Multi span cable 7

[zcp]

If a cable rests across 2 supports it will hang as a catenary and have a certain tension. What if a 2nd identical span is added (still only one cable passing over rollers on the middle support), is the tension in the cable 2X?

So if I have a cable going across 10 equal spans, is the tension in the cable equal to the tension for 1 span x 10?



ZCP

http://www.phoenix-engineer.com

 

[Denial]

If all loads on the cable are vertical (as is usually the case with cables) then the horizontal component of the cable tension will be the same everywhere, including both sides of any roller support. The rollers will roll until this condition is met.

This makes it a bit more difficult to solve the problem, because you do not know the lengths of the individual catenary curves, you only know the sum of their lengths. Mathematically, each roller adds one more degree of freedom to your problem (that degree of freedom being, in effect, how much the roller rolls), but also adds one more constraint (that constraint being the requirement that the horizontal component of cable tension be equal each side of the roller).

On the few occasions I have had a problem like this to solve I have set the equations up in a spreadsheet and used Excel's "solver" to home in on the solution.

 

[Dinosaur]

ZCP,

I think you should consult a structural engineer. Is your cable supported at each interior support or is it only supported at the abutments? If I had the geometry of your problem I could give you an answer, but that is why I think you should hire someone who knows how this is done. Good Luck.

 

[rb1957]

start with a single span ...
the simple analysis of a cable assumes pin ends; each end reacts 1/2 the applied load.

if you add a support in the middle ...
now the cable is essentially fixed (it would have zero slope across the support). the support would react 1/2 of the applied load, and the end supports would react 1/4 of the applied load ... then the cable tension would be reduced by adding a support. this is not completely accurate as the end conditions of the span aren't the same, but i think the general idea is ok.

if you instead you add another span, then i think the cable tension is pretty much unchanged from the single span.

 

[miecz]

You'll have nearly the same tension no matter how many spans.

Take your single span cable and apply a couple to the right end so that slope at that end is zero.

Take your second identical cable and apply the couple to the left end so that the slope a that end is zero.

The result will be an identical tension in each cable, though perhaps slightly different from the single cable.

Now attach the right end of the first cable to the left end of the second cable and release the horizontal support at the attachment point.

The deflected shape of the cable won't change. So the moment won't change, the tension won't change, and the reactions at the far end won't change.

 

[Dinosaur]

ZCP,

I would like to recommend once again that you talk, face-to-face, with a structural engineer about this. The solution to a cable problem is dependent on the geometry of the problem. The advice you have received here can not be applied to all cable problems and could therefore be misinterpreted. Good Luck.

 

[JStephen]

In the original statement, the tension will remain the same, regardless of how many spans are added. At the roller support, the force is balanced on either side, and the net effect is the same as if the cable wer fixed.

Of course, if you get unequal loading on different spans, and the intermediate supports have rollers, then you'll pull the cable from one side to the other. And if you don't have rollers, you'd get a lateral load at that point which you might not expect, based on uniform loading.

 

[zcp]

Thank you all for the replies. Dinosaur, I am a structural / mechanical engineer. I am looking for the opinions of other engineers without the cloud of my initial input.

My opinion on the problem is that the tension in the cable is constant across all the spans, no matter how many spans. But each new span comes with a new load from additional cable. I am looking for opinions on whether the additional cable ADDS to the total tension or if equal spans means equal amounts of tension in each and tension equals tension so the tension throughout the cable equals the tension in one span. In other words, if the tension in the cable for one span is T, is the tension in a 10 span cable T or 10T. Opinions?

Then we can move on to the actual problem I have where there are hanging loads on the cable in each span. In this case does the load in each new span simply increase the overall cable tension which will be the same throughout the cable (due to the rollers)?

So in the case of 10 equal spans with a load hanging from each span. If we find a tension in the cable due to its weight for 1 span (Ts) and then find a tension in the cable in one span due to the load (Tf), when we go to 10 spans, is the overall tension in the cable (Ts + Tf) or is it (10Ts + 10Tf) or is it (Ts + 10Tf)? Opinions?

ZCP

http://www.phoenix-engineer.com

 

[civilperson]

The tension in the cable is identical for same shape caternarys. The inner pulleys/support take a share of the vertical weight to make the tension equal when the curve is equal.

 

[rb1957]

"My opinion on the problem is that the tension in the cable is constant across all the spans, no matter how many spans. But each new span comes with a new load from additional cable."
i think we've agreed with your original thought that the cable tension is more or less the same for a given span length, and independent of the number of adjacent spans. there probably is a small effect due to the difference between the pinned ends on the cable and the continuous cable running over the top of a support. I think the additional load due to the additional length of cable is reacted by the additional supports.

"Then we can move on to the actual problem I have where there are hanging loads on the cable in each span. In this case does the load in each new span simply increase the overall cable tension which will be the same throughout the cable (due to the rollers)?"

"So in the case of 10 equal spans with a load hanging from each span. If we find a tension in the cable due to its weight for 1 span (Ts) and then find a tension in the cable in one span due to the load (Tf), when we go to 10 spans, is the overall tension in the cable (Ts + Tf) or is it (10Ts + 10Tf) or is it (Ts + 10Tf)? Opinions?"
I think your saying that the cables react their weight and another load, replicated in each span. From what we've agreed above i think the solution to one span applies to multiple spans (with superposition of support reactions) ... so Ts + Tf would be the answer.

