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Optimal structural design against elastic instability

In the classical Euler buckling problem, the critical buckling load can be increased by a factor of four if the first mode is suppressed by placing an additional simple support at the mid-point.

If we solve the more general problem where the additional support is placed at some different point z=a in 0<z<L, the critical load will be found to increase above that for the unspported first mode, but the maximum increase is achieved when the support is at the mid-point and buckling then occurs of course in what would have been the second mode of the unsupported beam.

It seems likely that this behaviour would apply to all elastic stability problems, but I have been unable to find a rigorous proof or even a formal statement of the result in classical stability texts. To promote some discussion, I propose the following theorem:-

"Suppose the stability problem for an elastic structure under a load P is formulated as a small-displacement linear eigenvalue problem with critical loads P1,P2,... such that P1<P2<P3,... etc with corresponding eigenmodes u1,u2,u3,.... If a single support is now placed at the (presumably unique) node of the eigenmode u2, so as to constrain the displacement there to zero, the critical load of the new system will be larger than that obtained by placing a single support at any other point in the structure.''

The new critical load will of course be P2 and the support will actually not be required to transmit any force. By contrast, if the support is placed at any other point, it will transmit a force when the structure buckles.

If anyone can give me a reference to a statement and/or a proof of this theorem or something equivalent or related, I would be very grateful. If not I propose it as a challenge, either to prove the result or disprove it by finding a counter-example. It is potentially a useful result for structural design, since efficient (low weight) structures tend to be thin-walled and hence limited by stability considerations. Also, increasing the stability threshold by adding supports in this way is an efficient solution, since the added supports theoretically carry no load and hence are not required to be particularly strong. They do however need to have some critical elastic stiffness.

These results are easily established for the classical Euler buckling problem of a simply-supported beam of length L loaded by an axial force. A more interesting case concerns the cantilever beam of length L1 loaded by an axial force P a distance b from the built in end, for which the optimal support is a distance (2b/3Pi) behind the force.


yawlou's picture

Hello Professor Barber,

That is indeed a very interesting problem.  You probably have looked in Timoshenko's book and I suppose you have the book, Zdenek Bazant and Luigi Cedolin, STABILITY OF STRUCTURES, Dover, 2003.  That is where I would look first for the answer.  I will take a look because it is an interesting problem that you pose.

Perhaps one could create a proof by an energy approach.

I like the topic and the insightful theorem.  Your theorem makes sense to me.  As you have mentioned, the question is, can we find a counterexample to your theorem, or can we prove your theorem?



Alejandro Ortiz-Bernardin's picture

Dear Professor Barber,

This problem is actually very interesting to me. If I understood well, you are implying that adding a support at the nodes where zero-displacement occurs for the corresponding eigenmode, then in an stability analysis we force the analysis to find the critical load for that eigenmode, and since P1<P2<P3< .... eventually the problem is not determined by elastic instability. Please, correct me if that is not what you meant. On the other hand, could you please give some 3D examples where this would be useful? For example, in stability analysis of 3D thin shell structures.



Dear Alejandro,

I interpreted the question as;

We know the buckling loads for a beam, and we know that the best placement of a support is at the stationary point of the 2:nd buckling mode, which will make the critical load will change from P=P1 to exactly P=P2.

Does this same behaviour hold for all structures? If so, how to proof this? Or alternatively, is there some structure where optimal placement (that will give a larger P) of a support is not at the 2:nd modes stationary points?

Such a theorem written by Professor Barber would of course be of great value. Optimal placement for any structure can be identified directly, simply by looking at only the higher mode, and if it doesn't hold true for any structure, then it's of more interest, as this could be a typical assumption of many engineers resulting in unoptimized structures.

I dont see anything indicating a vanishing stability problem, simple making the best out of as few supports are possible.

Edit; And the theorem right now only considers the first buckling modes. Even if you might see the supported structure as a new problem, and built on top of that, there is no proof that there isn't a better placement solution for the same amount of supports (outside two stationary points for u2 and u3) that would bring P > P3.

Edit 2; I reread your post and I realize that my post doesn't make much sense now, as we more or less interpreted the theorem in the same way.

Alejandro Ortiz-Bernardin's picture

Dear Mikael,

Thanks for your comment. I didn't want to imply that the instability problem would vanish. Just I was thinking in this problem: Assume you have a structure which possibly can buckle and you want to do a elastic design. Then, you start doing the stability analysis for which you determine the critical load that would cause the elastic buckling of the structure. Let's assume that you start putting supports at stationary points of the eigenmode u_i, thus forcing the analysis to find the critical load corresponding to the critical load for that u_i. Now, assume you start doing this repeatedly, and eventually you find a critical load that would cause a plastic failure (is this possible?). Because you are dealing with an elastic design, you are not interested in the plastic range, and thus your limiting case is a load that would cause a stress in the elastic range below the yield stress, so you can say that the design is determined by ductility rather than elastic instability (not sure if this is the best way to say it in english!).



