×
INTELLIGENT WORK FORUMS
FOR ENGINEERING PROFESSIONALS

Log In

Come Join Us!

Are you an
Engineering professional?
Join Eng-Tips Forums!
  • Talk With Other Members
  • Be Notified Of Responses
    To Your Posts
  • Keyword Search
  • One-Click Access To Your
    Favorite Forums
  • Automated Signatures
    On Your Posts
  • Best Of All, It's Free!
  • Students Click Here

*Eng-Tips's functionality depends on members receiving e-mail. By joining you are opting in to receive e-mail.

Posting Guidelines

Promoting, selling, recruiting, coursework and thesis posting is forbidden.

Students Click Here

Jobs

Simulating the Bending of Metal Plates?
4

Simulating the Bending of Metal Plates?

Simulating the Bending of Metal Plates?

(OP)
Hello Everybody,

I am trying to find a solution to predicting the curve a metal plate or sheet takes when we apply a compressive load on it to bend it or i should say flex it. IT is very well in is elastic limit and comes back to original shape later. We fix on end and other end is guided. I just want to simulate my assembly and get the final curve the basket takes.Assemblies contain a plate connected to an actuator which pulls the plate to bend it

I am not able to decide will FEA be necessary to give me the defelction. ( I am only interested in the defelection and curvature) Will any FEA software be able to do that. lease advise is FEA is teh way to go, If yes which software will be able to give me these details. Or if not which way will be able to give these details to me.

RE: Simulating the Bending of Metal Plates?

a simple by-hand calculation will be enough for this kind of problem...

Cyril Guichard
Mechanical Engineer Consultant
France

RE: Simulating the Bending of Metal Plates?

(OP)
Thanks Cyril,

But can you also advise what will be an appropiate eqaution to use for a plate bending under compressive loads. The end conditions are both end simply supported or one guided and other simply supported.

The Plates have complex geometry.

Thanks in advance for all the help

RE: Simulating the Bending of Metal Plates?

2
If it is in the elastic range, you can use the flexural relationship

M/I = E/R

M=Moment
I=Second moment of area about NA
E=Elasticity

R=Radius of curvature

The method assumes that the plate bends in a circular arc shape and is nearly true for deflections within elastic range.

RE: Simulating the Bending of Metal Plates?

(OP)
Thanks Flame,

Can you advise what equalition will be between the defelction and X location. Thus i would be able to get a curve ploted which is my main aim.

You are a great help!

Thanks in advance for all the help

Regards,
Pratt

RE: Simulating the Bending of Metal Plates?

It sounds to me like your analysis is fundamentally nonlinear, thus flame's equations do not hold.

Even if the material does not yield, there can still be very significant nonlinearity due to geometric nonlinearity. What you are describing seems to be almost like a "buckling" of the beam, if I understand it correctly. This is classic geometric nonlinearity.

Question--you stated "complex geometry"--in what sense?  Do you have a consistent cross section (thus they can be idealized as beams), or is it more complex than that?

Brad

RE: Simulating the Bending of Metal Plates?

Brad

Yes this is non-linear as in geometrically nonlinear which any nonlinear FEA package will handle with ease.

BUT Flame's equations are fundamentally CORRECT for deflection due to a moment!

Take a cantilever beam, and apply a pure moment at the free end. The shape of deflection is a true arc. Increase the moment and you can get the beam bent in a perfect circle! This is a test often adopted to prove how well geometric non-linearity behaves in packages.

RE: Simulating the Bending of Metal Plates?

Altough Flame's equations are fundamentally CORRECT for deflection due to a moment!  It will not help much for this problem.  This is buckling problem and has to be solved iteratively.

RE: Simulating the Bending of Metal Plates?

Since the plate "comes back to original shape later", I don't think it is a buckling behaviour and the moment equation is more CORRECT to apply.

RE: Simulating the Bending of Metal Plates?

The moment in the plate changes along the length and depends upon the actual deflection from the "FORCE LINE".  That's why I am saying the problem has to be solved iteratively.  The moment equation is CORRECT, but how can you get the moment without initial deflection?  Where the initial deflection comes from?  Due to the moment?  This is a loop.  You have to assume a deflection to start.

