Simulating the Bending of Metal Plates? 4

[pratt]

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.

 

[FrenchCAD]

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

Cyril Guichard
Mechanical Engineer Consultant
France

 

[pratt]

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

 

[flamby]

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.

 

[pratt]

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

 

[brad]

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

 

[johnhors]

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.

 

[Ding123]

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.

 

[flamby]

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.

 

[Ding123]

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.

 

[johnhors]

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!

 

[brad]

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

 

[johnhors]

Brad

Sorry to press this point but look at:-

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

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.

 

[prex]

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

 

[pratt]

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

 

[prex]

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

 

[johnhors]

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!!

 

[prex]

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

 

[pratt]

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

 

[prex]

pratt,
do the following in a new Excel sheet:
1)in [tt]A2[/tt] type 1 and copy down to [tt]A100[/tt] ([tt]A2:A100[/tt] is now filled with 1's)
2)in [tt]B2[/tt] type [tt]=(C3+C1)/(2-A2*(PI()/100)^2)[/tt] and copy down to [tt]B100[/tt]
3)in [tt]C1[/tt] and [tt]C101[/tt] type 0, in [tt]C2[/tt] type [tt]=B2[/tt] and copy down to [tt]C100[/tt], then in [tt]C51[/tt] 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 [tt]F9[/tt], 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 [tt]A[/tt]): you can easily check that those numbers represente a sine curve.
If you now replace the 1's in column [tt]A[/tt] 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 [tt]C[/tt] 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 [tt]A[/tt] 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