I think an iterative method of adding supports would be suboptimal.

For the beam above, to achieve P=P2 you would put a support at L/2, but to reach P=P3 you would place two supports at L/3 and L*2/3. So to achieve P=P_i you should just directly add supports on the modes of u_i.

I dont know how if this holds for other structures.

I agree that if we plan to add several supports (say n>1), we should calculate the mode corresponding to P_{n+1} and locate the supports at the n nodes of this mode. I am assuming here that we do not have the freedom to move the original set of supports corresponding to P_1. I don't know whether there is any theorem to guarantee the existence of the required number of nodes --- i.e. whether the nth eigenfunction always crosses zero n-1 times.

For the problem I posted today, the third mode has a double node in the range 0<z<L, where the mode shape is just tangential to the axis, so I suppose that is OK. In fact, we get the interesting result that for an even number of simple supports, these should optimally be placed in pairs (satisfying zero displacement and slope) at points in 0<z<L, whereas for odd numbers of supports, one is always outside in the region z>L.

Peyman Khosravi's picture

Prof. Barber;

This is an interesting problem, however it should be rephrased in order to consider all cases. For example, in the case of plate buckling, nodes (points with no deflection in the buckling mode shape) are actually in the form of nodal lines. So if you want to add a simple support at a nodal point corresponding to the second buckling mode of a plate, where do you exactly put it? You have many choices. You may need to add supports on all nodal points corresponding to the second mode.


Peyman Khosravi

Dear Prof Barber:


A very interesting thought. I am not sure how useful or relavnt the following may be to your question: 

An analogous problem was considered by Lord Rayleigh in the context of vibrations (Theory of Sound).  The question is to determine how natural frequencies change due to the imposition of a constraint. The result is called Rayleigh's  interlacing theorem or theorem of constraints, which, loosely put states that the constrained linear eigenvlaues (frequencies or buckling loads) are bound by the unconstrained linear eigenvalues.  For one dimensional problems in vibrations this works beautifully. However, I know cases in two dimensional structures (freqeuncies of circular saw) where the constraint theorem is not obvious. 


Do you know of any work that deals with the problem of transients involved in such a process, i.e. from the unconstrained state to one of the constrained states, say, for a 1D system or so? Pointers to simple or textbook kind of introductions would be most appreciated.

(I come to mechanical engineering from a metallurgical/materials background where such things are not treated in any depth; most everything about vibrations that I know comes from my first year course and, of course, some readings on the physics of vibrations, like, spherical harmonics.)

Thanks in advance

Dear Prof. Barber:

   The theorem you proposed is true. In fact, it can be proved that imposing the
   constraint on the deformation can only increase the value of P_i^old, i.e.

    P_i^new >= P_i^old, where P_i^new and P_i^old denoting the i-th eigenvalues of the constrained and original unconstrained system, respectively. But P_i^new will not exceed P_(i+1)^old, which is the (i+1)-th eigenvalue of the original unconstrained problem. In summary, we have

     P_i^old <= P_i^new <= P_(i+1)^old   for i=1,2,...

   Your conclusion is a direct of deduction this proposition.

   The following reference is about this topic:

   Weinstein, A., The buckling of plates and beams by the method of zero Lagrangian
                  multipliers and zero divisors. IJSS Vol.4 1968, No.5 p.579-583.

Dear Ajit,

I am not sure about constrained to unconstrained transient problems.

 Transient free vibration (without external forcing) is purely governed by natural modes. So Rayleigh's  interlacing theorem gives bounds on the eigenvalues of the constrained sytem interms of the unconstrianed system's eigenvalues (critical loads in buckling or natural frequencies in vibrations).

Guo's reply contains the formualic statement of the interlacing result for buckling problems. I have to read the paper cited. But the result in the most general form (linear eigenvalue problems) was known to Rayleigh (and perhaps to even Cauchy!).

It is hard for me to give a direct text book reference other than refer you to Rayleigh's theory of sound itself. If you are keen I can share some notes that I have.  Unfortunately, not many modern text books these days even acknowledge the works of Rayleigh and Love to many problems in mechanics.  But, Timoshenko's vibrations problems in engineering is a good one that I consult often.

 A recent paper with a colleague of mine, some what ralated to the discussion,  that looks at Rayleigh quotient in dissipative systems has appeared in Journal of Applied Mechanics. See Here we are asking if we can  get bounds for the complex eigenvalues of damped systems (both viscous and nonviscous type).