RE: Simulating the Bending of Metal Plates?

Just to try and clear things up a bit, this problem can be described as:-

i). Geometrically non-linear and thus has to be solved using incrmental/iterative techniques.

ii). Does NOT reach the critical buckling load, since it returns elastically to it's original shape, and will obey the normal flexural bending of beams as described by Flame. (Brad, buckling problems obey this law up to a point just below the critical load, just look up simple Euler buckling theory in any standard text book)

iii). It will only require an initial deflection (or a small applied offset moment) if the plate has no initial curvature. However Pratt describes the structure as "The Plates have complex geometry" , so it may not even require an initial imperfection or offset load.

I hope everyone can agree on these points!

RE: Simulating the Bending of Metal Plates?

johnjors--
I agree with (i) above and (iii) above

I disagree with an implicit assumption in (ii) above. Just because something returns elastically to its original shape does not negate the fact that buckling can occur (especially in the case of enforced deflection, which this appears to be, given that he is "guiding" the non-fixed end).

I don't know for sure whether buckling does or does not occur (not enough information either way).

I apologize for my earlier red herring--in hindsight I was incorrect state that it was buckling. I still contend that flame's equations do not hold, but it is due to this problem not being pure bending (as ding123 noted).

Brad

RE: Simulating the Bending of Metal Plates?

Brad

Sorry to press this point but look at:-

http://www.du.edu/~jcalvert/tech/machines/buckling...

the example here has the same boundary condition as Pratt's first load condition of both ends simply supported, his second condition with the one end guided will still use the same basic formula but with different constants.

And just to add a bit more confusion to this debate, look at:-

http://www.jlrcom.com/buckling.htm

And finally whatever theory or method you use, buckling only occurs when the structure can no longer support the applied load and you get catastrophic failure. Up to that point the structure is merely bending or deflecting under load. Thus as Pratt's plates return to their original shapes, then buckling does not occur.

RE: Simulating the Bending of Metal Plates?

The answer to pratt's question is in the first link given by johnors above: the deformed shape is a sinusoid (for two pinned ends and constant EJ).
Concerning the buckling-non buckling debate, it is clear to me that the condition that I understand pratt is speaking about is a post buckling state. If there were initial deformations sensibly impacting the deformation under load, then this would be an arch, not a straight beam.
Of course after buckling a catastrophic failure will occur, as found in all books on elastic stability, but this is true only if the load is kept constant. If on the contrary the axial displacement of the loaded end is the controlling parameter, as I suppose is the case in pratt's problem, then a true elastic return may take place.
A simple experiment will give the proof. Take a small plastic (straight) rule, and load it axially with your finger. Initially it will be fully stiff (pre buckling state), but increasing the load you will eventually obtain the buckling and the rule will bend sideways (with a nice sine wave, as told above). At this point if you continue pushing, you'll break the rule, but if you stop, the rule will be able to recover its straightness without damage.

prex

http://www.xcalcs.com
Online tools for structural design

RE: Simulating the Bending of Metal Plates?

(OP)
Hi Guys,

Thanks a lot for all your help and valuable inputs.

Prex here has very well defined my problem, it is defenetly like a ruler , we never go beyond the elastic limit so no failure occurs.

To clarify a few things:

"Complex Geometry" = The plates are not absolutely rectanglular but have different profiles like cuts , indentations and holes in them.

"End COnditions" = The plates have extention on the top and holes in them where they are guided along wires or tubes.
                Guidewire
              ___|___
             |   |   |
             |   |   |
             |   |   v
             |   |  Force to pull the plate
             |   |
             |   |
             |   |
             | __|__This end is also pulled up by a spring
                 |  attached to a yolk bar.

"Force" : As shown the actuator (Manual or motor) pulls one end of the plate and it takes a shape on deformation. I understand that the shape taken is influenced by the actually geometry of the plate (i.e where it has lesser metal)

"Intial Curve" = The plate may be straight or may have a pre-curve to start with.


