Scribd
Upload
438 views3 pages

Module 3: Cables Lecture 3: Application of The General Cable Theorem For Distributed Loading

This document discusses using the general cable theorem to analyze cables with distributed loading. It presents the theorem and shows how to apply it to a cable with uniform distributed loading w. The cable's shape is defined as a parabola with dip y at distance x from the left support. The cable tension T varies along the cable based on the slope. The maximum tension occurs at the ends and minimum at the mid-span, equal to the distributed load. The total cable length S can also be determined. However, a catenary under self-weight has a different shape than a parabola due to the non-uniform loading along the curved length.

Uploaded by

More's Kaushik
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
438 views3 pages

Module 3: Cables Lecture 3: Application of The General Cable Theorem For Distributed Loading

This document discusses using the general cable theorem to analyze cables with distributed loading. It presents the theorem and shows how to apply it to a cable with uniform distributed loading w. The cable's shape is defined as a parabola with dip y at distance x from the left support. The cable tension T varies along the cable based on the slope. The maximum tension occurs at the ends and minimum at the mid-span, equal to the distributed load. The total cable length S can also be determined. However, a catenary under self-weight has a different shape than a parabola due to the non-uniform loading along the curved length.

Uploaded by

More's Kaushik
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 3

Module 3 : Cables

Lecture 3 : Application of the General Cable Theorem for Distributed Loading

Objectives
In this course you will learn the following

Use of the general cable theorem for cables with distributed loading.

3.3 Application of the General Cable Theorem for Distributed Loading

We have seen that we can apply the general cable theorem to find the cable geometry under vertical loading
cases. The theorem also applies for distributed loading, since bending moment definitions for the
corresponding simply-supported beam ( and ) do not change. For a cable under uniformly
distributed load w , we have:

(3.7)

Let us consider the specific case of a cable AB under uniformly distributed loading w , with the cable's
supports being at the same horizontal level (Figure 3.5). Note that the system is symmetric about its mid-
span where the cable has its maximum dip. Let the span of the cable be L and its dip at the mid-span (point
C ) be . We can find, from the equilibrium of vertical forces and from symmetry, that the vertical support
reactions at both A and B are wL/2. Now, applying the general cable theorem (Equation 3.7) at point C , we
get:

(3.8)

Figure 3.5 Free body diagram of a cable under uniformly distributed load
Due to symmetry, we can see that the cable tension (axial force) is horizontal at the mid-span. This can be
observed also if we draw the free body diagram of either the right or the left half of the cable (Figure 3.6).
Figure 3.6 Free body diagram of the right half (CB) of the cable
We can also use Equation 3.8 to define a general shape of the cable in terms of the mid-span dip, . Thus,
the dip y at a (horizontal) distance x from the left support now is:

(3.9)

Let T be the axial tension in the cable at a distance x . This axial tension acts along the tangent of the cable
geometry. Let us measure the length of the cable by s , which is measured along the cable curve. Therefore

(3.10)

dy / dx is the slope of the cable and it can be obtained from Equation 3.9 which defines the shape of the cable.
Substituting in Equation 3.10, we get:

(3.11)

This equation also shows that the maximum tension occurs at the end supports, that is at x = 0 and x = L ,
which is also where the slope of the cable is maximum. The minimum tension occurs at the mid-span and is
equal to H .

The shape, as defined in Equation in 3.9, can be used obtain the total length of the cable ( S ) as well.

(3.12)

The expression simplifies if the dip becomes very small compared to the span, that is,

(3.13)

One should remember that Equations 3.8 to 3.12 are valid for only cables with both end supports at the same
horizontal level.

The shape of a flexible cable supported at two ends and hanging only under its self-weight is known as a
catenary . It is the shape that a cable attains under uniformly distributed vertical load (self-weight, in this
case). Therefore, the shape of the cable should be a parabola as per Equation 3.9 and this was what Galileo
claimed. However, Leibniz and other scientists later found the proper equation for a catenary to be different
from a parabola. This is because the self-weight of the cable is uniform along its curved length and not along
its span. The distributed loading w that we have considered for obtaining Equation 3.9 is uniform along the
span ( x ) and not along its curved shape ( s ). The equation of a catenary is:

(3.14)
Recap
In this course you have learnt the following

Use of the general cable theorem for cables with distributed loading.

You might also like