Point Load On Cable

[BenderLT]

Hello everyone.

The problem:

I need a way by hand the calculate the tension in a cable due to a central lateral point load neglecting any intial sagging effects and intial tensioning. So basically the intial stiffness is zero.

I have looked at many iterative solutions for similar cable problems but have not found any for this case.

Any help would be much appreciated.

 

[prex]

What about the self weight of the cable?
Anyway this is a simple statics problem: the cable elongates under load and it takes the shape of a broken line. The displacement of the load is calculated from geometry and equilibrium.

prex

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[GregLocock]

Just guessing, OP might be thinking of a massless elastic cable with an inital length = to the difference between the fixtures, and zero tension.

If it deflects sideways by y then the strain in each half of the cable is (sqrt((L/2)^2+y^2)-L/2)/(L/2). This produces a tension T, and you know that T*sin(theta)=F/2, and you know that tan(theta)=y/(L/2)

There is a missing step in there.






Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[rb1957]

the tension in the cable reacts the applied load, deflecting the cable to an angle theta to the horizontal ...
T*cos(theta) = P/2 ... T = (P/2)/cos(theta)
and (of course) T/A = stress
and (of course) the deflected length is L/cos(theta)
so that the strain is 1/cos(theta)-1
and (of course) stress/strain = E

so (P/2)/A/cos(theta)/(1/cos(theta)-1) = E

(P/2)/A/(1-cos(theta)) = E

cos(theta) = 1-(P/2)/(A*E)

no?

 

[asixth]

Ignoring gravity (gravity will give the cable a superimposed parabolic curvature).

The supports will need to react the point load applied, so this is an iterative solution until you can get the rotation of the cable away from the horizotal and tension due to the elongation to be equal.

I'm sure it's not impossible to write a spreadsheet that converges the two with goalseek.

 

[rb1957]

i don't see that the cable would deflect as a parabola ...

the tension in the cable is constant ...

a parabola has a changing slope, which implies that the vertical component of the cable tension is changing ...

but the only load is a central point load (so that the cable tension vertical component needs to be constant)

no ?

 

[asixth]

If the cable mass is considered the deformations from gravity would cause it to deform into a parabolic curve. Unless a cable is straight or circular, it cannot have a uniform tensile stress.

 

[SlideRuleEra]

For an anchored, level cable with a single load at the center, "the cable assumes two catenary arcs which intersect at the point of the load", but a parabola is a reasonable approximation. See page 44 of US Steel Wire Rope Engineering Handbook, which can be downloaded from this page of my website:

http://www.slideruleera.net/miscellaneous.html

http://www.SlideRuleEra.net

http://www.VacuumTubeEra.net

 

[rb1957]

i read "neglecting any intial sagging effects and initial tensioning." to mean assume the cable is massless (since the only way to prevent it from sagging is to pretension it).

 

[desertfox]

Hi BenderLT

If were ignoring mass of cable and any initial sag wouldn't this case just become a triangle of forces ie imagine a piece of string pinned to a board at each end of the string, then a central load is applied and the string deflects to a triangular shape and the tension in the string can be calculated depending on the angle of the string to the horizontal.
I believe rb1957 as done the maths relating to what I am describing.

desertfox

 

[csd72]

BenderLT,

with all those simplifications you state you would get zero deflection and therefore infinite tension.

Generally the simplest way is to assume a cable size and then calculate deflection based on this and therefore axial load. Unfortunately the iterative method is the only option here.

 

[rb1957]

IMHO, i don't think so

 

[BAretired]

rb1957,

You said: "T*cos(theta) = P/2 ... T = (P/2)/cos(theta)"

Shouldn't that be T*sin(theta) = P/2?



BA

 

[csd72]

My last post should have said to assume an initial reasonable deflection and then calculate the tension in the cable. Choose the minimum cable that can take this load and then recheck the deflection and load using the stiffness of this cable.

 

[BAretired]

If T = tension in cable, and theta is the angle below horizontal:

T.sin(theta) = P/2

delta = T.a/AE where a is the half length between supports

T = (L - a)AE/a where L is the slope length of half cable.

T = (1/cos(theta) - 1)AE = P/2sin(theta)

So tan(theta) - sin(theta) = P/2AE.

Solve for theta, then T.



BA

 

[rb1957]

@BAretired,
yeah, that looks better