I hope all this will help you gurus understand my problem better and can give me a direction to work for predicting the shape the plate will attain under load.

Thank you all for all the help.

Pratt

RE: Simulating the Bending of Metal Plates?

pratt,
of course there is no analytical solution (like the sine curve) to your problem, as your geometry (EJ) is changing along the axis of the beam. Moreover you seem to have different geometries, and each one requires its own solution. On the other side you can expect a good forecast of your deformed shape, as the bending equation of beams (M=-EJy') is still quite good at large deformations.
Personally would solve such a problem with a small Excel sheet performing the numerical solution of the differential equation. Unfortunately don't know of any commercial software that would do that without requiring some knowledge of calculus on the user.

prex

http://www.xcalcs.com
Online tools for structural design

RE: Simulating the Bending of Metal Plates?

Prex

"as the bending equation of beams (M=-EJy'') is still quite good at large deformations.
Personally would solve such a problem with a small Excel sheet performing the numerical solution of the differential equation"

It seems to me that you are advocating using the "graphical integration technique", something that I last saw being done about 25 years ago! I doubt if many FEA users today will have a clue how to implement it or even know about it.

Still, interesting that you think this is more appropriate than any FE package!!

RE: Simulating the Bending of Metal Plates?

johnhors,
nothing graphical, simply finite differences in Excel.
It is a method much older than 25 years, but still very adapted to small problems such as this one (monodimensional). Of course it requires knowledge of calculus for implementing it, as I already pointed out, so it is not for the average user.
Using FEA, this is a problem where large deformations must be activated and personally don't see very well how one can perform a post buckling analysis without adding an initial deformation that could impact on the results (unless this deformation is present in the real thing of course). I'm sure that with a good FEA package one can solve it, but the time required to set it up, at least for me, would be much bigger than with using Excel.

prex

http://www.xcalcs.com
Online tools for structural design

RE: Simulating the Bending of Metal Plates?

(OP)
Prex,

how will the equation :- M = -EJy will give me the shape? Whats the relationship of y to x (As you go along the axis of the plate).

I will be really obliged if you can explain a little more on : "Personally would solve such a problem with a small Excel sheet performing the numerical solution of the differential equation"

Thanks for all the help

RE: Simulating the Bending of Metal Plates?

pratt,
do the following in a new Excel sheet:
1)in A2 type 1 and copy down to A100 (A2:A100 is now filled with 1's)
2)in B2 type =(C3+C1)/(2-A2*(PI()/100)^2) and copy down to B100
3)in C1 and C101 type 0, in C2 type =B2 and copy down to C100, then in C51 overwrite the formula with a 1
4)under Excel options turn on the iterations and put a very small number in the deviation (or error) case; I also prefer to set the calculation to Manual
If you now repeatedly hit F9, you'll see all the numbers change and they should rapidly converge to stable values. Now the numbers in column C represent the deflection of your beam scaled down to an amplitude of 1. This solution corresponds to pinned ends and constant EJ (the 1's in column A): you can easily check that those numbers represente a sine curve.
If you now replace the 1's in column A with others that represent the product EJ of your beam every 1/100th of the length (you can use numbers scaled down to any suitable constant, e.g. 1 at one beam end), by recalculating you'll get in column C the deflection curve of your beam (again scaled down to an amplitude of 1, that you can rescale to your actual amplitudes).
Note that if your beam is made of a plate of constant thickness, where only the width changes, then the numbers to be placed in column A will be proportional to the widths of your plates at different locations along the length.

prex

http://www.xcalcs.com
Online tools for structural design

RE: Simulating the Bending of Metal Plates?

There a mistake in my previous post: the quantity in column A is proportional to 1/EJ, not to EJ (or you can change the formula in B2 to =(C3+C1)/(2-(PI()/100)^2/A2))

prex

http://www.xcalcs.com
Online tools for structural design

RE: Simulating the Bending of Metal Plates?

