 – 3 = 9 m
Fig. 15.12.
Let l 1 = Horizontal length AC, and
l 2 = Horizontal length CB