# Bending a cantilever

1. Aug 21, 2007

### vidar

Hello,

actually, Im looking for some data concerning horizontal cantilevers that bend under the influence of the gravitation force. The cantilever should be rectangle-shaped, with given young-modul, poisson-ratio and of course measures. In addition to that, the deformation of the cantilever should be large ( ratio of vertical displacement of the end of the cantilever / length of the cantilever > 0.05 ). I dont know wether the underlying differential equations could be solved analytically, but if so, it would be great if someone could show me that solution ( just the result, not the derivation), otherwise experimental results, respectively where to get those( webpages, books or fairly common magazines) would be fine. The main concern is to get results that do not rely on the assumption of small deformations. I know this is quite a special topic but I ve been looking for that for a few days now and asking does not hurt anybody;)

Michael

2. Aug 21, 2007

### Chris Hillman

Nonlinear elasticity?

Data, or a mathematical model? From the sequel it seems the latter. In any case, I'd suggest that the moderators move this thread to the "Mechanical & Aerospace Engineering" subforum.

(Edit: thanks, moderators!)

I take it you mean you envision a cantilever (a beam not simply supported but pinned at one end) with rectangular cross section? With given dimensions (length, width, height), density, Young's modulus, and Poisson's ratio? Most books on linear elasticity discuss such models, although unfortunately they tend to obscure the assumptions which enter into these models, which are not valid in all circumstances which might be of interest to engineers.

Case in point: this puts you well outside the limit of validity of linear elasticity. (Following Euler, one can model large deformations of a thin rod, so if you are interested only in the "deformed shape" and if the length of your beam is much longer than its width or depth you can look at textbooks on the theory of elasticity.)

(Edit: this refers to the Bernoulli-Euler theory of "elastica", which is mentioned below.)

There are various research papers and books discussing nonlinear elasticity, but this gets a lot trickier than you might anticipate, and concrete models are not easily constructed.
It sounds like you are trying to model a specific scenario and in that case, a numerical model using finite element methods is probably more suitable than plunging into the theory of nonlinear elasticity (such as it is) without first learning linear elasticity (much better developed and much easier). BTW, as background for another thread in different context, I once started to try to explain the latter at PF, but the thread died for lack of interest:https://www.physicsforums.com/showthread.php?t=171079
Among other things I gave citations to some standard textbooks.

HTH.

Last edited: Aug 21, 2007
3. Aug 21, 2007

### Gokul43201

Staff Emeritus
You may want to use an FEM package like Ansys for this...but that's probably taking the bazooka on your fly-swatting expedition. There are very likely simpler scripts out there that are good for beam deflection problems with simple (i.e., uniform cross-section) beams (I've used one called "Beam" - it ran on DOS - that doesn't seem to exist any longer).

4. Aug 21, 2007

### AlephZero

Not necessarily. For many "nonlinear" problems the strains (and stresses) are in the elastic range but the geometry becomes nonlinear because of large rotations. An extreme example is a clock or watch spring which can wind and unwind through several complete rotations, but the material's stress-strain relationship is completely linear.

You might find some closed-form solutions by searching for "elastica" which was the old name for this type of problem - though in Google you now have to filter out a few million references to a pop group with the same name.

In practical engineering this would be done with a finite element model. Look for a program that can handle "large displacement, small strain" problems, and "follower forces" where the geometry of the loading changes as the shape of the structure deforms. Any "big ticket" program like Ansys, Abaqus, Nastran, etc can do it - but if you select the wrong options, you will get the wrong answers of course. I would expect shareware/freeware programs would also be able to do it if they claim to solve nonlinear problems, since the relevant finite element formulations have been in the standard literature for 20 or 30 years now.

5. Aug 21, 2007

### Chris Hillman

I presume no-one is deflecting any actual cantilevers?

Hi again vidar,

After reading the responses by Gokul and Aleph0, I feel I should ask: why do you want to know? I hope you're just playing around, not trying to design a home-made diving board or anything like that!

Did you overlook the bit about "large deflections"?

I sure hope people don't assume they can grab any old thing off the internet and use it to design their own home or whatever!

I think you may have misread something I wrote.