(OP)
Thanks a Ton Prex,

But I have another query. In this calculation you are assuming that the maximum defelction will be at the center (C51) and setting that to 1. Can you explain why you are suggesting that.

This gives me a peak at c51 , but doesnot give me a smooth curve to show the shape of deformation. Remember that there is no point load at the center, but the load is applies by pulling both end of the plate towards each other.

Prex you have given a great path for me to follow, some light will help me find the destination.

Pratt

RE: Simulating the Bending of Metal Plates?

(OP)
Also Prex,

You say in your previous post that press F9 till values stabalize. They dont, they keep on changing. How will I know when are they stabalised?

Thanks for the help.

Pratt

RE: Simulating the Bending of Metal Plates?

Continue with F9 (or set the number of iterations to 100 or 1000 under Excel options). The numbers must stabilize (you don't see changes at recalculating with F9). To check compare the values with a sine curve: must match to the sixth figure or so.
The 1 in the middle is because this equation is homogeneous, so it calculates the values less a constant that must be assigned. However you won't necessarily have a maximum or a peak in the middle: I tried with column A equal to 0.5 over half length and to 1 on the rest: the maximum is 1.26 and it is located (as one would expect) in the 0.5 region.

prex

http://www.xcalcs.com
Online tools for structural design

RE: Simulating the Bending of Metal Plates?

(OP)
So Prex,

I can change the 1 in the middle after the first calculation.

How to change the equation if the end condition changes.

Can you give the actual equation with all the variable and explanation what becomes constant or zero with end conditions etc.

Thanks for all the help.

Pratt

RE: Simulating the Bending of Metal Plates?

john,
You stated:
"And finally whatever theory or method you use, buckling only occurs when the structure can no longer support the applied load and you get catastrophic failure. Up to that point the structure is merely bending or deflecting under load. Thus as Pratt's plates return to their original shapes, then buckling does not occur."

That is not necessarily true.  Even one of your cited websites notes snap-through buckling of a dome.  The keyboard which I am currently typing on exhibits such bifurcation on every single keystroke.  This is buckling, but it is neither catastrophic nor inelastic.  

Oil-canning is another classic example--it is not catastrophic, and in fact can be even be entirely linearly elastic. I can cite many other examples in which buckling is part of the designed behavior.

While I acknowledge my earlier error, I strongly stand by my last statement.

Please let me know if you do not agree after this clarificaiton; it may be that I am not making myself sufficiently clear.

Best regards,
Brad

RE: Simulating the Bending of Metal Plates?

Brad

Fair point, I had always associated buckling with a permanent failure of the component, collapsed beyond elastic limits, and that has always been the case with my experience within the automotive and aerospace industries. A buckled landing gear leg or side stay or car suspension arm will never return to it's original shape.

Out of pure interest, I would like to know of an example where buckling is part of the designed behaivour.

Best regards,
John

RE: Simulating the Bending of Metal Plates?

I've already cited keyboard "springs" as an example. Try to slowly depress a single key on your keyboard.  You will notice that there is a point at which it is very difficult to keep your finger at the same point.  This is because there is a load drop-off.  This is a designed buckling of the rubber "spring" underneath.  Most keypads exhibit this behavior.

There are also examples of electrical connectors and other "slide connector" components which undergo bifurcation during assembly.  Typically the part looks like a cantilever beam which is slid over another component.  These sometimes are designed such that they exceed a buckling load (but should not significantly yield).

I agree with your cited examples of landing gear and car suspension. However, a noteworthy characteristic of these is that they are "load-controlled". In a load-controlled situation, buckling typically does result in catastrophic failure.  However, some assembly situations are more akin to displacement-controlled. Thus post-buckling behavior is not necessarily catastrophic.

Given the initial description of this problem, my brain started viewing this an an assembly-type operation, which lead me down the path I went.  I'm not clear whether or not this is entirely applicable to Pratt's problem (Pratt has not provided enough information on which to judge this). IF this is relevant to Pratt's approach, then the cited formulae are inappropriate.

Brad

RE: Simulating the Bending of Metal Plates?

