Beam calculation doesn't add up. 1

[Biggadike]

I am using a piece of stainless steel sheet as a spring (small deflection, small load). In order to work out the effective spring rate I used a cantilever beam calculation, worked out the maximum deflection under a given load and took that as the effective spring rate.
I put all the calculations into Mathcad, I've checked multiple reference books to make sure its all correct and I can find no errors.
I have two problems:

1) When I check the radius of curvature against the deflection predicted, the geometry doesn't add up. Calculating the radius of curvature from the geometric data,
gives you an answer twice that predicted.

2) When I make a test piece and check the deflection, I get almost exactly twice the deflection predicted.

The only assumption I can make is that the two are connected but every text book I look at confirms the equations I am using are correct.

Has anyone come across this before or have any suggestions?

 

[GregLocock]

The mistake in your earlier post was that "

Radius of curvature is given as

R=EI/M = 0.802 (802mm)" is a correct equation mistakenly applied.


M/I=E/R is a local calculation, not applicable to the whole beam with one set of values.

The next problem is that you have a very wide plate, not a classic beam. if you apply the load at a point to the plate then the plate in the vicinity of the load will deflect a lot, and the rest of the plate will deflect less. Classic beam theory assumes that the whole of the end of the beam moves together (plane sections remain plane).

Note that the deflection> thickness, so linear theory may be in trouble. Also note that the root fibres will be getting up towards their yield stress (sigma =150)

I think your problem is most likely to be that you have a plate, not a beam, so you need to analyse it with plate theory not beam theory. You may even need to include membrane effects to get good correlation.

A competent FE analysis might be the way to go, since Roark has no cookbook solution for a cantilever plate under a point load. Cheers

Greg Locock

 

[Biggadike]

Excellent stuff Greg. Thats very useful. What you say about the radius of curvature makes complete sense. In my textbooks its just thrown in as an aside without proper explanation so that helps me a lot.

Chris Biggadike

PS. I wish I had the luxury of consulting an analyst over probelms like that one!

 

[kafeel]

I completely agree with greg. The plate/beam might have had plastic deformation. Once it has become plastic the bending equation does not hold good. A good check would be to load the beam in steps and see if the load vs. deflection curve is linear, If its not , you have a problem.

All the best

 

[GregLocock]

I did a quick and nasty FE model last night. 3.75 mm!, 160 N mm-2 Von Mises stress

So, the beam is still sufficiently beam like. Next question is - how do you know the clamping of the strip is robust and effective. I agree that a factor of 2 is unlikely, but we are grasping at straws now.








Cheers

Greg Locock

 

[Biggadike]

I'm not suprised the beam is 'beam-like', Although it sounds like its thin and 'floppy' , when you see the test sample it doesn't look like its got much flex in anything other than the desired direction.
I did test to see where the yielding occured and no yield was seen until a deflection of 24.5mm so I'm confident I'm well withing the elastic region.
I think the problem lies not so much in the clamping arrangement moving but in the geometry at the clamping end:

Originally I tried a square clamping device on a parallel strip. This seemed to generate a stress point as you get a sharp (90deg) corner effectievely pressing into the beam at its point of greatest stress. This lead to early yield.
I then changed the arrangement to a radius after the nominal beam length (R5/R8 approx) leading to a wider section which was then square clamped.

I'm sure that there is a problem with that arrangement too: although the wider section must bend much less than the beam, it experiences a greater bending moment and, due to the leverage effect, only needs to flex a small amount to move my deflection point by quite a distance.
My gut feeling is that this only goes some way to explaining the entire discrepancy but without FEA its difficult to prove.

 

[gunsmith]

Biggadike, are You really sure that Your beam is stainless steel and not something like Ti13V11Cr3Al with E=100000 N/mm^2 and G=39000 N/mm^2. Greg pointed out that the beam is "beam-like" and we are also sure that there is not enough tension (160 N/mm^2) for a plastically deformation. Only the clamping conditions (think about Kiran´s angle calculation of the 2 degrees) and material properties are left to be wrong.

