Structure Analysis I Chapter-9

Session No: 09 Date: 2078/02/27

Time: 10:30 A.M. Class : BCE/II/II
Subject: Structure Analysis I

Unit No : 09 CABLE STRUCTURES

 INTRODUCTION TO CABLES
Cables are flexible structures that support the applied transverse loads by the
tensile resistance developed in its members. Cables are used in suspension bridges, tension leg
offshore platforms, transmission lines, and several other engineering applications. Cables are
flexible structures in pure tension. The distinguishing feature of a cable is its ability to take
different shapes when subjected to different types of loadings. Under a uniform load, a cable takes
the shape of a curve, while under a concentrated load, it takes the form of several linear segments
between the load’s points of application.

 CATENARY AND PARABOLIC CABLES

A cable or a chain hanging freely and acting under its own weight, supported
only at its end points (not in a vertical line) and with no other load, takes the form of a catenary.
This is different to the loaded cables supporting a suspension bridge, which take the form of a
parabola.
Both forms are always in tension and may be described as funicular forms, which are the forms
assumed by cables (or chains) under any given load: the catenary is funicular because it bears only
its own load, while the parabola is funicular because it carries load, as in a suspension bridge.

 ELEMENTS OF SIMPLE SUSPENSION BRIDGE

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Basically, a suspension bridge consist of the following elements:

i) Cable
ii) Suspenders
iii) Decking, including the stiffening girder
iv) Supporting Tower
v) Anchorage

 ANALYSIS OF PARABOLIC CABLES

i) Equilibrium of Light Cables : General Cable Theorem

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ii) Analysis of cables under different loads at same level

a) Cables subjected to concentrated loads

b) Cables subjected to uniformly distributed loads

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iii) Analysis of cables under different loads at same level

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iv) Forces on anchor cable and tower

The forces on anchor cables and towers depend upon the type of support given to the
cables . As there are two types of supports namely, guided pulley and roller support.
a) Guided pulley system

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b) Roller Support

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Example No: 1 :- A bridge cable is suspended from the towers 80 m apart and carries a load of
30kN/m on the entire span. If the maximum sag is 8 m, calcuate the maxumum tension in the
cable. If the cable is supported by the saddles which are stayed by the wires inclined at 30 to
the horizontal, determine the forces acting on the towers. If the same inclination of back stay
passes over pulley, determine the forces on the towers.

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Example No: 2 :- A bridge cable of span 120 m and dip 10 m carries a load of 6 kN/m on the
entire span. Find the maximum tension in the cable and the inclination of the cable at the
support. Find the forces transmitted to the supporting pier if the cable passes over smooth
pulleys on the top of the pier. The anchor cable is at 30 to the horizontal. Determine the
maximum bending moment for the pier if the height of the pier is 15 m.

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Example No: 3 :- A cable is suspended from the points A and B which are 80 m apart
horizontally and are at different levels, the point A being 5 m vertically higher than the point B
and the lowest point in the caable is 10 m below A. The cable is subjected to a uniformly
distributed load of 30 kN/m over the horizontal span. Determine
i) the hoizontal and vertical reaction at each end
ii) the position of lowest point
iii) the maximum tension in the cable.
iv) length of the cables
Solution : -

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2 ℎ12 2 ℎ22
 Length of the cables (L) = l + +
3𝑙1 3𝑙2
2 𝑥 102 2 𝑥 52
= 80 + +
3x 46.863 3x 33.137
= 81.925 m

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 ANALYSIS OF THREE- HINGED STIFFENING GIRDER

When suspension bridge is subjected to a moving (or rolling ) load system the funicular polygon
will change its shape and hence the shape of cable is also change.
In suspension cable bridge it is necessary to maintain the parabolic shape of the cable to satisfy
this condition the load transmitted to the cable should be uniformly distributed load.
This is achieved either two hinged or three hinged stiffening girder.
The main purpose of providing these stiffness girders to reduce the sag under the rolling load.
The girders are suspended from the cables through hanger cables. Hence, the uniformly
distributed dead load of the roadway and stiffening girders is transmitted to the cables through
hanger cables and is taken up entirely by the tension in the cables. The stiffening girders does not
suffer any S.F. and B.M. under dead loads as girders is supported by closely spaced hanger cables
throughout.
Any live load on the bridge will be transmitted to the girders as point loads. The stiffening girders
transmit the live load to the cable as uniformly distributed load. While doing so the stiffening
girders will subjected to S.F. and B.M. throughout the entire length.