(OP)
Hello Brad,

Your vision is very applicable in my problem here. In my scenario it is a displacement controlled system. We know our part will never exceed their elastic strength , because we never flex them so much. Reason is that the flexing is something that these part are deisgned for as the main purpose.

To explain it more. Take a plastic scale and put it verticle on a table (like a slender column), apply compressive load on the top, you will find that after the intial resistance the scale flexs quiet a bit and if you stop applying more load and release it, scale comes back to it original shape.

This is just like what we do in our systems.

As per the geometry is concerned, the cross section is consistant, as the part are made from sheet metal or abs plactic sheets. But the shape it self changes. For example you might have a rectangular plate with cuts on it to give it fin like structure.

I hope this gives you a better idea of the problem.


If you can advise your point of veiw in solving this kind of problem to determine the shape the plates will take. What will be the factors governing the shape they take etc.

Your input will be very helpful

Thanks
Pratt

RE: Simulating the Bending of Metal Plates?

pratt,
what you ask is not that simple: the form of the differential equation changes with the end conditions. In fact the equation is M=-EJy' with M=Py for a pinned end but M=Py-Mo for a clamped end, where Mo is the unknown end moment.
One issue with finite differences is that every problem requires a different setup: I tried a few minutes to find a setup for the end moment, but failed (no convergence).
Another method found in the literature is by assuming a polynomial with unknown coefficients as the deflection curve, then minimizing the expression of the critical load that leads to a linear system of equations. However this one too requires a quite vast analytical effort.
There is in fact a much simpler method, if you can accept some error in the result. I've checked that, for pinned ends and uniform section, the post buckling deflection curve (sinusoid) and the deflection curve under a uniform lateral load (a polynomial of the fourth order) differ by less than one percent of the maximum deflection (that of course is set equal in the two equations). So I guess that this will remain valid (though a somewhat larger error can be expected) even for a variable section or for other end conditions. So, as the deflection curve of a beam of variable section with uniform lateral load is much simpler to calculate, my suggestion is that you adopt that one.

prex

http://www.xcalcs.com
Online tools for structural design

RE: Simulating the Bending of Metal Plates?

This seems like a fairly straightforward nonlinear FEA problem.  I have solved many problems very similar to what you are describing. A closed-form solution would be rather difficult.

Do you have FEA software at your company? If not, do you know of any consultants that do FEA?  If this is pretty straightforward, you may able to "job-shop" this to an FEA house for a fairly small amount of money (a few thousand US).

Brad

RE: Simulating the Bending of Metal Plates?

(OP)
Hey Brad,

Can you suggest a few good FEM software which will do this Job. If you can provide price information also , it would be really geat.

Thanks for all the help

Pratt

RE: Simulating the Bending of Metal Plates?

This is a classic problem for a nonlinear implicit code.  The three dominant nonlinear implicit codes in the US are ABAQUS, ANSYS, and MARC.  They are most typically sold on a "subscription" basis, meaning you pay a recurring fee to keep the software. For all three of these, expect to pay over $10K/year (probably closer to $20K).  They are generally outside the reach and budget of those who are looking to solve "occasional" problems.  Hence my suggestion to consider outsourcing this project to a consultant.

If you are contemplating building an internal analysis capability after hearing this price, let me know and ask more questions. I'm happy to provide answers.
Brad

RE: Simulating the Bending of Metal Plates?

(OP)
Hey Can any guys help me decide that my problem is flexural or bucking. As I am not able to decide, as I am getting to know more and more about these things. If anybody can give me a website or tell me the basic difference between bending and buckling. That will great. Also Can anybody tell me how to perform flexural analysis.

Red Flag This Post

Please let us know here why this post is inappropriate. Reasons such as off-topic, duplicates, flames, illegal, vulgar, or students posting their homework.

Red Flag Submitted

Thank you for helping keep Eng-Tips Forums free from inappropriate posts.
The Eng-Tips staff will check this out and take appropriate action.

Reply To This Thread

Posting in the Eng-Tips forums is a member-only feature.

Click Here to join Eng-Tips and talk with other members!


Resources