I was talking about beam deflection. In the thread I cited, I gave bibliographic information for a half dozen textbooks which discuss such problems; several authors offer some approximate guidance on deflections too large to be reliably modeled by the "analytic theory" in such textbooks. A fundamental point concerning the analysis of beams found in such well known textbooks as Timoshenko and Goodier, Theory of Elasticity is that this analysis ignores the distinction between Lagrange and Euler coordinates (see Sokolnikoff, Mathematical Theory of Elasticity), which is not valid for beams subjected to large deflections for geometric reasons.

I should have mentioned that the term "linear elasticity" is potentially misleading. The "linearity" in this term refers to the standard definition of the strain tensor, in which we consider only first order terms. This enormously simplifies the theory. The "theory of beams" which ME students are taught in engineering schools (based on standard physics textbooks by Landau & Lifschitz, Sokolnikoff, etc.) assumes small deflections in this sense; the most fundamental definitions are invalid without this assumption.

In the case of beams, unfortunately, it is the net deflection at the free end which can cause trouble, because of the misalignment of Lagrange and Eulerian descriptions of the "deformed" shape of the beam wrt the "undeformed" shape. Specifically, in the classical treatment of a deformed cantilever, the putative "gravitational acceleration vector" will not wind up pointing in the right direction at the free end. This error can be ignored only if the net deflection is small. This problem was brought out very clearly (as I thought) in the thread I cited.

It is true that the equations of continuum mechanics, which ultimately underlie this theory, are nonlinear, but this doesn't contradict what I just said. "Nonlinear elasticity" doesn't refer to introducing new nonlinear terms in the Euler beam equation or anything like that, but rather to adopting a notion of strain which includes "higher order terms". This leads to nontrivial conceptual and technical difficulties. In principal one could try to apply nonlinear elasticity to devise a workable theory of beams, but I haven't seen that done, and given the complexity of even the linear theory, I tend to think most MEs would better invest time in mastering linear elasticity plus finite element methods.

As I mentioned in the thread I cited, the theory of "elastica" is due to Jakob Bernolli 1705, but the much improved version found in (some) modern textbooks is due to Euler 1744. This theory is concerned with the shape assumed by a bent thin rod and is not valid for thin plates or for cantilevers (unless the beam is essentially a thin rod); see Landau and Lifschitz, section 14.

The linear theory of elasticity was not created until the nineteenth century, with essential pieces not in place until rather late in that century. The classical theory of beams (including distribution of stresses and strains) belongs to this theory, which implicitly assumes small deflections.

Physicists began to study the foundations of nonlinear elasticity in the second half of the twentieth century, and there are now various competing formalisms. I think it is fair to say that none of them appears to be anywhere near as workable as the linear theory for making crude but reliable estimates of the kind useful for engineers.

That's precisely why I am just a bit worried about what the OP has in mind.

Last edited: Aug 21, 2007
6. Aug 22, 2007

### vidar

In fact, I use Comsol for modelling the scenario that is described above. There are two scenarios which I got the "right" data to compare with for. The point is that the deflection is about 10% too large for that specific problem compared to the data I got and I wanted to verify that the error is made within my simulation and not in the data ( Comsol uses the FEM -method, and I have been playing around with the solving parameters for quite a bit of time now. Despite that the parameters do not seem to have a great influence on the solution). That is why I am actually looking for more, trustable data. Data for scenarios ( and analytical solutions) for small deformations has been received by me and it seems that my scenarios do quite a good job when staying within the "small deflection range". I am actually neither familiar with the mathematical description for small nor for large deformations, so I do not know wether analytical solutions for that problem are available, but obviously if so, those would be quite great to get. I had some not too close looks into some ingeneering books about statics but they seem mainly to deal with small deformations and the already mentioned linear approximation to create the underlying differential equation.
But none provided any kind of information about how the whole thing looks like at the appearance of large deflections, but I am actually quite shure that there *is* some information about that topic. As I stated measurements would do well as well...
BtW: I dont try do compose any kind of "evil machine" to break my or someone else's neck with;)

Last edited: Aug 22, 2007
7. Aug 22, 2007

### Gokul43201

Staff Emeritus
No, but I did interpret it as "large (meaning not necessarily linear) elastic deflections". Programs like Beam usually work for all deflections up to yield. The package typically has a database of stress-strain curves for different materials which it "looks up" to calculate deflections.

