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Can an elastic structure buckle under tensile dead load?

Davide Bigoni's picture

We all know Euler buckling of a beam under axial thrust, but can buckling occur in an elastic structure in which all elements are subject to tensile dead loading?

We provide a positive answer to this question, see http://www.youtube.com/user/RoyalSociety#p/u/0/EKngs1vvcJU

 

More information about my research activity can be found in http://www.ing.unitn.it/~bigoni/

More information about our experiments can be found in http://ssmg.ing.unitn.it/

jason Zhu's picture

Dear Prof. Davide Bigoni,

How interesting it is.

Is there a torsion spring in the left side? If it is true, I do not think  all elements are subject to tensile dead loading over the beam. The bottom fibre in left side may be in compression in inital stage.

Best Regards,

Davide Bigoni's picture

The spring is called "rotational", maybe you call it "torsional", no problems. All elements are subject to tensile loading in the straight configuration before buckling. In the post-bifurcation regime, the structure is under normal force and bending. Therefore, no compressed elements are present before buckling.

Mike Ciavarella's picture

can you send us the ref for the actual paper?

I guess the novelty comes from the slider which makes the system search for the alternative configuration... How we can formulate in general?  

Michele Ciavarella, Politecnico di BARI

 

Davide Bigoni's picture

Hi Michele, nice to hear from you!

The article is: "D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni - Structures buckling under tensile dead load. Proceedings of the Royal Society A, 2011, in press."  and you can find a preprint on my website http://www.ing.unitn.it/~bigoni/

You are right, the novelty comes from the slider! We are in fact generalizing the results both theoretically and experimentally. We are not far from submitting a new article, but this is another story ...

interesting , i'm wondering if laoding rate affects the results. :)

Davide Bigoni's picture

For the moment the structure is purely elastic, so that it is unaffected by the loading rate, but the model can be generalized...

Dear Davide,

0. I browsed with great interest the material at your Web site, including the preprint of the paper. Allow me to make a few observations, perhaps critical in nature.

1. Buckling of a single rod

One may think of a simpler case of a single elastic rod, hinged at one end to a fixed support, and hooked into a slider at the other. Assume that the rod is aligned with the x-axis and the slider, with the y-axis. A tensile load is applied directly to the slider. Will buckling be observed for such a set-up? Two cases can be distinguished:

1.1 In imparting the tensile load, if the slider is constrained to always move parallel to the y-axis, then it would appear that there wouldn't be any buckling of the elastic rod. 

1.2 However, if the slider is allowed to rotate in the xy-plane (by attaching it to a hinge, and then applying the tensile load to this hinge), then, buckling of the elastic rod can be expected.

The reason is that as the slider rotates, one of its end-points comes closer to the other end of the rod (which is hinged to the fixed support). If such shortening is allowed in the prescribed kinematic constraints, then the system can settle (snap) into a lower-energy configuration. No doubt there would be the bending resistance developed in the rod too. However, the kinematic constraints now do allow the system to develop multiple (an infinity of) paths, out of which the rotated-slider configuration has a lower energy.

If the kinematic constraint is such that the slider must remain parallel to the y-axis, then the straight, unbuckled state has a lower energy.

2. The comparison with the Euler buckling: Structure vs. Mechanism

For the usual compressive buckling, let's call it the Euler buckling, no slider is required. For the tensile buckling, it is. This distinction is significant. Hence, a thought crosses one's mind: Is it really apt to compare these two situations?

Another way to look at it is to say that one is a structure and the other, a mechanism. For instance, imagine substituting simple tension springs in place of the elastic rod in your arrangement. Springs are assumed to be unable to bend. However, they will still slide under tension provided the slider was allowed to rotate.

3. Tensile load cannot set up a pulse without a slider, compressive load always can:

This is just an implication of the point discussed in 2. You can always set up a transverse "wave"-like pulse in a long thin rod in a compressive loading. But, you cannot do that in a tensile loading. This distinction remains unaffected by the (otherwise wonderful) study you report.

4. The Zeeman machine:

The arrangement you report on is similar to the Zeeman machine, which also displays a buckling effect under tensile loading. And, it, too involves a mechanism---it's a machine.