There would be a difference if different loads applied to different spans, with the cable was continuious over the supports (rollers) which would probably be a very messy calc. I think you can see something abot this problem by thinking of a three span cable. starting with weight loads, then apply a hanging load to one span; this'll increase the cable tension and change the deflected shape of the cable ... the cable in the adjacent spans would be shortened (cable would be pulled through by the load, no?) so in these adjacent bays the cable would be essentially pretensioned ...

 

[csd72]

This is fundamental physics.

There are two components to the reaction of a single span cable, the vertical reaction the sum of the two ends being equal to the applied load, and the horizontal reaction which is equal and opposite at each end (from basic statics sum of resulting horizontal forces=sum of applied horizontal forces=0) The Horizontal force is usually several times the vertical force.

The force in the cable is the vector sum of these.

Now as the cable is not accelerating (wel I hope not) then the force in the cable and the support reaction are equal and opposite.

Add a second span, with exactly the same loads, the horizontal reactions of the two spans are then exactly the same. Therefore at the middle support the left span iss pulling left and the right span is pulling right with equal and opposite force.

The same applies to more equal spans, the forces do not accumulate, they oppose each other. The reactions at the end supports is exactly the same regardles of the number of equal spans.

 

[rb1957]

agreed, but if there are different loads in different spans their combined effect on the tension of the continuous cable is not obvious, and i don't think you can get there by superposition.

hanging a weight in one span changes the tension in the cable and the deflection of the cable. add a second weight to the cable, the tension and the deflection changes again, and this time you are also changing the deflection of the first load (ie doing work).

BUT, if you doing the same thing in several spans, this can be analyzed as a single span and superposition applies (mostly at the support reactions).

 

[prex]

zcp,
think of it in this way.
If all the spans are equal, including the loads, then the cable won't move on the rollers, and these will behave like simple supports. It is evident in this case that the tension in each span is independent of that in the others.
As you depart from this simple situation, of course the same conclusion does not hold, but the deviation will be small if the spans differ each other only a little.
So the statement 'when the cable goes across 10 equal spans, the tension will be 10x' is totally wrong.

prex

http://www.xcalcs.com

: Online tools for structural design

http://www.megamag.it

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http://www.levitans.com

: Air bearing pads

 

[miecz]

Agree with the others, the force does not accumulate. That said, once you add concentrated loads, determining the tension is no easy matter, even for a single span.

The solution depends on the sag in the cable, which is a function of the support structure, cable stretch, temperature, creep, wind, birds... The tension in the cable is the reaction divided by the sin of the angle at the support. If there were no sag, the tension would be infinite. The greater the angle, the less the tension in the cable.

For multiple spans with different loads, equilibrium is reached when the tension can support the load in each span. The cable will shift until the required angle is achieved. I'm not sure it's possible to calculate the tension, though.

 

[rb1957]

I don't think prex's post ... "It is evident in this case that the tension in each span is independent of that in the others." is correct (as written) in that the tension in the cable is going to be reasonably constant (as the cable is continuous) and not independent of the other bays (you wouldn't have a tension of 1,000 lbs in one span and 2,000 lbs in another.

 

[csd72]

Ignoring friction at the roller supports, the tension in the whole cable is theoretically equal.

The higher loaded spans will just sage more to resist the load.

 

[swearingen]

The reaction at the ends of the cable will not change just by adding another span - it will be in equilibrium, as stated several times above. As long as the new spans are the same in every way, the end reactions will be the same.

Notice I say "end reactions". Looking at catenary theory, the tension in the cable is definitely NOT constant in the whole cable. The HORIZONTAL reaction is constant, regardless of the slope of the cable. I just wanted to point out that the actual cable tension VARIES along its length. The only way you could neglect it's own weight (which is the cause of the varied tension) is to provide that the weight that you hang on the cable far outweighs the total cable weight of the span.

 

[prex]

rb1957,
I could have said it more clearly:"It is evident that in this case each span behaves exactly as a single isolated span". Of course with equal spans (and equal loads) all the spans have the same tension and this tension is exactly the same that you would have in an isolated span.

prex

http://www.xcalcs.com

: Online tools for structural design

http://www.megamag.it

: Magnetic brakes for fun rides

http://www.levitans.com

: Air bearing pads

 

[zcp]

Thank you all for the replies (and stars for each of you). I agree with the consensus, the tension in the cable is the same for an individual span. They can not add or our power industry would be up a creek with overhead power lines.

Now that this is established, let's move on to my actual problem. The cable is not a cable at all. It is the fabric in a retractable fabric roof.

Imagine a rectangular section of fabric (15 ft deep, 4 ft wide between supports) attached to frames on each side (so the rollers are not really rollers). Continue adding sections until you have the 10 spans.

The question is 'what force is required to pull this structure tight'? Is it the force required to pull 1 span tight or 10 spans?

Now what if we add weight in pockets in each section so the fabric falls down as the structure is retracted. What is the force required to pull this structure tight? Opinions? Any thoughts on the line of logic here?

ZCP

http://www.phoenix-engineer.com

 

[rb1957]

frustratingly, i think the "real" problem is very different to the cable analogy.

i guess it not much of a question about tightening a fabric roof, but more about when there are weights on the panels. If you're in the elastic region for the fabric, tightening the fabric is going to do work against gravity (regarding the weights). lifting one weight will take a force; lifting 10, will need 10x the force (assuming you're adding the force only at the edges of the roof).