Andreas

 

[Biggadike]

Gunsmith,
The material is 3 series stainless steel, 1.2mm thick. It was made from standard company stock and is easy to identify. The value for E cannot be the culprit.
As I have said, I do suspect the clamping conditions to be either largely or wholly to blame. I have described my dilemma intrying to organize the clamping arrangement to give accurate results and I'm of the opinion that a clamped strip is almost impossible to set up in a way that reflects the spirit of the common model for a cantilever beam. (A welded T assembly using square section would perhaps be very close).
My hands are tied at this moment or I would have conducted further tests to try and verify this. I suspect a CDI at the interface between radius and wide clamped section would show a change of position if the 2degree theory is correct.

I have achieved a great deal in this forum so far, namely:

1) The clarification of use of the radius of curvature equation
2) Elimination of several potential areas of error (red herrings)
3) Reconfirmation that the maths are sound in this application.
4) Reconfirmation that the value for Youngs mod is correct.
5) The discovery of the most likely location for the problem and a realization of the potential sensitivity of cantiler beams to their mounting method.

I consider that a significant success and thank everyone who has contributed wholeheartedly.

 

[GregLocock]

Yeah, but we're still screwed. Some more thoughts

(1) are you loading the centre of the end of the beam?

(2) How sure are you that you are applying 19 N - is that being measured directly by a load cell? I have a horrible vision of you converting imperial to metric and doing a NASA!

(3) I'm glad you've thought about the cantilever clamping system, sounds like you have a good handle on it. Unfortunately your curved jaws will tend to reduce L under loading, leading to a reduced deflection, if anything. A good check fro the foundation stiffness is to measure the deflected shape along the beam, which will show you if rotation at the root is occurring.





Cheers

Greg Locock

 

[Biggadike]

The measuring set up was:

A digital force guage set to read in Newtons was used to deflect the beam (at its centre) at the appropriate length (taken from the beginning of the parallel section of the beam). When the guage read 18N, it was held in place.
Reference straight edges had been placed along side the beam. The deflection was measured from these.
The beam I tested was 95mm x 50mm x 1.2mm which is a nice manageable size. The deflection was great enough to be measured with a reasonable accuracy.

As I have said, the beam sample flared out at its end, the flared bit being clamped. This increases the effective beam length to (say) 105mm but the concept was that the flared section would see minimal deflection. Even if it had seen the same deflection as the rest (which is impossible) I still calculate that the maximum deflection should be just over 5mm. This suggests to me that whilst the clamp/end geometry is partly responsible for the deflection I have seen, the may be some other factor involved.

 

[gunsmith]

Deflection stresses have the maximum at the root of the beam and they consist of compression stress at the top of the beam and tensile stress at its bottom. Could it be that the surface pressure caused by the jaws is so high that an additional compression stress occurs on the top of the plate an adds to tensile stress component of bending because of its identical direction! The line in profil having no stress moves downwards in the intersection and the deflection of the beam increases. It´s only a thought coming to my mind at the moment.

Andreas

 

[Biggadike]

I would have thought that the stresses at 90deg to the primary stresses (Hoop type - crudely speaking) which would be caused by poisson type contraction under tension and vice versa would be no different in this application to any other (ignoring the clamping).
The clamp would then add to the compressive 'radial' stresses and reduce or eliminate the tensile 'radial' stresses. I would have thought that the main influence of this would be to force a greater extension in the longitudinal plane (squash the plate).
I'm struggling to see how this would affect the deflection on bending.

 

[gunsmith]

If You work on a metal beam by anvil and hammer hitting only the right side of beam the beam will be bended to the left side by a force working 90 degress to direction of hammer blast. If You were working at edge of anvil hitting the beam on its whole width - what will happen? Bending it down the edge or bending it up? I´m not sure of the direction, but the beam won´t keeping straight direction. Suggesting that the hammer with lower force (without plastical deformation of the beam) could be a jaw affecting the same forces. Then these forces are not able to bend the beam standing alone, but they could cause additional effects working together with a force of 19 N. My suggestions may be wrong - it is only an idea.

Andreas

 

[Biggadike]

I'm sorry Andreas but I don't buy it..

