Multi span cable
Multi span cable
(OP)
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?
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?






RE: Multi span cable
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.
RE: Multi span cable
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.
RE: Multi span cable
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.
RE: Multi span cable
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.
RE: Multi span cable
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.
RE: Multi span cable
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.
RE: Multi span cable
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
www.phoenix-engineer.com
RE: Multi span cable
RE: Multi span 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 ...
RE: Multi span cable
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.
RE: Multi span cable
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).
RE: Multi span cable
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 : Magnetic brakes for fun rides
http://www.levitans.com : Air bearing pads
RE: Multi span cable
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.
RE: Multi span cable
RE: Multi span cable
The higher loaded spans will just sage more to resist the load.
RE: Multi span cable
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.
RE: Multi span cable
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
RE: Multi span cable
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
www.phoenix-engineer.com
RE: Multi span cable
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).
RE: Multi span cable
For the cable analogy, I think of the power line. What if a weight was added to each span? What is the new tension in the cable? From that perspective it leads toward it is the same as a one cable / one load span.
You could also picture 10 spans of pulleys with a weight hanging from the rope between each. If one end of the rope is fixed and I am holding the other end, am I holding 1X or 10X the weight in each span?
I think I need to give more credit for the friction at each support.
ZCP
www.phoenix-engineer.com
RE: Multi span cable
In either case, as the fabric pulls from "full dangle" to "almost horizontal", the horizontal force required increases, and it would be a function of total weight being lifted (and other factors).
I can't quite come up with an appropriate numerical model for this on the fly, but the roller supports could be assumed frictionless, so all of the folds flatten at the same rate, to keep the tension in each segment equal. I'm having a hard time reconciling how I can hold up 10 weights with the same force required to hold 1 weight, since the force required to lift one is propotional to the weight lifted.
Exactly how "flat" (or horizontal) are we talking?
RE: Multi span cable
So on one side of the discussion I have a power line analogy. If I hung a small weight on each span between LA and NY, would it break the cable? Seeing all of the powerlines running up and down the road, it doesn't seem to make sense that the weights will add.
On the other side, the pulley / sling analogy, which says the weights add so that 10 spans means 10X. So if we have spans of pulleys between LA and NY and a weight on each span, we would be holding a billion pounds force at the end of the cable.......
I am also struggling with the notion you could hold an infinite amount of weight if you broke it into enough spans....
ZCP
www.phoenix-engineer.com
RE: Multi span cable
"ignoring ... work in moving the weight of the ... weights"
i don't think you can ignore this (and i think you realised that towards the end of your post).
RE: Multi span cable
Respectfully disagree. Lifting 10 weights will take 10x the work, not 10x the force. The force will be the same. The distance required to get the fabric taut is 10x the distance for one span. Hence, 10x the work.
RE: Multi span cable
I see that the force required would be roughly the force to lift one weight on one span, plus friction and other "losses".
RE: Multi span cable
i think that if you had 1 span of this fabric, secured at both ends, and then put a weight in the middle of it it would sag somewhat. if you had two spans with two weights, each span would sag pretty much the same; so that the tension in the fabric for the two span arrangement would be the same as the 1 span.
now if you try to raise the weights (flatten the fabic) by pulling on it you have to pull out twice as much fabric with two spans as you do with one span (just by geometry).
so yes, it would be the same Force, but double the distance travelled, and so doubling the work done (against gravity).