multi span cable

huangyhg

multi span cable
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
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.
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.
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.
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.
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.
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.
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
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.
"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 ...
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.