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Slender tensioned rod be approximated by cable calculations?

Slender tensioned rod be approximated by cable calculations?

Slender tensioned rod be approximated by cable calculations?

(OP)
Greetings everybody - I'm a first time poster, long time lurker here on the forums, and hopefully I have an easy question for you structural guys and gals.

Say I have a solid steel rod (44" length x .5" dia) tensioned horizontally between two fixed points, then apply a load perpendicular to the rod somewhere along the length, for this example we'll say right in the middle. Is it reasonable to approximate the deflection at the point where the load is applied by using a cable tension/deflection calc? The tension force and the applied load would both be much larger than the weight of the rod itself. Really, what is a cable other than a bunch of tiny rods twisted together?

I have done the calc assuming that this rod is a simple beam, from what I can tell, that doesn't account for the affect the tension in the rod has on the deflection. I'd like to minimize the deflection while maintaining the smallest diameter permissible.

If this is not a good approximation, perhaps someone can point me in the right direction...digging through old textbooks and google searches have not been very fruitful thus far.

Any input is much appreciated.

RE: Slender tensioned rod be approximated by cable calculations?

Will the rod be pre-tensioned?

RE: Slender tensioned rod be approximated by cable calculations?

If the only parameter of interest is deflection, then treating the rod as a cable should give an upper bound estimate. Keep in mind that cable like behaviour probably means flexural yielding at the point of application of the load.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

RE: Slender tensioned rod be approximated by cable calculations?

(OP)
Trenno - yes, the intention would be for the rod to be pre-tensioned.

KootK - do you mean upper bound as in max deflection that will occur? The load will be applied over approx a 1"-1.5" distance, so it will be distributed at least a little bit.

RE: Slender tensioned rod be approximated by cable calculations?

Yup, upper bound estimate of deflection. Flexural resistance of the rod will make deflection less than the pure cable model would predict. 1.5" inches isn't much at all. Plan for the rod to yield in flexure.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

RE: Slender tensioned rod be approximated by cable calculations?

Maup:
If you apply the point load onto a saddle which then applies the load to the rod, you can reduce the deflection and probably eliminate the rod yielding at the point load (the sharp change in direction or shape of the rod). This saddle might be a 1.25" sq. by 8" long stl. bar, with a .5" dia. semi-circular groove machined, along its length, in the bottom of the bar. This groove would also have some circular shape, in side view, a function of the deflected shape you wanted right under the load. In these kinds of calcs. wire rope doesn’t have much bending stiffness, so we ignore it except where extreme bending occurs. The cable calcs. should be a reasonable upper bound on the deflection. But note, any point load on this system induces very high lateral loads at the end supports, so they are difficult to restrain, and this causes very large tension loads in the rod or cable. If you control the sharp angle change immediately under the load, where you could get relatively large plastic elongation, take a look at the high tensile stress causing rod elongation, and this through trig./geometry should give you are fairly accurate deflection in the region of the load. You might have to do some support groove shaping to prevent yielding at the two end supports (plastic elongation, yielding again), or at least set those supports at a slope matching the deflected shape of the rod.

RE: Slender tensioned rod be approximated by cable calculations?

If you need to load something like this, why not just use an actual cable? Either that, or put a rotating pin of some sort at the point of load application so the rod has a hinge there. Then you can detail the supports to rotate and you'll get pretty close to true tension only behaviour.

dhengr, I'm pretty sure you're still going to yield while mating to the saddle if this is any significant load. You're talking about large deflection behaviour here.

RE: Slender tensioned rod be approximated by cable calculations?

an approximate approach might be as follows:
a simple span beams model as you have indicated will give you a maximum deflection, a curved shape and calculable deflection, but maximum as that deflection will be reduced by the actual presence of tension in the member.
I envision the application of tension to that deflected shape functioning like a post tensioning cable under a beam with a harp at midspan, the tensioning of the cable wanting to straighten the cable and thereby creating an upward force on the harp at beam midspan. That upward force applied to the simple span beam will reduce the deflection. the summation would be the minimum deflection. the post tension analogy, of course, creates triangles which can be solved trigonometrically, whereas in your case, a portion of the force would be expended in attempting to straighten the curvature of the rod rather than exerting an upward force. that calculation is above my pay grade.

RE: Slender tensioned rod be approximated by cable calculations?

