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Bending and 2D Elasticity: Going Back in Time

The following is a (relatively minor) question which had occurred to me more than two decades ago. By now I have forgotten precisely when it was... It could have been when I was in my TE (third year engineering) at COEP. ... Or, perhaps, it was later on, when I as at IIT Madras (studying stress analysis on my own). ... I don't remember precisely when it occurred to me, only *how* it did---it was when I was poring over the first part of Dieter's book.

IMHO, a matter like this should have been explicitly dealt with by the undergraduate texts on solid mechanics / elasticity. But, none does. Without straining your curiosity any further, let me tell you what that (minor) problem is:

Consider a horizontal cantilever beam as shown in the accompanying figure (A).

Image of Cantilever and a Plane Stress Problem

The beam has the  length of L. Suppose that it has a uniform rectangular cross section, say of height h, and thickness t. Suppose the beam is loaded by nothing but a point load P at its free end.

Analysis of stresses/deflections in a cantilever beam like this involves considering the bending moments existing along the length of the beam. Bending moment is nothing but another name for torque. The simple Euler-Bernoulli theory for such a beam is given in any introductory book on solid mechanics.

Now, suppose you increase h such that its magnitude becomes comparable to that of L, say, h = L. This circumstance is shown in the figure (B).

Suddenly, the beam problem now looks like one from the plane elasticity.

Three closely related questions follow:

(A) Now, checking the formulae or detailed derivations from 2D elasticity theory, we find no mention of the term "bending moment" anywhere in them. Why is it so?

(B) Why do torques seem to be present in the beam, but not in the plate? Don't the forces in the plate (say those associated with stresses) also form couples? After all, these forces also do act across finite moment-arms, right? If so, precisely where, in the act of "stretching" the beam into the plate (or of "compressing" the plate into the beam), do they torques get vanished (or introduced)?

(C) To make the matter even more confusing: Does the beam theory include couple-stresses as in contrast to the Cauchy definition (which, obviously, doesn't)?

What would be your own answers to the above questions (A), (B) and (C)?

Note that despite the length of the description preceding these questions, one-line answers are possible (though by no means mandatory!)


A little more on it all

Surprising, but I haven't ever found a single person thinking along the above lines---neither a professor, nor a postdoc, nor a student. My personal interactions with mechanicians have been limited, and so, in a way, this is not a big deal.

But, still, I found it surprising that no textbooks write about such matters either. Neither Beer (of Lehigh, and guru to more than one Timoshenko winner), nor Popov (of Berkeley, a student of Timoshenko's, I suppose), nor Shames (of SUNY Buffalo, a winner of several outstanding teacher awards) nor Crandall (MIT(?)), nor Timoshenko himself (later, of Stanford), nor AEH Love (of the 19th century, the author of what is probably the longest in-print title in the solid mechanics field) mention any such relation or contrast between these two theories directly and explicitly.

I could be wrong, but at least I don't remember having run into a comparison like this during my browsing of any of these books...

So, the question also becomes: Why don't textbooks mention the above matter even if they do cover the two topics separately in great detail and depth?

Is it the case that the matter behind my questions is so trivial and obvious that any competent engineer could be assumed to have known and mastered it if he has mastered the these textbooks?

Or is it that what we bank on, in engineering education, is an indirect implication, namely, that if the student knows how to work out solutions to numerical (i.e. mathematical) problems from each of the two areas taken separately, then all must be well with the state of his overall theoretical integrations, too? ...

Comments on this more general issue, as well as answers to the specific questions (A) through (C) above, are both welcome!

Also, if you remember having seen something like a comparison of the two theories in one of the books mentioned above, or any other book, then do feel absolutely free to correct me---I will appreciate your help.

And also, no, I won't mind being told (even very bluntly) that I was making a mountain out of a mole-hill, if that's what you honestly feel about this issue...

Thanks in advance for your answers/comments!

