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).
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.)