8. Aug 22, 2007

### Chris Hillman

Quick clarification

That at least has a short half-answer: there are most definitely "ready-made" solutions to problems concerning the small net deflection of a cantilever of various shapes under various loadings (including the type of cantilever which you described), which you can find in several of the more detailed textbooks such as S. P. Timoshenko and J. N. Goodier, Theory of elasticity or Chih-Teh Wang, Applied Elasticity or even L. D. Landau and E. M. Lifschitz, Theory of Elasticity.

Just reviewed the thread I cited and unfortunately it looks at though I left the planned section dealing with deflections of beams unwritten. Still, I'd be interested to know whether what I wrote (sorry it got so disordered due to technical limitations of Physics Forums) is attractive/friendly/readable for engineering/physics students.

Sounds like you are being systematic and careful (whew!). Your FEM model no doubt makes assumptions, and the obvious guess, it seems to me, is that at least one of them is violated in the scenario you are studying (assuming the data is good). I'd look into that; if you can't find any violations of assumptions, I think you can reasonably begin to question the available data.

As you can probably tell I don't know anything about such programs; my knowledge is limited to the classical theory of "linear elastostatics" which is presented in standard textbooks. From what you say, if Beam can cope with nonlinear stress-strain relations, it must in effect be approximating (using FEM, yes?) "nonlinear elastostatics". I don't know much about FEM, but that's what I would have expected. I presume that FEM can cope with the assumptions of the classical beam theory which are violated as per the above (from what little I know about FEM, the problem I mentioned wouldn't arise).

Last edited: Aug 22, 2007
9. Aug 22, 2007

### Q_Goest

Maybe I’m missing something, but this isn’t that hard. You don’t need FEA or solve differential equations. Just look for equations for a linearly loaded cantilever beam – or just derive them, it’s not that difficult.

The linear load is simply the weight of the beam. For example, a 1” square beam made of stainless steel will have a ratio of 0.05 at a length of roughly 148” which results in a stress of only 18,600 psi. Well within the elastic range of any stainless I can think of. Since deflection is still relatively small, this should be reasonably accurate.
(Note: I used 0.283 lbm/in3 for density, 27,500,000 psi for modulus )

Consider then the deflection when maximum stress equals 100,000 psi. Deflection ratio is now 0.624. Many different types of stainless can handle that, and much more. Granted that with this much deflection, error in these equations is fairly significant. However, somewhere between the ratio of .05 and .5, the equations are reasonably accurate. If you want a more accurate result, you could potentially ‘chop’ the analysis up into smaller segments just like any FEA program might do, and solve the equations locally for each chunk, taking into consideration the change in direction of the load due to bending. This could all be done fairly easily using standard equations for a cantilevered beam and a spreadsheet to do the iterations.

10. Aug 22, 2007

### AlephZero

Yes it is easy to derive the equations assuming small displacements. For large displacements you have to consider other effects - for example the position of the loads changes as the beam deforms. As an extreme example, a very flexible "horizontal" cantilever loaded by gravity would hang down approximately vertically, with axial stress varying along the length and almost no bending stress, except near the attachment point.

To put numbers to this, I recall somebody once attempting to do a "linear beam" analysis of a jet engine fan blade, including gas pressure loads and rotation. Linear beam theory said the blade tips moved about 3 feet backwards, not a fraction of an inch! There was nothing wrong with the software, or with the input - except for assuming the structure behaved linearly.

The practical thing to do is find a FE program, and check it is doing the right things. You can do some good checks without "exact" solutions, by making sure the answers are consistent with the approximations you are trying to use. For example:

1. Model a cantilever with a shear force at the tip. For a nonlinear solution, the tip should move in the axial direction as well as vertically, (axial movement roughly proportional to vertical movement^2, for "small" displacements) because the curved length of the beam should remain approximately constant. The reaction moment at the fixed end should be smaller for the nonlinear solution compared with the linear solution, because the axial tip movement reduces the length of the moment arm. If you don't see those effects, either you haven't successfully asked for a nonlinear solution, or there are bugs in the software. Checking the reactions are in equilibrium with the loads is a very good test for errors in FE formulations which can put artifical restraints into the model.

