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

 

[zcp]

If the weights are bars running parallel to the supports (in and out of the paper with supports being to the left and right), could that dimension be summed up to the 2d analogy of a cable with weights?

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

http://www.phoenix-engineer.com

 

[grandcasita]

The analogy is close, but the force required (ignoring acceleration and work in moving the weight of the fabric and weights) is a function of the angle of the fabric and of weight. If the weight used to pull the fabric down is greater than the fabric weight, a better analogy might be slings used in crane rigging - the force required is a trigonometric function of the angle of the leg (thinking of essentially straight legs.)

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?

 

[zcp]

The catenary analogy says 1/4" to 3/8" for a 4 foot span of fabric only. That seems reasonable for water drainage, etc.

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

http://www.phoenix-engineer.com

 

[rb1957]

grandcasita ...

"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).

 

[miecz]

rb1957

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.

 

[grandcasita]

Yes, you can't ignore the friction or work, I simply made that statement to simplify the force question. And work v. force is where I was having trouble making an analogy to the use of pulleys.

I see that the force required would be roughly the force to lift one weight on one span, plus friction and other "losses".

 

[rb1957]

jmiec,

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).