Advanced Methods of Structural Analysis

5.4 Cable with Self-Weight 133

since the lowest pointCis located beyond the curveAB. Let the distance between

support points in horizontal and vertical directions beD 0 andh 0 , respectively;q 0 is

weight per unit length. It is required to determine the shape of the cable, thrustH,

and tensionNAandNBat supports.

Fig. 5.11 Design diagram of

the catenary; saddle pointC

beyond the span

x

xA

B

A

y

yA

yB

D 0

xB

h 0

a

O

L 0

C NA RA

NB

qB

qA

H

For this length–thrust problem parameteraDH=q 0 cannot be calculated right

now because thrustHis unknown yet. However, this parameter may be calculated

analytically having the total length of the cableL 0 and dimensionsD 0 andh 0 .For

this some steps should be performed previously.

1.The length of the curve from pointC to any point (x, y) according to
(5.25)isLCx D asinh.x=a/, so the length of curvesCBandCAare
LCBDasinh.xB=a / ; LCA D asinh.xA=a /. Therefore, the total lengthAB
of a cable is

L 0 DLCBLCADasinh

xB

a

asinh

xA

a

: (5.31)

2.Equation of the curve assumed by cable according to (5.27)isy.x/Dacoshxa.
Therefore, ordinates of pointsBandAareyBDacoshxaB;yADacoshxaA
and vertical distance between two supports is

h 0 DyByADacosh

xB

a

acosh

xA

a

: (5.32)

Equations (5.31)and(5.32) present relationships between parametersaDH=q 0 ,

L 0 ,h 0 ,andD 0 DxBxAand contains two unknowns parameters. They areaand

xB(orxA).

For given geometry parametersL0;D 0 ,andh 0 the analytical solution of (5.31)

and (5.32) leads to following results.

Parameterais determined from transcendental equation

cosh

D 0

a

D 1 C

L^20 h^20

2a^2

(5.33)