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Considering the moment at the central hinge of girder C’ , we, the uniformly distributed load
exerted by the suspenders can be determined. Then, the beam may be analyzed for the given load
𝑤𝑒 𝑥
along with we . Note that due to we alone, there is a bending moment of (l –x ) at the section
8
𝑤𝑒 𝑙 2
x-x and a maximum moment, i.e., at C. This moment is a hogging moment. The shear force
8
𝑙
at that section due to we is - we (2 − 𝑥) . The cable can be analyzed for the uniformly distributed
load we .

S.F AND B.M. diagram of three hinged stiffening girder

Figure above shows a three hinged stiffening girder supported at the ends connected to the cable.
Let the girder be subjected to any external load system w1, w2, w3. Let we per unit run be the
uniformly distributed load transferred to the cable. Obviously, the girder will be subjected to an
upward uniformly distributed load we per unit run.
Now, let us consider the equilibrium of cable and stiffening girder separately.

Considering the cable

This is subjected uniformly distributed load we per unit run transmitted by the hangers. The
equation of cable say A as the origin is;
4ℎ𝑥
∴ y= (L-x)
𝐿2
𝑤𝑒 𝑙
Vertical Reaction at each support (V)=
2
𝑤𝑒 𝑙 2
Horizontal reaction at each support (H) =
8ℎ
 Maximum tension (Tmax) =√𝑉 2 + 𝐻2

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Example No: 4 :- The three hinged symmetricaal stiffening girder of suspension bridge of 100
m span is subjected to two point loads of 100kN each placed at 20m and 40m respectively from
the left hand hinge. Determine the maximum tension in the cable which has central dip of 10m.
Calculate the length of the cable and also B.M. and S.F. in the girder at the section 30 m from
left end.

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Example No: 5 :- A suspension bridge, 100 m span, has two three-hinged stiffening girders
supported by two cables with a central dip of 10 m . The dead load is a uniformly distribuyed
load of 30kN/m for the entire span and in addition, it supports three point loads of 200 kN each
placed along the centre line of the roadway, dividing the span in four equal parts. Calculate the
maximum tension and minimum tension with their locations in the cable and the length of the
cable; also draw the shear force and bending moment diagram for the girders.
Solution: Given , Span of cable (l) =100 m
Dead load (D.L.) = 30kN/m

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Example No: 6 :- A suspension bridge, 120 m span, has two three-hinged stiffening girders
supported by two cables with a central dip of 12 m . The roadway has a width of 6 m. The dead
load on the bridge is 5 kN/m2 while the live load is 10 kN/m2 which act on the left half of the
span. Determine the shear force and bending moment in the girder at 30 m from the left end.
Also find the maximum tension in the cable for this position of live load.
Solution:

Given that,
Width of the bridge = 6 m

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 INFLUENCE LINE DIAGRAM OF THREE- HINGED STIFFENING GIRDER

i) ILD for equivaent cable load we

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ii) ILD for Bending moment H

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iii) ILD for Bending moment M in the girder

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iv) ILD for Bending moment SF in the girder

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Example No: 7 :- A suspension cable of 90 m span and 9 m dip is stiffeined by a three-hinged

girder. The dead load is 10 kN/m. Determine the maximum tension in the cable and the
maximum bending moment at the section 30 m from the left support due to concentrated load
of 100 kN crossing the girder, assuming that the dead load is carried out by the cable without
stressing the girder.

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 TOWER STRUCTURES, WIND CABLES AND TIES

THE END

Task No:- 08
Solve the questions of Cables from your question bank of year 2015 ( spring and fall both)
,2013 fall and 2014 fall.
(Note : the task must be submitted next week of today class )
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