Advanced Methods of Structural Analysis

5.2 Cable with Neglected Self-Weight 113

The concept of “shear” at the any section of the reference beam will be helpful. The
portion length ofxis loaded by uniformly distributed loadq, concentrated forceP,
reactionRA, thrustH, and axial forceN.x/as shown in Fig.5.2b. The beam shear
isQ.x/DRAPqx. At the sectionx(pointK) three concurrent forces acts.
They areQ.x/; N.x/,andH. It is obvious that for any section

N.x/D

p

H^2 CQ^2 .x/: (5.6)

Now let us derive the differential relationships between loadqand ordinate of the
cabley. The anglebetween the axial line of a cable andx-axis (Fig.5.2b) obeys
to equation

tanD

dy

dx

D

Q

H

: (5.7)

To derive a relationship betweenQand intensity of the loadq, consider a free body
diagram of the elementary portion of the cable (Fig.5.2c). The axial forces at the
left and right ends of a portion areNandNCdN, respectively. Their components
areHandQat the left end, while at the right endHandQCdQ.
Equilibrium equation

P
YD 0 leads to the following relationship: dQ=dxDq.
Taking into account (5.7), the required relationships betweenyandqbecomes:

d^2 y

dx^2

D

q

H

: (5.8)

The first and second integration of (5.8) leads to the expressions for slope andyordi-
nate, respectively. The constant integration should be determined from the boundary
conditions.

5.2 Cable with Neglected Self-Weight....................................

This section contains analysis of the perfectly flexible inextensible cables subjected
to concentrated loads, as well as distributed loads along horizontal foot of the cable.
For both loadings (concentrated and distributed), the thrust–shape and length–thrust
problems are considered.

5.2.1 Cables Subjected to Concentrated Load.....................

In this case of loading each portion of the cable between two adjacent forces presents
the straight segment. The simplest design diagram of a cable, subjected to one con-
centrated load, is presented in Fig.5.3a. Parameters of the system areaD 10 m,
lD 25 m,PD 20 kN.