(Update on March 12, 2009 only: Made better my use of the English language, and streamlined the writing.)


Yuanli Bai's picture

I think the reason is that the Euler-Bernoulli beam theory is valid only for slim and long beams, which requires a large ratio of L/h. (for example, assuming zero through thickness stress) .For the 2D plate, the bending moment, force, stresses are still there, but the classical beam theory doesn't apply.

Rest assured that there are others who think along these lines.  They're just taciturn. :)

(A) Professor Dr. Vijay K. Varadan used to teach "Theory of Elasticity"
at Penn State University.  In this course, he showed how the bending
moment of a cantilever could be used as a basis for establishing the
order of the Airy stress polynomial.  I believe he studied at Madras in
India, too.

(B) For beams, moment has long been a convenient stepping stone to
obtaining approximate displacements and stresses, courtesy of
Euler-Bernoulli beam theory.  For walls, a more general elasticity
theory is necessary to find displacements & stresses.  A sound
knowledge of membrane theory is prerequisite to establishing a zone of
distinction, and, thanks to student debt and lack of funding, no one's
working in this field.  Well, almost no one. ;)

(C) A stress is a force per area, so a couple-stress would be a moment
per area?  In that case, a uniform couple-stress across a surface would
amount to forces canceling each other out at every point except the
ends, and a non-uniform couple-stress would essentially be a force
distribution.  The result is perhaps a beam in transverse compression
with a few moments here and there.

(D) Well, who do you think wrote the textbooks in the first place? 
People who had the understanding but not the funding, or people who had
the funding but not the understanding?  Perhaps they were the select
few with both and they just wanted to keep it that way.

EDIT: Hooke's Law seems to suggest that
another problem can arise for *wide* beams.  Hint: it has something to do with Poisson's ratio.

Hi David,

Interesting points you have raised... Here we go in brief...

-- Do you know if Mr. Varadan's notes are available on the Internet? I would have loved to have at least browsed through them. ...

-- About couple stresses, my own knowledge also is very limited, but I know enough about them from Sadd's book that I can fake things around a liitle bit as you can evidently see Wink... But, on the Internet, I found this one page by Patrizio Neff to be extremely informative, comprehensive, and an excellent starting point to many downloads

-- About writers... I think that rather than any other attribute such as funds or talents, the folks who write books only have a very special kind of patience... I think that is one requirement that is special to writing of books, above anything else...

I know of people in Pune who will write "textbooks" for the local engineering colleges market, for as little as Rs. 50000/-  or USD 1000/- one-time royalty... They address these books to the merit-wise downscale student population who must, somehow, pass their examinations, and get that BE degree from the University of Pune, and thereby enter either the job market or the marriage market (whichever pays first or better) ... There are more than 20 engineering colleges in Pune city alone, and a majority of student population often requires books like these... These books are sort of like diluted version of Cliff's Notes... Similar authors exist in every other major university of India, e.g., JNTU. When these engineering colleges advertise their "library holdings", they mean these books. .... Since such books are specifically addressed to a particular syllabus of a particular university in a particular time-period, the market they have is strictly local in both space and time. But yes, people are there to supply such books...

So, funding is not an issue relevant to authorship... Neither is talent... Anybody can write... Even I am thinking of writing one, though it won't be for that local market.

It's hard to write a hierarchically well-ordered book, though, speaking in general terms.... 


Kindly accept my apology for referring to information that, as I'm just now realizing, is practically impossible to share.  If it is any consolation, Varadan's solution to that particular problem suffers a strain compatibility problem; sigy should be non-zero at the support for non-zero values of Poisson's ratio.

Thanks for the links, though this is far too much information for me to take on.  I've seen "three additional, independent degrees of freedom, related to the rotation" used in (matrix methods of) space frame analysis.