But, yes, yours is a very wonderful, and, as far as I know, very novel contrast to the well-studied Zeeman machine. Before you reported, one could not have easily imagined something like this---unlike in the Zeeman machine, the loading motion remains aligned to the x-axis alone, it has no y-axis component. This is truly wonderful.

5. New possibilities for catastrophe-theoretical studies

It is obvious that your machine is bound to give a lot of food for thought to the catastrophe and bifurcation theorists---the areas in which my own knowledge is next to nil. (Reading of a few popular science articles does not make for knowledge.)

All in all, an enjoyable read.


Best wishes,

--Ajit

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[E&OE]

Davide Bigoni's picture

Dear Ajit,

Thanks for your thoughts and comments, very interesting!

I agree with almost all your comments, although I think that a structure may have several constraints at his edges (for instance: it can be clamped, the clamp may also slide, it can be free) of which the hinge and the slider are only two examples. The distinction between structure and machine is not clearly cut. Our way of thinking is that a structure is governed by the equation of the elastica and at buckling it becomes inflexed. Another important observation is that the disk in the Zeeman machine is under compression and is the element responsible for the instability, whereas in our system all parts are subject to tensile load. The key of our structure is that all parts are subject to tensile load. Thanks for your nice comments, hope to meeting you one day!

Best wishes,

Davide

Dear Davide,

Yes, I agree that the disk in the Zeeman machine is put under compression whereas in your arrangment, all parts are tensile. Also, the other point you highlight, namely, that many different kinds of constraints are possible, and their effects on an assemblage (i.e. a general term to denote a structure/mechanism/machine/whatever) are, despite centuries-old history of mechanics, still open to vigourous new investigations.

While on this line, let me also point out the work done on tensegrities; see, for example, Tim Poston's Web page: http://geometeer.com/geometeertensegplaceholder.html

Studies such as Zeeman's, Poston's, and yours, show how much we still have to learn about even "simple" assemblages of simple elements like rods, springs, strings, etc. Obviously, contrary to what many people imagine, mechanics---even classical mechanics---is far from being a dead discipline.

On the point of meeting. Sure, it would be great to meet you in person when you visit India, too---provided that, in the first place, our bureaucrats in New Delhi don't end up denying you a visa or, as necessary, its extension (the way they seem to have done, I gather, in Prof. Tim Poston's case.)

Best,

--Ajit

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[E&OE]

Davide Bigoni's picture

Dear Ajit,

In fact I am just back from India. I have participated in the twelveth event (CMDS-12) of the International Symposium series on "Continuum Models and Discrete Systems (CMDS)", organized in Kolkata by Professor Bikas Chakrabarti and later in the workshop "Fracmeet" organized in Chennai by Professor Purusattam Ray. It has been a beautiful experience! Best wishes, Davide.

Dear Davide,

In that case, it is obvious that you overestimate me!

Best,

--Ajit

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[E&OE]

binoddhakal's picture

It is surprising to see buckling in tension......

Interesting effect.

I agree with the comments of Ajit R. Jadhav.

I note that the capillary surfase, pendulum, classical string, bending of beam and cylindrical membrane have identical equations.

S.S. Antman, M. Schagerl, Slumping instabilities of elastic membranes holding liquids and gases // International Journal of Non-Linear Mechanics 40 (2005) 1112 – 1138

 

Dear Alex,

I simply forgot to add earlier, but that is a helpful observation you make there.

Best,

--Ajit

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[E&OE]

nithesh's picture

It seems like a simply supported beam having horizontal and vertical reactions at both the supporting ends; and also having bending moments in the same direction.

That way, a free body diagram at the centre of the beam would theoretically show "buckling"!

 

Pu Zhang's picture

This work is quite interesting. I am thinking about to change the slider shown in your paper into an elliptic shape, as illustrated in the figure here. When perturbation occurs, the slider would transit from type (a) to (c).

Pu Zhang's picture

I do not know why the figure cannot be shown. In Fig. (a), a couple of tensile forces are applied on the short axis direction of the elliptic slider. In Fig. (c), the couple of tensile forces are applied on the long axis direction. If you are interested, I can send the figure to you personally.

Davide Bigoni's picture

Dear Dr Zhang,

I don't see the figure, but I guess that what you are pointing out is very close to the subject of a paper that we are terminating in these days, so that I prefer not to see the figure...Thanks for the comment! Best wishes. Davide

buckling instability

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