Advanced Methods of Structural Analysis

5.2 Cable with Neglected Self-Weight 119

The maximumy-coordinate of the cable in terms of total lengthLand spanlis

ymaxD

p

3

2

p

2

l

r

L

l

1: (5.16d)

Exact Solution of the Length Determination

Integrating equation (5.13a) leads to the following exact expression for total length
in terms ofq,l,andH

LD

l

2

s

1 C



ql

2H

 2

C

H

q

sin h^1

ql

2H

: (5.16e)

Ta b l e5.1contains results of numerical solution of approximate equation (5.16a)for
different total lengthLof the cable. Three and two terms of this equation have
been hold. Results of numerical solution compared with solution of exact equation
(5.16e). The problems are solved forqD 2 kN=mandlD 30 m.

Ta b l e 5. 1 Cable under
uniformly distributed load.
ThrustH, kN of the cable vs.
the total lengthL

Total length of a cableL(m)

32 34 36

Two t e r m s 47:4342 33:5410 27:3861

Three terms 45:8884 31:1147 23:9464

Exact solution46:0987 31:7556 25:3087

We can see that even two terms of (5.16a) leads to quite sufficient accuracy.
Moreover, sinceH 3 terms<Hex:sol<H 2 terms, then two-term approximation is more
preferable for design.

Example 5.1.Design diagram of flexible cable with support pointsAandBon
different levels, is presented in Fig.5.5. The cable is subjected to uniformly dis-
tributed loadq. Find shape of the cable and determine distribution of internal forces,
if thrustHof the cable is given. Parameters of the system are:lD 30 m,cD 3 m,
H D 40 kN,q D1:8kN=m. Use two approaches (a) integrating of differential
equation (5.8) and (b) the concept of the reference beam.

Solution.

(a)Differential equation of a flexible cable is d^2 y=dx^2 Dq=H; its integration leads
to following expressions for slope and shape

dy

dx

D

q

H

xCC 1

yD

q

H

x^2

2

CC 1 xCC 2