Advanced Methods of Structural Analysis

126 5Cables

H

q

N

H

W

N

q

C

W=q 0 s

D

Free body diagram

O

x

a

y

Base of catenary

s

C

ds dy

A

B

dx

D

H W=q 0 s

Design diagram N

Fig. 5.8 Design diagram of the catenary. Free body diagram for portionCDand force triangle

The final results may be conveniently expressed in terms of additional parameter
aDH=q 0. This parameter is measured from the lowest pointCof the cable and
defines an originOand the lineO-xcalled a base of catenary (or chain line). In
terms ofaandsthe normal force, accordingly (5.21) becomes

NDq 0

p

a^2 Cs^2 : (5.22)

The relationship between coordinatesxandsin differential form is

dxDdscosDds

H

N

D

aq 0

q 0

p

a^2 Cs^2

dsD

ads

p

a^2 Cs^2

:

Integrating this equation from pointC.0; a/toD.x; y/allows us to calculate the
x-coordinate of any point of the cable in terms ofsandaDH=q 0

xD

Zs

0

ads

p

a^2 Cs^2

Dasinh^1

s

a

: (5.23)

The inverse relation, i.e., the solution of (5.23) with respect tosis

sDasinh

x

a

: (5.24)

This formula allows us to obtain some useful relationships in terms of intensity load
q 0 and thrustH.
The length of the cable from lowest pointCtoD.x; y/is

LCDD

H

q 0

sinh

q 0

H

x: (5.25)