2. Model the cantilever with a distributed pressure load. If there is a "follower force" option, the load should change direction to be normal to the deformed shape. That will give an axial component of reaction force, and non-zero axial stresses in the beam as well as the bending stresses. Of course for your gravity load case, you want that option switched OFF. If you re-do problem 1 applying a pressure load over a small area at the tip of the beam instead of a point load, you can check whether the option is turned on or off.

3. One simple problem with an analytic large-displacement solution is pure bending. A constant bending moment along the beam means the curvature is constant, and for large displacement theory means the shape should be a circular arc. For small displacent theory, it will be a polynomial curve. Apply a moment to the tip of a cantilever, and with state of the art nonlinear FE software, you should be able to load up the beam to deform it into a complete circle - but don't be too disappointed if your model falls over before you get that far!

If you run checks like those and are happy with the results, you should be fairly confident about how to model your your own problem correctly. If I was using an unknown FE program for this type of analysis, however impressive its track record, I would still run models like that first, so I was sure I knew I was using the correct options in the program.

Final tip: correct nonlinear formulation of solid elements is simplest, shells/plate elements are more complicated, and beam elements using sophisticated beam theories are even more complicated Another good check is to model a cantilever with solid elements and compare with a beam model.

Hope this helps.

Last edited: Aug 22, 2007
11. Aug 22, 2007

### Q_Goest

Hi Aleph,
I fully agree that the typical equations such as I've suggested using are inacurate for large deflections (especially one which "would hang down approximately vertically" as you say), which is why I was careful to point that out. But I think everyone is missing the point - the OP said:
This isn't very much. For example, the end of a cantilever beam 148 inches in length and 1" square in cross section only needs to deflect downward at the tip by 7.4 inches which results in a ratio of 7.4/148 = 0.05 and this amount of deflection results in relatively small stresses. Would you agree that conventional equations predicting this amount of deflection are accurate?

What I see is more and more students that don't bother to try and do analysis by hand. They loose the ability to model things and predict results accurately because of this. They go directly to a computer so they can model something. We see students predicting the end of mechanical engineering because a computer will do it all. It seems the mentality these days is that FEA and CFD will do everything, and everything is too difficult to model otherwise.

12. Aug 22, 2007

### Chris Hillman

Plug in some values? Or perturbations, perhaps?

This has become a rather interesting discussion!

I probably should have suggested this earlier: Vidar, if you give us the exact dimensions and physical parameters, sounds like there are several here who can plug into a standard analytical model of a cantilever loaded by gravitation (using what I called "linear elastostatics"), and then we can simply compare with your data.

Getting ahead of myself, but just to mention another powerful body of theory which can be useful in "almost-linear" scenarios (which might be what we have here): perturbation theory is your friend. See e.g. Bellman, Perturbation Techniques in Mathematics, Engineering, and Physics, or Murdock, Perturbations: Theory and Methods, the books by Nayfeh, with cautions as per Murdock.

Last edited: Aug 22, 2007
13. Aug 23, 2007

### AlephZero

In some circumstances it's enough to make the problem significantly nonlnear. For example, if a flexible beam constrained at both ends, a displacement / length ratio of 0.05 wouldl cause a significant change in length, a significant axial force in the beam, and a significant stiffening effect against bending.

Granted this doesn't seem relevant to the OPs model - but the model wasn't fully specified, so we don't know for sure.

Certainly this is a common reason for inexperienced modellers getting the wrong results when modelling thin plates, which usually are constrained so motion out of the plane of the plate causes significant tension and compression in the plate, not just bending.

In practice the consequences usually aren't serious when the wrong model gives completely nonsense answers (like my fan blade example), but if answers are "only" wrong by a factor of 2 or 3 times (!!) and that fact is not obvious, they CAN be serious.

With a good FE package, running a model that has an almost-linear response through a nonlinear solution algorithm doesn't increase the cost of the run much compared with a linear analysis. So it's a pretty cheap insurance policy.

14. Aug 23, 2007

### uiulic

Professor Hillman,

A theoretical physicist and applied mathematician (at least potential) has extended his interest in engineering!