When you work on a piece of metal with a hammer you are plastically deforming it locally, this bends the metal because you have changed the shape on one side.
That is a very different situation to applying a clamping pressure with a vice to stop a tets piece moving around. If what you are suggesting was true, I'd have seen a deflection of the piece when the clamp was tightened, which I didn't.
The forces required for clamping are pretty small really and I'm not convinced they are responsible.

Chris

PS - of course, I still don't have an answer...

 

[gunsmith]

Greg was talking about straws were grapsing at. And he is right - nothing else we are doing now. There is a force causing the additional bending of the beam and we have to look for it.

What about the manufacturing process of the beam? Is it 100% stress free coming up to Your working table? What would happen if You take the beam to a furnace for stress-relieving anneal! Does it look like a corkskrew after four hours of annealing? There could be "hidden" forces from manufacturing spoiling Your experiment.

Did You try to turn around beam by 180 dgrees (upside down)?
Perhaps Your measurement results are not the same! This is the easiest way for a validity check for hidden forces and their directions.

Andreas

 

[Biggadike]

Now that I do buy..

Its a piece of rolled steel so it will not be at equilibrium. There should be resisual stresses in the steel.
If the outside of the sheet is in compression and the core is in tension then the material will bend very easily. The early part of the bending simply taking up the compressive 'slack' before tensile forces begin to apply.
It makes a great deal of sense that the sheet would have such a balance of residual stresses in its cross section after rolling. The best way to test this idea is to gently machine away one half of a strip too see if the strip bends when its symmetry is broken. I shall see if I can arrange this.

 

[Biggadike]

Early results...

The strip seems to have its outer surfaces in compression and its centre in tension. I haven't any data at the moment for the compressive modulus of steel or any other differences to be seen between compression and tension but this may provide the solution.

Tomorrow I will test bending with and across the grain.

Chris

 

[Biggadike]

OK, I think this could be the answer:

The residual stress in the material is compressive at the outer surfaces and tensile in the centre. A reasonably rough measurement of the magnitude of the stress (by measuring strain from geometry change after sheet is divided by machining) puts it at a higher value than the maximum seen during my test (something approximating 19x10^7 Nm^-2 where I was only calculating 15x10^7 Nm^-2 as a maximum stress)
By the rule of stress superimposition, this pre-existing stress profile, when put on top of the bending stress profile gives a shifted profile with maximum compression at the inner bend surface which falls to neutral at about 1/3 of the way through the section. This then has a shallow tensile peak at about 2/3 through the section followed by a small amount of residual compression or neutral stress at the upper surface.

Also, the bending stress only reaches the maximum at the point of clamping while the residual stres is equal along the length. This makes for quite a complicated transition between one stress state and another which is difficult to quantify.

My best guess of an analogous situation would be a composite material which has a high modulous thin strip as the lower compressive surface and a low modulous material as the upper 2/3. I would suggest that this would be easier to bend than a medium modulus homogenous beam.

The most important aspect of this is that this residual strain is only in the direction of the grain. I shall repeat the original experiment with a piece cut across the grain (unfortunately, next week is the best time for this) which should see a return to the expected values.

Chris Biggadike

 

[Biggadike]

Now I'm not so sure... (I'm also aware that I'm having this conversation with myself).

I just repeated the bend test with two samples, one cut with the grain, one across it. They behaved exactly the same in bending and both produced deflections well in excess of those predicted by the standard beam calculation.

Hmmmmm.

 

[gunsmith]

Chris,

do You mean that both tests showed the double deflection than calculated? Those things occuring in gun-production we call gunsmith-voodoo in Germany. That means nothings else than looking for "straws" to graps at in the haycock (instead of the pin). I have no idea at the moment - but I will return to this board having one.

Andreas

 

[vonlueke]

Biggadike: I'd first assume your end fixity and residual stresses are fine. It appears you're not testing what you're calculating and instead have two large triangles of steel going along for a so-called "free ride." If you haven't already done so, place a rigid bar (spanning across the width of your cantilever tip) underneath the load application point of your digital force gauge. Now your measured cantilever tip deflection should hopefully approximately equal your beam hand calculations. Let us know what deflection value this produces. Note: Use digital force gauge to apply a downward force F, where F = 19 N - mg, where m = mass of above-mentioned rigid bar (kg) and g = 9.80665 m/s^2.