Advanced Methods of Structural Analysis

122 5Cables

5.3 Effect of Arbitrary Load on the Thrust and Sag

So far we considered behavior of the cables that are subjected to concentrated or
distributed load only. In case of different loads acting simultaneously formula (5.4)
is most appropriate. However in engineering practice another type of loading is
possible, mainly: the cable is subjected toany dead load and after that additional
live load is applied. How will this change the shape and state of the cable? This
problem may be effectively solved by knowing the expression related to the total
length of the cable, the thrust, and external loads.
Consider a cable supported at pointsAandB and subjected to any loads
(Fig.5.6). Assume that a cable is inextensible, and support points of the cable do
have the mutual displacements. The covered span isl, while a total unstressed length
of a cable isL.

Fig. 5.6 Cable carrying
arbitrary dead and live load

q(x)

x

l

L

H

H ABx

The total length of the cable is defined by exact equation (5.13). For gentile cable
a slope is small, so s

1 C



dy

dx

 2

Š 1 C

1

2



dy

dx

 2

(5.17)

and expression forLbecomes

LD

Zl

0

s

1 C



dy

dx

 2

dxŠ

Zl

0

"

1 C

1

2



dy

dx

 2 #

dx: (5.18)

According to (5.4), the shape of the cable is defined by expressionyDM=H,
whereMis a bending moment of the reference beam, andHis a thrust. Therefore
the slope can be calculated as follows

dy

dx

D

d

dx



M

H



D

1

H

dM

dx

D

Q.x/

H

;

whereQ.x/is shear at any section of the reference beam. After that expression
(5.18) becomes

LD

Zl

0

"

1 C

1

2



Q.x/

H

 2 #

dxDlC

1

2H^2

Zl

0

Q^2 .x/dx: