Concentric Tubes in Bendine
Concentric Tubes in Bendine
(OP)
Hello all, I am analyzing a design that involves a stainless steel tube in bending. This tube is reinforced with a smaller "doubler tube" snugly fit inside (its OD being equal to the larger tube's ID). The tubes are not bonded, and are free to slide, ignoring friction. I have scoured the internet and my textbooks to no avail - how is the section modulus for this configuration calculated? The area moment of inertia of the cross section is the same regardless of the tubes being bonded or free to slide. I assumed, similar to composite beams, that two sliding tubes would be less stiff than one thick tube... Is this not true?
Many thanks
Mike
Many thanks
Mike






RE: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
Since they are "with" each other in terms of deflection and share the same neutral axis, an equivalent shape would be a shape with the sum of the two moments of inertia...similar to two beams side-by-side but forced to deflect the same amount.
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RE: Concentric Tubes in Bendine
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RE: Concentric Tubes in Bendine
Thanks for your reply. My hangup is this - a shape with an equivalent moment of inertia is just the same two tubes, considered as a solid cross section (unlike a stack of rectangular layers, where the MOI of the composite beam is larger than the sum of the layers). Is it possible that the tubes would not tend to slide under bending, meaning there would be no difference if they were bonded?
RE: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
That being said, with the assumptions made in beam bending derivations, if the load is proportioned so that calculated deflection of both tubes is the same, there wouldn't be any sliding or shear between the two tubes.
RE: Concentric Tubes in Bendine
1) The two sections will behave compositely in flexure in the sense that matters: they will share a common strain diagram.
2) The composite section properties will be identical to the non-composite section properties (I_in + I_out).
So there you have it. You get composite behavior but, in this particular case, it's of no particular benefit.
Assume that the section behaves compositely (or non-compositely as it's irrelevant here) and calculate S = (I_in + I_out) / c
No, that assumption is incorrect in this instance.
You can simply add the two tube wall thicknesses for equivalent flexural properties.
That is indeed correct and is probably the key thing to understand about this situation.
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: Concentric Tubes in Bendine
The Mc is treating the tube as one, and looks at the stress at extreme fiber of the outer tube and extreme fiber of the inner tube. the results are the same whether you proportion the load by stiffness or treat it as a composite shape.
RE: Concentric Tubes in Bendine
http://weldingdesign.com/processes/using-gussets-a...
RE: Concentric Tubes in Bendine
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: Concentric Tubes in Bendine
you may have a point load on the outer tube, but the inner tube would load up from a non-uniform distributed load (as the outer tube bears against the inner).
my sense says that the two tubes should be less stiff and a single tube, because you have a shear slip boundary between the tubes.
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RE: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
1. Check the same two tubes independetly....assume an equal(same) deflection for each and find it's
corresponding applied load. Add the 2 loads to get total load.
2. Check the capacity for the same defection of an equivalent tube with a total I equal to the sum
I1 & I2 and with an OD of the larger tube and wall t equal to t1 + t2...
3. Compare the loads of method 1 with method 2....
RE: Concentric Tubes in Bendine
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RE: Concentric Tubes in Bendine
as a sliding fit, there is an obvious difference and the interface of the tubes between the two tubes and a single thick tube. I'd expect the two tubes would deflect more than a single thick tube.
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RE: Concentric Tubes in Bendine
1) The vertical positions of the neutral axes for both members are coincident and;
2) The neutral axes of both members are made to have matching curvatures.
1 + 2 = identical strain profiles for both members at all locations. And identical strain profiles, by definition, means no slip.
I feel that structSU10 pretty convincingly demonstrated that I_composite = I_non-composite. Do you dispute that? If not, why would you think the composite section to be any stiffer?
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: Concentric Tubes in Bendine
do you see a different if the inner tube is interference fit ?
in the single thick tube, is there no stress on the parting plane ? (I don't think so, but there is in the two tube case)
in the two tubes, a single point load on the outer tube would not load the inner tube as a point load (IMHO) but the inner tube would be loaded (and the outer tube relieved) by a distributed load (as the outer tube bears against the inner one).
I'd analyze as the outer tube loaded by the point load, reacted by a distributed load and end reactions. At the same time the inner tube is loaded by the distributed load. now calc deflected shapes and tune the distributed load so that deflections of the two are the same.
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RE: Concentric Tubes in Bendine
Force follows deflection and visa-versa. Hooke's Law.
No matter how they are loaded, the two tubes are deflecting in the same curve/pattern/degree and so they would indeed (as KootK mentioned) share the same strain diagram.
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RE: Concentric Tubes in Bendine
I have not run the numbers, but my intuition tells me that the stiffeness of the 2 tubes would be approx 75% of that of an equivalent tube as described in the previous posts...then , again, I would not bet my shirt on it....
RE: Concentric Tubes in Bendine
surely in the thick tube there is some stress where the parting plane is in the two tubes ? surely this affects the results.
surely two tubes with an interference fit would behave differently compared to two tubes with a sliding fit ?
two tubes with an interference fit would behave like a single thick tube.
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RE: Concentric Tubes in Bendine
In theory no. Again, because the strain diagrams are identical, there's no need to engage friction whether it's available or not.
In reality, I would expect a tight friction fit to change stresses a bit. Intentionally, I've just been discussing flexural behavior assuming that plane section remain plane. In the wild, the strains at the extreme ends of the two tubes would probably differ a bit due to shear lag up the sides of the tubes. That difference would be, at least partially, ironed out by the action of the friction generated at the interface.
I would expect there to be some stress at the faux-interface plane in the solid section case. This is conceptually similar to what I mentioned above with respect tot he interference fit. Continuity in the material would tend to enforce strain compatibility more convincingly than shear lag.
No doubt, this phenomenon would have an effect on local bearing stresses etc. As JAE described, however, it wouldn't impact theoretical flexural strains.
That's precisely the point, it's not necessary in certain cases. Regarding the wood example, consider:
CASE 1: tube and tube as described here.
CASE 2: 4X10 main with with a pair of 2x6 sides for reinforcing. The center lines of all member align.
For both cases, I would submit that the following is true:
1) Both members need some mechanism by which vertical load can be transferred from the directly loaded member to the member not directly loaded. Without this, the curvatures of the various members cannot be made to match.
2) For the tubes, the vertical load transfer mechanism is some complex bearing situation, as intimated by rb1957.
3) For the wood, it's vertical shear transfer in the bolts.
So the bolts in the wood are necessary, just not for horizontal shear transfer. Their role is solely to transfer vertical shear and enforce curvature compatibility among the plies. Fundamentally, this is why we don't have to bother with VQ/It forces when we laminate multi ply wood beams of the same depths. There's just no demand for horizontal shear transfer.
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: Concentric Tubes in Bendine
rb1957: you confirmed that you agree that I_comp = I_non-comp. With that being the case, how do you justify deflection being different for the two cases? EI is the same for both. How is it that you see the beam behaving differently from our normal Bernoulli assumptions? Is it more than an intuitive feel?
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: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
It can't be because the horizontal shear transfer would be required. With horizontal plies, the important feature of this thread's example is lost: the neutral axes of the plies, acting independently, are no longer aligned vertically.
The mechanism is web shear (and potentially lag as I mentioned above). Same as how strain compatibility is enforced between the flanges of a stand alone I-beam. With tube in tube, the two tubes do this job independently but the net effect is the same.
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: Concentric Tubes in Bendine
I_composite = I_non-composite. Do you dispute that? If not, why would you think the composite section to be any stiffer?
Quote (rb1957)
"Do you dispute that?" no, the math is straight forward; but will be beam behave that way is a different question.
I was using "lawyer speak" ... I agree with the math presented, I just don't think the underlying proposition (that deflection of a single thick tube = deflection of two tubes) is correct.
We agree that there is some stress happening on the "faux parting plane" in the single thick tube. I think two tubes with interference fit would exhibit the same behaviour as I think there's something there that'll allow the stress to develop (the interference between the two tubes allows them to talk to one another). If there's a sliding fit then this stress can't develop and so the two tubes are different to the single tube.
How big a difference ? As I've proposed, if the outer tube has a point load, the inner tube will have a distributed load, maybe something like a sine wave. If the outer tube is loaded by a distributed load then I think the difference is very small.
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RE: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
I maintain that the two pipes do deflect together (in unison), but they share the load in proportion to their relative stiffness’, functions of their MoI’s, E’s and lengths. Since E and L are the same for each, the MoI is the controlling factor. The outer pipe will carry more of the load because of its greater MoI. The section modulus (Sx) must be figured for each pipe, and because of the slip btwn. them there is no single element equivalent for this part of the problem. On the 6 - 12 o’clock axis the stresses might look something like this (no real numbers were run as structSU10 did): near 6 o’clock, at outer o.d. (3"/2) σ = +50ksi; at outer i.d. (2.75"/2) σ = +45ksi; at the inner o.d. (2.68"/2) σ = + approx. 50ksi; at the inner i.d. (2.5"/2) σ = + approx. 45ksi; at the N.A. σ = 0ksi; near 12 o’clock, at the inner i.d. (2.5"/2) σ = approx. -45ksi. Note that at the (2.75"/2) or at (2.5"/2) levels, or any other level the stresses are not the same in the inner and outer pipes, but the deflected shape is.
RE: Concentric Tubes in Bendine
the stresses in the smaller tube must be higher than the larger one as their deflected shape is the same and the smaller tube has a smaller I/y (proportional to OD^2*t)
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RE: Concentric Tubes in Bendine
Rolling on his side to make room for some clapping? Just foolin'... I know what you meant.
As I see it, the sharing of a common strain diagram is what puts the "composite" in composite behaviour. A cross section could be comprised of interplanetary dust particles seperated by miles for all the difference it makes in Bernoulli flexure. So long as all of the parts follow the same strain diagram, an equivalent section is possible for stress and deflection calcs. And simple as well. All equivalency requires is the ignoring of boundaries / empty space in recognition of strain matching there.
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: Concentric Tubes in Bendine
I believe that stresses will be higher in the larger member. Consider:
1) again, that common strain diagram and;
2) the the outer tube attracts more moment which will increase stress.
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: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
I haven’t put any real numbers to the real problem yet. I am suggesting (I believe) that the two pipes will be at about yield (50ksi, whatever) at their own extreme fibers at about the same time. This is due to their own/individual M/Sx, and due to their different loading proportion, sharing, which is due to their different stiffness’. I agree with you that there will be some funny localized loadings or stresses; your example, the point load on the outer pipe, being somewhat distributed to the inner pipe at the same general location, some length on the beam. For the slip fit, the two pipes are more or less forced to deflect the same, +/- a few hundredths or thousandths of an inch, by their almost intimate contact. But, these are smaller than the straightness or ovalty tolerance in the pipe, thus some interference in a long length. I also agree with Koot that the interference fit will introduce some new triaxial, very complex, stresses into the picture, that none of us really want to try to define here. The point that I was trying to make is that the two pipes act independently from the normal bending stress standpoint and that except at the N.A. level, where σ = 0ksi in both pipes, the stresses are not the same at any other level in the beam system. Certainly not like someone might be inclined to think if they went too far down the single composite section ‘pipe line,’ er path. I’ll leave the more complex secondary affects to another day.
RE: Concentric Tubes in Bendine
If both tubes are bending along the same deflection curve then stress is proportional to distance from the NA (as M/I is constant) ... not what I posted before ... I still think that losing what is happening on the "faux parting line" should have an effect.
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RE: Concentric Tubes in Bendine
What do you mean by "assuming that plane section remains plane". Are you referring to the cross section relative to the NA? Why would it be out-of-plane? Also, where does shear lag come into play? At the released ends of the tubes?
This is now beyond the scope of my analysis, but it is good to know/understand nonetheless.
RE: Concentric Tubes in Bendine
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: Concentric Tubes in Bendine
The larger tube just gets there first.
RE: Concentric Tubes in Bendine
Interesting academic discussion though.
RE: Concentric Tubes in Bendine
practically, since we are talking about a close tolerance fit, i'd probably freeze the inner tube, having verified that it won't be loose in it's final position. maybe scan the ID of the large tube to get a true representation. and, yes, I realise that these ideas are not normally practical.
another day in paradise, or is paradise one day closer ?
RE: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
another day in paradise, or is paradise one day closer ?
RE: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
Hey, even a blind squirrel finds a nut once in a while. Thanks for doing the leg work and for reporting back to share the results.
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: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
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: Concentric Tubes in Bendine
BA
RE: Concentric Tubes in Bendine
First one, the smaller tube is a little bigger than specified and then it won't fit without getting a lot of residual stresses.
Secondly, the members perfectly size each one another and then it will behave as described.
Third, probably the most common situation will be to have a gap.
RE: Concentric Tubes in Bendine
In the interest of precision, let's revise that to:
It'll work so long as the respective neutral axes of each piece are at the same location and are made to travel in unison.
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: Concentric Tubes in Bendine
another day in paradise, or is paradise one day closer ?
RE: Concentric Tubes in Bendine
rb1957's intuition is correct. Without bonding there is no composite action.
The logic that because the neutral axis is the same for both tubes whether they are bonded or not proves that there must be composite action is flawed. Let's take the case of three point bending. If you take the two tubes separately and bend them the same amount they will each have a certain load in the center of the span. If you place one tube in the other and deflect them the same amount, the measured load will be the sum of the loads measured for each tube. This is non-composite action. So why is this?
The factor that has been overlooked in all of the discussion so far, are the hoop stresses in the tube. When a tube is flexed it wants to ovalize where the tension and compression sides move closer together, which results in a lower "I" in axial bending. In an I-beam the web prevents this. In a tube only the walls can do this and they behave as two "C" shaped springs. So, with a tube in a tube that are not bonded you have four thin springs. When they are bonded they act as two thicker springs, which are stiffer than the four (composite action).
A unique feature of fiber reinforced composite material is that you can tailor material properties by changing the fiber orientations. One way to make a carbon fiber tube is by pultrusion, where fibers and resin are pulled through a shaping die. All the fibers are axial in the tube wall (think fiberglass tent poles). When flexed too far, these tubes fail by splitting axially because they ovalize, and there is no fiber reinforcement in the hoop direction. They have low axial stiffness because of the low hoop stiffness.
Tubes can also be made by table rolling where two sheets of unidirectional preimpregnated fibers (prepreg) are laminated together in a 0/90 orientation and then rolled around a steel mandrel and cured in an oven. Interestingly, a tube that is made from 50% 0 and 50% 90 fiber is stiffer in axial bending than a tube made from the same amount of fiber that is all in the 0 degree direction.
RE: Concentric Tubes in Bendine
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RE: Concentric Tubes in Bendine
Actually, in this instance, that is composite behavior. That was one of the interesting findings above. Here, IE_comp = IE_non-comp.
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: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
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: Concentric Tubes in Bendine
BA
RE: Concentric Tubes in Bendine
RE: Concentric Tubes in Bendine
I would characterize the two tube setup as "trivially composite". That, implying:
1) Sure, it's composite but then the composite properties are no improvement over the non-composite properties. A bit like concentric sistering in wood construction.
2) Demand for shear flow shear transfer to achieve "composite" is nominally zero unless one introduces a meaningful gap or drills down deep into more granular effects.
I believe that #2 is the important insight from the perspective of OP's original question.
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.