Is this a structrual member taught in structrual mechancis (looks like a beam depending on the length and section sizes)? You want only the solution (not the process for obtaining them), indicating that you are (?) designing something; but you did not give sufficient parameters (as Prof. Hillman requires), then you must be assuming that the readers would assume the parameters for you (and show you). If the readers assume the parameters in terms of symbols a,b,c,d, then I must say the solutions are almost impossible. If the readers assume all the values of the parameters for you, and only give you the solutions, then the solution will not help you at all.

I would imagine Prof Hillman is going to compare his analytical solutions with those obtained by others via FEM and check how well they mimic. Then Prof Hillman would decide whether FEM is worth studying in detail or not.

Linear (Nonlinear) has become a confusing item and I would make a big guess as what they are here (I am not sure whether my guess is right). "Nonlinear" usually in engineering seems to refer to two INDEPENDENT types(the boundary between these types are not clearly stated), i.e. material nonlinear (elastic or plastic, stress-strain) and geometrical nonlinear (finite strain or not). When you refer to elastic linear/nonlinear or linear/nonlinear elastic, you know the linear/nonlinear item must refer to geometric (the deformation is a geometric property not a material property, as Prof Hillman said it is about strain tensor). We often do not speak material nonlinear/linear, but use elastic/plastic? So we can get four types of such problems. Prof. Hillman of course refers to geometric nonlinear/linear. Also, for elatic nonlinear, the stress-strain is still Hooker's law (still linear), sij=cijklekl. Geometric nonlinear plasticity may be too complicated for your problems (and your material does not need plasticity). But even for finite strain (geometric nonlinear) problems, people may still use engineering strains as their measure of deformation like your deformation (>0.05) ) (it is!), which is acceptable.

FEM packages like ANSYS, ABAQUS can sort out your problem (I used them many years ago). I think nonlinear elastic solution is what your want, and some FEM package should have ready-made subroutines.

15. Aug 23, 2007

### Chris Hillman

Corrections

Did you confuse me with mathwonk? He is a math professor, not I.

Actually, I am trained as a mathematician (pure as the driven snow). My Ph.D. concerned a topic in symbolic dynamics, the most abstract part of ergodic theory, the most abstract part of dynamics, but my mathematical interests have always been broad, including information theory, Lie theory, symmetries of differential equations, invariant theory, algebraic geometry (especially computational geometry), combinatorics, category theory and even, when absolutely unavoidable, philosophy and history of mathematics. I have always been most fascinated by (1) unexpected theoretical connections between apparently unrelated phenomena, (2) the interface between mathematics and non-mathematics (the problem of interpretion of a theory is sometimes acute, as in statistics). Thus, I have always been interested in a range of applications, including but not limited to physics.

The models I refer to are not "mine"; they are described in detail in several textbooks including the ones I mentioned. These models date back to the nineteenth century.

No-one is questioning the utility of FEM, which is very well established. However, as I said, I don't know much about FEM so I certainly won't say much about it! Rather, what I suggested was that the OP give us enough details so that one of us can simply plug into the standard analysis of a horizontal cantilever loaded only by gravity; the textbooks even give a formula for the "droop" of the free end, so this doesn't involve any creativity! I suggested that comparing with his data might shed some light on the discrepancy he noticed. I did point out that FEM models constructed using popular software packages contain assumptions of their own, which might be violated in particular scenarios. For example, as came up yesterday in an another thread, one suggestion currently bruited in the aftermath of the I35W bridge disaster is that bridge design models should take account of the long-term corrosive effects of pigeon poop, which involves electrochemistry. I also suggested that perturbation methods might prove useful, should his scenario be sufficiently close to the regime of "linear elastostatics". The goal of a perturbation analysis is to find analytical results which are approximate but should have computable error bounds.

Actually, a number of limitations have been mentioned (not explained). An explanation would require a long thread, but as I have already found out, no-one is willing to read anything demanding, complicated, subtle, or lengthy, so...

I didn't even mention the term "geometric nonlinear", which I have never seen before. In fact I'm not sure I agree with anything you said.

I made a half-hearted attempt to summarize some complicated issues which I was just beginning to hint at in the thread I cited when I realized no-one capable of benefiting from it was reading it, so probably I can't complain overmuch that I have been misunderstood here. But I can declare myself disappointed.

