Tightrope Tension Theory 1

[engreg]

See attached image.

I am confused by the fundamental principle of defelction-derived tension in a "tightrope". I have to design a similar (but pretensioned) zipline system, and I want to make sure i am usign the correct design principles.

In the image attached, I highlight my confusion.

Is the tension in the rope truly the same (and its connection requirements (similar)), if a rope deflects a ratio's amount to the load applied. I understand why the math says it is. But I am imagining a string across a room. If I apply 10 lbs of force it will deflect (x) amount. If I apply 5x the force. It will deflect more, maybe not (5x) more, but enough to make me question the math. The math would say the tension is more similar than I would expect it to be.

Similarly, if we apply this logic to a steel beam that deflects 1" over 25 feet. We would determine a near-ifinite tension in the connections at the end. Which obviously isn't the case.

Any explanation of how the phenomena should be handled, or advice on a pretensioned zipline conenciton design would be appreciated.

 

[canwesteng]

You can't apply this logic to a steel beam since it carries the load in bending. If you let the cable go more slack, you will reduce the tension in the cable, so that makes sense. The calculation of the actual deflection will be iterative, so the scenario you described will not happen.

 

[rb1957]

I think you need to think of the problem differently. What angle will the cable deflect to ?

With 100 lbs applied, 5 deg deflection means 573 lbs tension developed but 4 degree deflection means 717 lbs tension. so which is it ? What you need is cable strain ... 5 degree deflection means the length of the deflected cable is 2*573 (if the original length was 100), so how does this strain(146/100) compare with the elastic strain from the tension (573/AE*100)) ?

Another way to think of it is to say if the tension in the cable is 1000 lbs, how much will it deflect with a 50 lbs load applied. As a RoM answer the cable will deflect so that the component of tension reacts the load (the idea being that the angle is so small that the increase in tension (due to the strain of the cable) is small.

 

[ANE91]

Start by reading about funicular structures. Schodek has a good chapter on this topic.

A lot of the basic math assumes that the cable is inextensible, meaning that we neglect small elongations relative to the cable length. This is obviously not always true, like with your string example or for membrane structures. The other major assumption is that the cable is perfectly flexible, so the internal force is always tangent to the cable.

Your zipline is probably made of 7-wire strand. There are creep principles and temperature effects that must be considered. Additionally, self-weight will not be negligible.

A maximum sag must be predefined, as the problem is statically indeterminate. A specific deflected shape can then be interpreted.

 

[HTURKAK]

Any explanation of how the phenomena should be handled, or advice on a pretensioned zipline conenciton design would be appreciated.

This is a typical geometric non -linearity problem. The solution could be with iterations.
The steps are ,
- Assume a deflection (D1) and find the tension T1 in the ropes
- then calculate the deflection(D2) corresponding to T1
- Try for the average deflection D3=(D+D2)/2 and find the tension T2 in the ropes
- then calculate the deflection(D4) corresponding to T2
Continue the iteration up to Dn reasonably equal to Dn-1

...

 

[human909]

While it has been somewhat pointed out by others, I'll try to be specific.

Is the tension in the rope truly the same (and its connection requirements (similar)), if a rope deflects a ratio's amount to the load applied. I understand why the math says it is. But I am imagining a string across a room. If I apply 10 lbs of force it will deflect (x) amount. If I apply 5x the force. It will deflect more, maybe not (5x) more, but enough to make me question the math. The math would say the tension is more similar than I would expect it to be.

Where you are going wrong with your mathematics there is that you are looking at one equilibrium (geometric force static equilibrium) without considering another important equilibrium, force/strain equilibrium.

You diagram shows that your cable/string is increasing in length. Which breaks one of the most fundamental rules of basic static equilibrium mathematics. But that is ok, we can deal with that as long as you consider the strain in the cable.

 

[JStephen]

There can be axial loads in a beam with restrained ends due to transverse loads. Most beam applications don't have the necessary end restraint and/or bending flexibility for that to be an effect. But see Table 12 in Article 7.7 of Formulas for Stress and Strain, 5th Ed., for example. Similar effects occur in plate bending.

 

[rb1957]

I get "- Assume a deflection (D1) and find the tension T1 in the ropes" (as the OP has down ... tension has a vertical component by geometry)
but "- then calculate the deflection(D2) corresponding to T1" ... ?? how ??