Triangled:
That’s about what I had in mind, the post tensioned tendon analogy may be a better way to express/explain it. And, that is about the approach I would have taken to start a set of calcs. I just didn’t want the kinks (changes in tendon shape) to be too drastic at the end reactions and at the point load, or you could expect certain yielding and plastic elongation, due to bending, fairly quickly and fairly large in extent. This is not a structure which will redistribute much load, once a plastic location forms, you pretty much will have a failure/collapse mechanism. You probably won’t have instant failure as long as the loading stops (doesn’t grow further) as yielding starts, since you still have to climb the strain hardening slope of the stress/strain curve, and the geometry improves as rod elongation increases.

RE: Slender tensioned rod be approximated by cable calculations?

(OP)
Thanks for the replies, I will have to try and absorb the information further later today/tomorrow AM - duty calls out on the factory floor today it looks like.

To give you guys a little more insight - I'd be looking at a 250-300lbs worst case near the ends. Basically its a circular array of rods resisting a torque as it travels along the length, where the torque requirements are higher near the end than mid-span. I can control the load on each rod by adding rods, but I can only add so many before it I run out of room. I can also change the diameter rod to help reduce deflection, but this in turn effects how many rods I can reasonably fit. The load requirements in the middle of the span are typically much less than near the ends, but I do have 1 load scenario requiring close to the full amount at about 3/4 of the way through the travel.

I'd love to increase the saddle length/bearing distance, but I am under some pretty tight envelope constraints. I might get 2.5" at best.

Typically our efficiency in the system is much better so that 250-300 lbs is down to 50-75 lbs so we usually don't have issues with deflection for the size/span rod I am looking at.



RE: Slender tensioned rod be approximated by cable calculations?

(OP)
Sorry it has been so long, this project was put on the back burner for a bit.

Is the tension from any preload purely additive to the tension caused by the weight hanging in the center of the span? From the equations and another thread I was reading either here or on the physics forum it seems that way but I wanted to be sure...

Thanks again.

RE: Slender tensioned rod be approximated by cable calculations?

yes...modified by the angle created by the deflection

RE: Slender tensioned rod be approximated by cable calculations?

You might like this OP: Link

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

RE: Slender tensioned rod be approximated by cable calculations?

A direct solution to the lateral deflection of a tensioned rod can be found in "Strength of Materials", Volume 2, by Timoshenko. The equation is on page 39 in the second edition and on page 42 in the third edition for the case of a concentrated load applied at any location along the span of a simply supported rod. I checked the equation for the case of a concentrated load at mid-span with a very small axial tension force against PL3/48EI and the results were in close agreement.

RE: Slender tensioned rod be approximated by cable calculations?

(OP)
Hokie93, that equation is for pinned ends, yes? Both ends were fixed so I was thinking 192 would be more appropriate than 48, unless we are talking different equations here.

RE: Slender tensioned rod be approximated by cable calculations?

Yes, the equation I referenced in my previous post was for pinned ends and, hence, the comparison to PL3/48EI. If the ends of the rod are fixed, then the coefficient in the denominator should be 192. Timoshenko does not provide a direct solution for a tensioned rod with a concentrated lateral load at mid-span with fixed ends but it can be arrived at by the method of superposition from the equations he does provide (simply supported tensioned rod with concentrated load at mid-span and a tensioned rod acted upon by end moments).

RE: Slender tensioned rod be approximated by cable calculations?

if the rod is being considered as equivalent to a cable, why fixed ends ? the simple reactions for the applied loads are the rod/cable defelcts into straight line segments, and the component of tension reacts the applied load (so you need a lot of tension to react the load). Preload should be accounted for by saying the tension in the cable is Preload + X, so that you'll need a smaller angle to generate the required component.

clear as mud ?

another day in paradise, or is paradise one day closer ?

RE: Slender tensioned rod be approximated by cable calculations?

(OP)
the rod is threaded on both ends, one end into a housing up to a shoulder, and the other end will have a nut on it (the method of pretensioning), perhaps you could call that end pinned if it isn't shouldered up to anything besides the nut.

The link KootK directed me to with the post tensioning equations might do the trick...using equation 3 section 3.3. Only difference is my load technically isn't moving at any real velocity, so instead of using MV^2 for kinetic energy portion, I'd have to come up with another energy term. Still working on that one.

RE: Slender tensioned rod be approximated by cable calculations?

how much moment can a thin cable-like rod support ?

another day in paradise, or is paradise one day closer ?

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