Advanced Methods of Structural Analysis

118 5Cables

Therefore, the total lengthLof the cable in terms of sag according to (5.13) becomes

LD

Zl

0

s

1 C



4f

l

 2 

2x

l

 1

 2

dx: (5.13a)

Approximate Solution of Length Determination

The binomial theorem to expand radical in a seriesp
1 C"Š 1 C^12 "^18 "^2 C:::



allows presenting (5.13a) as follows:

LD

Zl

0

(

1 C

1

2



4f

l



2x

l

 1

    2



1

8



4f

l



2x

l

 1

    4

C:::

)

dx: (5.15)

Integration of this relation leads to the following approximate expression:

LDl

"

1 C

8

3



f

l

 2



32

5



f

l

 4

C

256

7



f

l

 6

:::

#

(5.16)

This expression allows calculating the sagfof the cable in terms of total lengthL
and spanl.SincefDql^2 =8H, then length of the cable in terms of thrustHmay
be presented as follows

LDl



1 C

1

24

q^2 l^2

H^2



1

640

q^4 l^4

H^4

C:::

(5.16a)

This expression allows calculating thrustHin terms of spanl, total length of the
cableL,andloadq.

Gentile Cable

Iff < 0:1l, then a cable is called the gentile one. Taking into account two terms in
equation (5.16a), we get the following equation

LDl



1 C

1

24

q^2 l^2

H^2



(5.16b)

Therefore, thrust in terms of total length of the cableLand spanlmay be presented
in the form

HD

ql

2

p

6



1

p

l 0  1

;l 0 D

L

l

: (5.16c)