Two dimensional vibration problems, and buckling problems (presumably?), could be worth looking. The nodal diameter/nodal circle patterns of a circuar plate with non uniform density is a case in point where the eigenvalue pattern seems to counter what we expect from constraints theorem.

For a buckling problem one could consider critical  buckling loads of an anisotropic/composite plate.  

 I  leave it at that.


Dear Srikantha,

If it's Rayleigh and Love, I do have easy access to both at the British Library in Pune. Also others. ...

Yes, it's true that Americans of recent times, to put it plain and simple, hate to acknowledge non-Americans' work even while making full use of the latter in many different ways... At least for as small a value as to garner publicity of one's own Web site if not for anything else. (And then, they do oftentimes go far beyond too, outright stealing even valuable work/ideas/suggestions/credits too. One has learnt to remain on the guard all the time. (All of this applies to today's Americans---not the Americans even as late as a generation or two ago, though exceptions will probably always exist at all times---I mean, the distribution changes, and these days, for many years now, the distribution has been peaking near "stealers".)) The stealing tendency gets spilled over in the "principled" denials of a Rayleigh here, or a Griffith there, and so on... 

Anyway, my interest in the matter of the abovementioned transition (and now come forward, Americans; come forward to pursue at least the side-threads suggested by this writing of mine if not also its main idea---something which did not strike you earlier---so as not to give me any money and/or credit for it; it would be so perfectly like you) was from the following viewpoint:

If you take an empty blackbody radiation cavity and introduce monochromatic radiation and then shut the cavity window, over a period of time, you would still get a spectrum---the monochromatic energy would get redistributed to give you a spectrum which follows Boltzmann's distribution. (Theoretically, it would be possible even if you introduce exactly one photon into the cavity.)

I was thinking along these lines just when you happened to mention your (unrelated) point. Naturally, I got curious if there can be any leads into the mechanics of the transients at least for the classical case. ... Apparently, as you say, there aren't any easy sources on it... OK.


(I now expect many Brits etc. to misunderstand me and make a show of their support to the poor hurt Americans. (LOL!) (Carry on...))

- - - - -
Even as you read this, I remain jobless (as I have, for years)

 Buckling of thin walled shell under hydrostatic pressure also exhibited the same behavior.A simply supported stainless steel  shell resulted in 1st buckling mode at 0.8 MPa pressure. Now the same shell is constrained radially at three points 120 deg apart.This resulted in 1.6 MPa critical buckling pressure. Thus by adding supports,we can design stronger shell structures against buckling.




As background to my original suggestion, here is a link to the solution of a buckling problem, (i) using the theorem and (ii) not using it.

Also, here are a few additional comments and/or guesses:-

1) With regard to the application to two and three-dimensional problems, I agree that to be effective a support may need to consist of a line on the surface of the body, rather than a point. I'm not sure whether there would be anything optimal about placing a support at one point on the nodal line and if so whether all points on the nodal line give the same result. I suspect not for both these questions.

2) There are clear parallels between the vibration problem and the buckling problem. Thus, we should expect that placing supports at the nodes of the second vibration mode will raise the frequency of the first mode by the maximum possible from one support, and similarly for the effect of bearing location in raising the whirling speed of a shaft (which might carry a number of heavy gears).

3) I read the paper by Weinstein. It is clearly relavent to the problem, but I confess that I was unable to draw from it a proof of the theorem. I wonder whether Professor Guo could explain it, preferably by applying the Lagrange multiplier argument to the simple problem that I link above. I recognize that adding a support anywhere will raise (0r at least not reduce) the critical load, but I didn't see the step in Weinstein's paper proving that the maximum that can be achieved is the second mode critical load. [What I am hoping for here is something simple enough to explain to my students!]

4) I hope Professor Jadhav is not suggesting that I am trying to claim ownership of this result. If true it seems sufficiently fundamental and important for it to have been discovered by the pioneers of elastic stability. I just want to track down a clear proof from the literature and maybe translate it into a format that is easier to comprehend and use by a novice!

5) There are certainly numerous results of great interest and value in the classical works by Love, Rayleigh, Lamb etc. and there is a serious danger that many will be lost because of the often archaic notation and the fact that the frenetic pace of modern research leaves little time for exploring such sources.


Teng zhang's picture

Dear Prof. Barder

Your third comment may be explained by the interlacing theorem which says that for the system with one linear constraint, its eigenvalue interlaces between the two neighbor eigenvalues of its corresponding system without constraint.

To draw this conclusion, one should first derive the Min-max theorem, which can be found in the wikipedia The interlacing theorem is also given as an application of the Min-max theorem.

