Advanced Methods of Structural Analysis

5.4 Cable with Self-Weight 127

Formula (5.22) for tension at any point can be rewritten as follows

N.x/DH

r

1 Csinh^2

q 0

H

xDHcosh

q 0

H

x: (5.26)

The minimum tensionNDHoccurs atxD 0 , to say in the lowest point.

In order to obtain equation of the catenary, we need to expressyin terms ofx

dyDdxtanD

W

H

dxD

q 0 s

aq 0

dxDsinh

x

a

dx:

After integrating this equation from pointCto pointD,weget

Zy

a

dyD

Zx

0

sinh

x

a

dx!yjyaDacosh

x

a

ˇ

ˇ

ˇ

x

0

:

The equation of the curve, assumed by the cable, and corresponding slope at any

point are

y.x/Dacosh

x

a

D

H

q 0

cosh

q 0

H

x

tanD

dy

dx

Dsinh

q 0

H

x: (5.27)

Below is presented analysis of typicalcases of cables carrying a uniformly dis-

tributed load along the cable itself:

1.Supports located at the same level
2.Supports located at different elevation and saddle point within the span
3.Supports located on the different elevation and cable has not a saddle point within
the span

5.4.2 Cable with Supports Located at the Same Level

In this case a cable has the axis of symmetry. It is pertinent here to derive the simple

and useful expression for maximum tension for the cable in terms ofq 0 ,f,andH.

Maximum tensionNmaxoccurs at the supports.xD ̇l=2/,then

NmaxDH

r

1 Csinh^2

q 0 l

2H

!sinh^2

q 0 l

2H

D

Nmax^2

H^2

1: (5.28)