Thanks also for reminding me not to generalize so quickly.  Pune sounds like a very different place than, say, where I went to school in Pennsylvania, where engineering students pay $1000+ each year on books.  Funding is relevant to authorship if the level of complexity of the topic requires one to devote all of their time and attention to understanding, organizing, writing, and formatting the content.  This might just be me, since I took everything but the kitchen sink in my undergrad years.



In "Karl Girkmann, Flächentragwerke. Einführung in die Elastostatik der Scheiben, Platten, Schalen und Faltwerke, Vienna, Springer 1946" (it's in German, a book about theory of plates and shells) an example is given, where a plate simply supported at left and right end under a sinus-shaped loading on its top is treated by using an ansatz for Airy's stress polynomial.

Furthermore, it is shown that even for a side ratio of l/w = 2 the solution for stress distributions is very similar to results of beam theory (parabolic shear stress, linear normal stress in cross section).

Hi Manfred,

Thanks.... BTW, even in Shames (or Beer and Johnston---I forgot which book) there is this example where they take the L/w ratio up to 3 or 2. But then, it occurs only in the context of highlighting the fact the contribution of shear stress in producing the final displacements is much smaller as compared to that due to the normal stresses, due to the difference of the 4th order and 2nd order... Unfortunately, the authors don't notice the point that I meant to highlight... Pl. see my general reply below too...

(A) Moments and forces are vectorial (6-scalars in all) characterizations of distributions of vector valued (traction/body) loadings.  When the aspect ratio of the plate "becomes" beam-like then one finds that this low order charaterization is effective in describing the behavior.  When you have plate-like dimensions this is not true so no one bothers with it; Though you are free to define it if you want; it just is not that useful.

(B) See answer to (A)

(C)  A beam theory is a special case of a Cosserat medium so it does contain "couple stresses" (in a manner of speaking) but they are not reductions of couple stresses from the 3D/2D theory.


Prof. Dr. Sanjay Govindjee
University of California, Berkeley

Dear Sanjay,

Thanks... Pl. see my reply below...

ramdas chennamsetti's picture

I have a couple of things to add:

Theoy of elasticity deals with deformations and forces. Even if there is a moment applied, it is decomposed into forces (statically equivalent).

Bending and plate theories have been developed making some assumptions (like neglecting some stress components).

In plate theory also, there are moments due to bending stresses (like in the beam). These are expressed per unit width.

With regards,

- Ramdas

Hi Ramdas,

Thanks... On the first point, I almost agreed with you except that I am not sure: Aren't moments supposed to be as "primitive" as forces, for static equilibrium? (In the sense, doesn't conservation of angular momentum stand on its own, without any reference to conservation of linear momentum?) So, can you decompose moments? Probably, "decomposition" is not the term... But, of course, I got the main direction of your point that the moments just translate into stresses... Also pl. see my general reply below.

Thank you all very much for your replies, and also let me say sorry for the delay from my side... I was reading your replies as they came, but also thought it best to wait just a while longer before jotting down my replies and clarifications...

(1) First of all, what I wanted to point out was not, really speaking, the mathematical relation between a beam and a plate. What I wanted to emphasize was something more conceptual in nature than that...

It was: this big difference of terminology that a typical student runs into---a difference which is never explained to him at all.

Consider, just for example, either Popov's book or Beer and Johnston's. ... Some 1/2 to 2/3 portions of these books involve a very prominent usage of the term: "bending moment." Most other books are similar in terms of emphasis.

Just a semester or a year later, the same student enters the class-room (or is referred to some other advanced books), and bingo! Now, none of the stress analysis theories he reads will involve anything like a "moment" in them. There are potentials and stress functions and complex analysis and path integrals and multiply connected domains... But no moments. The torques simply disappear. In principle. In fact, he is sternly reminded that the moments, of course, cancel out---wasn't he attentive in the first lecture?....

Why this difference of treatment? That was the crux of the matter I wanted raised.

Sometimes people explain it away as the "Strength of Materials" approach vs. the "Solid Mechanics" approach. But this still is beating around the bush, I felt.