On the basis of interlacing theorem, W.H. Wittrick and F. W. Williams established a method named as Wittrick-Williams (W-W) Algorithm to solve the eigenvalue problem of the structure. The orignial paper is:

W.H. Wittrick and F.W. Williams, A GENERAL ALGORITHM FOR COMPUTING NATURAL FREQUENCIES OF ELASTIC STRUCTURES. The Quarterly Journal of Mechanics and Applied Mathematics, 1971 24(3):263-284.

The W-W algorithm is only related to the displacement, while Zhong wanxie et al extended it to the mixed variables (displacement and force), which can be refered in the following paper:

Z Wanxie, FW Williams, PN Bennett, Extension of the Wittrick-Williams algorithm to mixed variable systems. Journal of Vibration and Acoustics, 1997.

Hope this can be useful for your questions.

Best Regards

Teng Zhang

Dear Jim,

If you were to claim ownership of any result of your own discovery/invention, I would only heartily support you and welcome such a move on your part. I believe in Ayn Rand's philosophy including individual rights, including property rights, including intellectual property rights, including, of course, the intellectual credit part of it. All the way through. 

What you have given in this thread is a formulation of fundamental importance. Also, it's very beautiful because of its simplicity. One comment I actually wrote and then somehow did not post here pertained precisely to this part. In it, I had also noted that I am enjoying this thread (as most all previous threads by you). I had also noted that the formulation was so neat that it had to have been discovered/invented earlier and that one only wondered how come such a formulation had not been pointed out earlier. After writing such a comment, in a hurry, somehow, I then clicked somewhere else on the Browser's page, and so, lost my draft comment. Then, somehow, I lost enthu. to write it again, and so didn't post it.  (This happened about a couple of days back).

(BTW, I call it a "formulation" rather than a "result" because I would like to see at least an outline of a proof---the flow of the essential argument, even if only in words and only in 4-5 sentences. I will wait for a proof before I call it a result. I would like it if the "proof" is unlike most mathematical proofs, but instead is written the way you usually write.)

If it turns out to be your own original formulation, I would be happy about it. I carry a good impression about you starting from that thread in which you had mentioned what kind of singularity can/cannot be permitted and why. Coming back to this thread, I would like to know if it's your own original formuation. (May be, it will later turn out, to be an reinvention. But even if reinvented, it can be original.)

Finally, I am not a professor. I get paid Rs. 200/- (about USD 4/- ) per lecture hour at COEP, and the arrangement is sufficiently limited that I cannot use COEP's library. Hope this clarifies the "visiting faculty at COEP" status appearing on my profile here at iMechanica.

About other concerns perhaps indirectly suggested by your writing, this is not the right forum to discuss these; please see my other blogs. But briefly:

Not all Americans are bad, and Brits anyway can always go to US without visa or immigration botheration. If as a white (i.e. non-Indian-skinned) your experience of USA or of being American is good, good for you. And I don't feel bad out of it, either---I only wish your type of people increase in the USA so that it becomes a better enough country that I could consider dealing with an average American as an equal (say, from the moral viewpoint). (The statement is to be taken in a slightly loose or general sense; there can be none such as an average American. When one uses such terms, one talks about the integrated effect of dealing with the citizens of a given country, a powerful country like the USA.)

No, I couldn't have written this post shorter. Anyone else is welcome to write a shorter version and send it to me by email---and I will point out all the points he missed in making it shorter than necessary.

Yep, some of what I wrote in this thread has been a bit of digression. But what do you do when the attacks are systematic, and looking the other way whenever one wants to point out the injustice being suffered by oneself, is the style followed by so many Americans---including those like Harry Binswanger---that the general statement, allowing for exception, has to be condemning in nature? What does one do? Isn't everything inter-related? If not even once in 11 (or about 20) years, then when are Americans going to find the time to do justice to my (specific) case? For how long do you expect a non-American like me to keep goodwill for Americans, even those like you, if the hounding me out and attacking me happens on a daily basis? Sure it's a digression, but do think about those aspects too... You need not answer me here. If a man like you gives an honest thought to it, then that, too, can be enough for the time being. But only for a little while---would you have any greater patience? Why?



- - - - -
I remain jobless---and I remain being targeted by the Americans on a daily basis, including psychically

Dear Prof. Barber,

There is a lab in Keio Univeristy (Japan) under the supervision of Professor Takahashi Kunihiro using the similar method to make structural analysis such as buckling analysis, crashworthiness analysis.

The innovation of Prof. Takahashi's method is the ratio of energy method to represent the character of structure under both static and dynamic loading conditions.

This kinds of applying constraint (0 displacement) to an arbitrary node in a structure to get relative connectivity and eigenvalue have already been studied by Professor Takahashi for couples of year. Related research idea and method could be helpful.

The detailed information can be found in this web link.

Hope this information is helpful to everybody.

(P.S. I am a PHD student under supervision of Prof. Takahashi.)


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