16. Aug 23, 2007

### uiulic

Lovely Chris,

Thank you for sharing your experiences. I mean you will be a a Prof as I described.

17. Aug 25, 2007

### Chris Hillman

A warning, for future reference

Looks like this thread has died (perhaps deservedly so), but since people do sometimes site previous threads in subsequent discussion, I would be remiss not to add a caution to something I wrote:

The unforced Duffing's equation (modeling the vibrations of a cubic nonlinear spring released from a strained state, say) $\ddot{u} + u + \epsilon \, u^3 = 0$ provides the classic example of one of the (many!) potential pitfalls of naive perturbation: the appearance of spurious secular terms, which ruins any chance of modeling a periodic solution except for very short times. The cure is Lindstedt series, which allows us to approximate periodic solutions with simpler periodic functions (in this case, sums of cosines, but with a new frequency), valid over a reasonable length of time. When we try this with Van der Pol's equation (modeling a nonlinear oscillator), we would find that only one periodic solution exists (the limit cycle), if we didn't know this already from dynamical systems stuff! See the book by Bellman, Perturbation Techniques, Dover, 2003 (reprint of 1966 classic). See the classic by Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983 for the modern theory of nonlinear oscillators.

18. Aug 27, 2007

### vidar

Sorry that I have not answered til now. I have been abroad.
In fact, my scenario contains the cantilever, with a length of 0.35 meter and a heigth and thickness of 0.01m, together with a density of 1000 kg/m^3, a Youngs modulus of 1.4*10^6 kg/(ms^2) and a poissons ratio of 0.4. the cantilever lies with its long sides parallel to the ground,its one small side has been fixed(no displacement/ the normal vector is constant throughout the whole side), the other one is free. the gravitational acceleration is given by 2 m/s^2. the displacement of the point in the middle of the free small side is ~ 77*10(-3)m according to the simulation results, according to the reference data it should be only about 66*10(-3)m. There is a second scenario with the same data except for the Youngs modulus, that is then given by 5.6*10^6, resulting in a displacement of the point mentioned above of about 20*10(-3)m, whereas it should be 16*10^(-3)m according to the reference stuff.

19. Aug 27, 2007

### Astronuc

Staff Emeritus
I just came back to this thread. Interesting.

1000 kg/m3 is the density of water, so the material is something like a dense plastic?

And a local g of 2 m/s2 is much less than 9.8 m/s2, so this is like a moon or small planet.

Seems like a hypothetical problem of a light structure in space or on a small mass (small compared to earth).

What yield point is one using? Does the FEM model outside the elastic range?

What is the reference?

20. Aug 27, 2007

### Chris Hillman

Do tell!

Hi, Vidar,

I just checked a half dozen books, and several treat the problem of finding the deflection of a cantilever (with rectangular or elliptical cross section) loaded at one end or uniformly along the top, with gravity neglected, but none of them treat the case of a cantilever loaded only by its own weight. So I guess I'll have to work out the required deflection formula myself when I get a chance.

In the mean time, let me read back your data in cgs units to make sure I have the numbers right. In the notation of the thread "What is the Theory of Elasticity?":

$$\begin{tabular}{l} \ell = 35 \, \rm{cm} \\ w = h = 1 \rm{cm} \\ E = 1.4 \times 10^{11} \, \rm{dyn}/\rm{cm}^2 \\ \nu = 0.4 \\ \rho = 1 \, {\rm gm}/{\rm cm}^3 \\ g = 200 \, {\rm cm}/{\rm sec}^2 \end{tabular}$$

Hang on, what's this?! That's about [itex]0.20[/tex] Earth gravity.

Houston, is Vidar one of yours? Or have extraterrestials shown up at PF?

Hmm...

$$\begin{tabular}{ll} \rm{Mercury} & 0.38 \\ \rm{Venus} & 0.91 \\ \rm{Mars} & 0.39 \\ \hline \rm{Earth} & 1.00 \\ \rm{Moon} & 0.16 \\ \hline \rm{Io} & 0.11 \\ \rm{Europa} & 0.13 \\ \rm{Ganymede} & 0.14 \\ \rm{Callisto} & 0.13 \end{tabular}$$

OK, I give up: Vidar, where do you live? Or if you don't actually live wherever this place is, who took that data?!