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?

 

[Biggadike]

Vonlueke,

I see your thinking and to some small extent I'm sure you're right but we're talking major discrepancies here. If the triangles to the sides of the pushing point weren't being moved at the same time it would look like a V section. The width of the test piece is only 50mm and the curve produced during bending makes it very difficult for the side edges to stay behind while the middle deflects. The test piece is vertical so its weight and any other weights I might add have negligible effect - in theory this should produce a slightly lower level of deflection than predicited, at the moment I'm measuring about twice that predicted.

Gunsmith,

I'm well into the clutching at straws arena. All the ideas I and everyone else have had would account for small discrepancies, the kind of stuff you might ponder over if you had very accurate test equipment and just couldn't account for that last 2% error. This is massive and very obvious. Only something fundamantal can be responsible but I just can't find it.

 

[kafeel]

Hi,
It has developed into a real interesting problem. Like all, I too dont have any solutions. However you could rule out clamping errors by checking with another material like aluminium. You could also do a tensile test to ascertain the young's modulus, if you dont have the resources, can you do a hardness test? You can get approximate values of young's modulus from them.

I take it that you have tried finding deflection for the material under pure bending(simple supports at both ends). see if the formulae match up.

Keep us posted

 

[GregLocock]

If you plot the deflected shape along the beam and compare it with the theoretical result it will show you wheteher you have a clamping problem, as it will hav the wrong shape if you do, as it will have a rotation superimposed on the deflected shape.

I want to fast forward to the end of this thread to find the answer out! Cheers

Greg Locock

 

[vonlueke]

Biggadike: Yes. I agree now that the incorrect boundary condition I mentioned earlier creates only an additional 1% error in this particular problem. So that's OK and not the problem.

Rethinking the situation, and provided you are certain you have regular stainless steel, I would next suspect the measurement of your plate thickness is incorrect. Reobtain the average plate thickness of the final test article very carefully from two or more independent sources using different brands of calibrated, certified equipment. E.g., take measurements with two different brands of micrometers at several locations inside the plate, not on the cut edges.

Using half of your fixed-support fillet radius, and therefore an effective beam length of about 100 mm, the actual test article average plate thickness should be anywhere from about 1.043 to 1.070 mm thick, not 1.200 mm thick. However, this beam is slightly outside small deflection theory, so expect an additional error of about 2.7% in your hand calculations due to this.

If the above fails to uncover the problem, I'd next recommend a quick reality check of your digital force gauge (e.g., by lifting known volumes of water).

 

[Biggadike]

Well,

I think I have the complete answer following some final experiments I did yesterday:

Originally, when I was interested in finding the yield point of some of our stock material I found that testing done on parallel sided strips in square jaws caused premature yielding beacause the effective attack angle at the clamping point shifted during bending and produced a stress raiser.
With this in mind, I changed the profile of subsequent test pieces to include a radius to a wider area which was itself clamped.
This is where the problem was. We had susupected it but when I calculated the deflection of an extended beam it still didn't give the results so I dismissed it.

The final conclusions are:

1/ At high levels of stress, a square clamped sheet beam will yield prematurely at the clamp point due to the clamp acting as an edge.

2/ At low levels of stress, a square clamped sheet beam (with parallel sides up to the clamp) will exactly follow standard beam claculations.

3/ Flaring out a sheet beam before the clamping point has two effects, both of which heavily distort beam calculations:
i) The beam is made effectively longer despite the wider section. This is because the beam is thin and the support afforded by the wider section is limited to the sides.
ii)The supported sides distort the beam. The outer edges flex less putting an arc into the transverse section.

4/ There are high residual stresses in rolled sheet but, as they are in equilibrium they seem not to effect the bending of the sheet.