Advanced Methods of Structural Analysis

112 5Cables

denoted asrAandhAfor supportAandrBandhBfor supportB. Summing of all
forces on the horizontal axis leads to the relationshiphADhBDh.
The vertical component of the reaction atB

rB!

X

MAD 0 W rBl

X

Pixi

Z

l

q.x/dxD 0 !rBD

P

PixiC

R

l

q.x/dx

l

(5.1)

This expression coincides with general formula for reaction of simply supported
beam. Therefore if axial forces at supports are resolved as shown in Fig.5.2a, then a
vertical component of axial force at support equals to corresponding reaction of the
reference beam.
Now we can calculate ordinateof the cable at the sectionCwith abscissac.
Since the bending moment at any sectionCis zero, then

MCDrAchAf1C

X

Pi.cxi/

Zc

0

q.x/.cx/dxD 0

f1CD

rAc

P

Pi.cxi/

Rc

0

q.x/.cx/dx

h

(5.2)

wheref1Cis a perpendicular to the inclined chordAB.
It is obvious that nominator presents the bending moment at the sectionCof the
reference beam. Therefore

f1CD

MC^0

h

: (5.3)

To determine the vertical coordinate of the pointCwe need to modify this formula.
For this we need to introduce a concept of a thrust. The thrustHis a horizontal
component of the axial force at any section. From Fig.5.2b we can see that equilib-
rium equation

P
XD 0 leads to the formulaHDconst for any cable subjected to
arbitrary vertical loads. At any supporthDH=cos'.Sincef1CDfCcos',then
expression (5.3) can be rewritten asfCcos'DMC^0 =.H=cos/, which leads to
the formula:

fCD

MC^0

H

(5.4)

wherefCis measured vertically from the inclined chordAB. This parameter is
called the sag of the cable.
The formula (5.4) defines the shape of the cable subjected to any load. The sag
andy-ordinate are equal if and only if supports are located on the same elevation,
the origin is placed on the horizontal chordABandy-axis is directed downward.
Now the total vertical reactions of supports may be determined as follows:

RADrAChsin'DrACHtan';

RBDrBhsin'DrBHtan': (5.5)