Thus, I wanted to highlight this abovementioned confusing part. The addition of the couple-stress related thingie was just to confuse the reader a little bit---just to add a little bit of spice to this main question, that's all!

So, while everyone's reply was valuable, IMHO, it was David who really addressed the crux of the issue(s) that I sought to highlight...

(2) Now, here are my answers to the technical part of it

The beam theory does not include couple stresses. Not at least the beam theory for the simpler homogeneous class of materials like metals. (Sanjay, I wasn't talking about micropolar materials, composites, or metals with extensive presence of micro-voids or microcracks in them... Thus, I didn't really have a Cosserat medium in mind.)

For the normal homogeneous (metal-like) materials, couple stresses are absent in the beam theory just the way they are absent in their 2D/3D elasticity theory.

The sole purpose of bending moments in the beam theory is to act as a vehicle or an intermediate concept (or a link) to translate the load boundary conditions into stresses---esp., the normal stresses. That's all! 

A main likely confusion here is the following. Students see moments present across the sections of the beams, and so, inadvertently, they might conclude that couples exist in the sense of couple-stresses. This is wrong. Books should highlight this. But none does.

Here, it's useful to distinguish between an infinitesimal element and a finite section. Couple-stresses involve resistive torques across infinitesimal elements. The couples which the beam theory considers, actually, are considered only across finite sections. The infinitesimal elements inside a beam do not carry torques. Bending moments are just a convenient short-hand for a special pattern involving the usual stresses in the vertical cut.

(3) The concerns I expressed in the second part also are relevant.

In fact, to go further, I would say that the teaching of solid mechanics has actually suffered because it has traditionally been considered a responsibility of the Civil departments (yet another controversial statement from me) and not to a separate TAM department (see my comments related to the recent closure of the Cornell TAM).

Civil folks think teaching of beams is important. But, it isn't. Not if the response of solids to a variety of mechanical forces is your real aim. A 50% to 66% weightage to the beam theory in the first (introductory) courses is summarily uncalled for. It only helps pace out the Civil curriculum better---but hampers the preparation of mechanical, aerospace, electrical, metallurgical and other graduates.

The absence of topics like Airy's function from the introductory (first) courses on solid mechanics is hard to explain.

The absence of plasticity theories also is very hard to explain---and very immediately required by the metallurgical/materials students.

(A similar thing happens to teaching of fluids, too. Give it to Civil engineers, and they will unnecessarily over-emphasize the flow through channels... Give it to Aero engineers, and they will over-emphasize external flows wherein the solid body is tiny compared to the fluid... Give it to mechanical engineers and they will overemphasize the empirical performance characteristics of pumps and turbines... So on and so forth...)

So, what I am thinking aloud here is about changing the sequence and emphasis of topics for teaching of Solid Mechanics... I will post my thoughts again, later on... 

Thanks again for reading and do let me know if I am going wrong in any technical part...

There's no such thing as a "concentrated moment load" at the fudamental level.

Whatever you see in approximate analyses, pointloads, moments, are idealizations.

All we have at the "basic" or "primitive" level are

1. surface tractions (surface loading),

2. body force distributions (volume loading) and

3. couple stress distributions (as someoen described these are surface/volume force distributions that cancel out at every point but produce a turning effect.\

Surface loadings (surface tractions and surface couple stresses) are the direct result of "bodies" interacting with each other in direct contact, while body force loadings (body forces AND volume couple stresses) are the direct result of bodies interacting with each other NOT in direct contact.


 The moments yous ee at the ends of beams are "idealizations" of the axial stress integrated over the whole cross section (taking orientation of cross section) into account :d


Not sure what the confusion here is. Whenever you try convert a distribution into a concentrated load, these bending and torsional moments pop up. Not to say that couple stresses are not present. If you have the E-B approximation, there is nothign that states that the end moment load ISN't  due to a couple stress distribution.

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