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Beam calculation doesn't add up.

Beam calculation doesn't add up.

Beam calculation doesn't add up.

(OP)
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?

RE: Beam calculation doesn't add up.

(OP)
This is news to me. My value for Youngs modulus (E) is based on a general stainless steel value of 210GNm-2.
I was under the impression that as the youngs modulus is based the atomic attraction rather than any other forces, the condition of the steel and its composition makes very little difference. I have never seen a significantly different value for E than the one I have quoted above in any of my reference data.

RE: Beam calculation doesn't add up.

(OP)
I've just check Matweb and they have the same value for E for every stainless steel (20Gpa) regardless of condition. Are you sure your aren't thinking of yield?

RE: Beam calculation doesn't add up.

A factor of two error is the most common kind.

Double-check your input dimensions (did you use width where you were supposed to use half-width, etc).  I assume that your spring is indeed a long, slender beam, like your equations are probably intended to address?

RE: Beam calculation doesn't add up.

(OP)
I can't find any errors in the input data at all. I've been through the whole thing with a fine toothed comb with another engineer and we couldn't fault any of it. The good thing with working in Mathcad is that the whole thing is very transparent.
I agree that there must be some kind of error somewhere but I can't see where. I even double checked the whole thing by putting the final stresses and so on into the engineers bending equation and it added up.

RE: Beam calculation doesn't add up.

sure you're calculating bending radius and not bending diameter?

RE: Beam calculation doesn't add up.

(OP)
I thought of that but in all the text books I have it is referred to as bend radius. The wierd thing is that at least two or three books have the same equation.

I also tried doing the calculation for a three point bend and got the same results (with allowance for the obvious differences in geometry and forces).

RE: Beam calculation doesn't add up.

(OP)
I wonder whether the standard maths are actually a poor reflection of the reality:
It seems to me that a cantilered beam would not bend in an arc but would bend in a parabola. The bending moment at the anchor being greatest, reducing along the length to the tip where it is zero. This would explain the discrepancy between the features produced in reality and those caluclated.
The probelm with the maths (or rather my brain) is that its rather convaluted and conatins a lot of integrations of fundamental maths. This makes it a bit difficult to work out exactly what the equations are modelling.

RE: Beam calculation doesn't add up.

You are right, in response to a force at the end it won't be a circular arc, it will bend far more at the root than at the tip. The standard beam equation does not predict a circular arc, therefore you are either using the wrong equation, or you are not just applying a force perpendicular to the tip of the cantilever.

So, which is it? It would probably be helpful if you posted a few more details, ie the exact dimensions and the equations you have used.

Cheers

Greg Locock

RE: Beam calculation doesn't add up.

(OP)
OK, the details of the beam are as follows:

Using a standard one point catilever beam calculation
(I convert to SI units for calc. then back to mm)

Width = 100mm
Depth = 1.2mm (Centroid = 0.6mm)
Length = 95mm
E= 201,000,000,000
I=1.44 x 10^-11
Force = 19N
Maximum deflection=3.752mm
Bending moment (M) = 1.805

Radius of curvature is given as

R=EI/M = 0.802 (802mm)

The problem is that when I measure the delfection I get 6.75mm at 19N and I'm looking for the discrepancy.

RE: Beam calculation doesn't add up.

Why not do a tensile test on the stainless steel and measure the actual Youngs Modulus? At least that will rule that error out!    

Speedy

RE: Beam calculation doesn't add up.

(OP)
I have no tensile test equipment but as far as I'm concerned the young modulus of steel is well established and unchanging from steel to steel. I tried seeing if I'd got the number of zeros wrong but that puts in errors by a factor of ten each time and my measured error is a factor of two.

RE: Beam calculation doesn't add up.

Just a thought. The cantilever formula is based on the assumption that the end is fully built in (i.e no rotation at the end). If the end has any kind of flexibity, then the deflections will be higher than those calculated from a cantilever formula. In your example, I estimate that a 2deg rotation at the fixed end support is enough to account for the 3mm discrepancy in the measured end deflections (ARCTAN(3/95)= 2deg approx). What does the end support of your test setup look like ? Any play in the gaps at the supports ?
Just a thought.
Kiran

RE: Beam calculation doesn't add up.

(OP)
This is indeed a good thought. The design of the spring and of the test piece was a bit like a tensile test sample in that it had a broad radius to a wide section. It was this wide section which was clamped. The assumption was that it was much harder to deflect the broad section and so the bending there was negligible.
This may not have been the case as the bending moment at that area is at its highest. I would be slightly supprised to see 2deg of bend there but it is certainly possible.

RE: Beam calculation doesn't add up.

It's really odd the fact that you calculate a deflection that exactly doubles the true one, coming from the well known formula
f=Pl3/3EI
so now you are faced with a discrepancy factor of 4!
I agree with Kiranpatel (and you): a problem is certainly in the clamping area, you must be 100% sure that no slope change occurs at 95 mm from the tip, otherwise big changes in deflection may occur. By the way that condition is so difficult to achieve and to check that you should definitely use a simply supported scheme with load in the middle if you want to check the properties of your spring.
Another factor you could consider is that the front border of the sheet won't stay straight id you apply the load as a concentrated force in the middle of it.
I don't understand what use you want to make of the radius of curvature, anyway keep in mind that the actual value is changing along the beam length from the value you quote to infinity at the tip.

prex
motori@xcalcsREMOVE.com
http://www.xcalcs.com
Online tools for structural design

RE: Beam calculation doesn't add up.

(OP)
Thanks, the on line tools look handy.

I went for the cantiler beam as that is what the design had to be. The measured value from the test piece is accurate to what I will actually get from the component. I was expecting some discrepancy but I was suprised by the error. I think the concensus that cantilever beams are difficult to reproduce without stress raisers or flex is accurate. I will do a three point bend check on the calculations just to see what that gets me.
The radius of curvature was listed with the standard equations in a number of text books. I realised quite early on that the curve was not an arc. I've no idea why the radius of curvature is quoted - text books rarely justify themselves. It has been a red herring in my search for the discrepancy.

RE: Beam calculation doesn't add up.

I agree with PREX. He has the right result - you are dealing with the factor 4 now. The formula is working well since over 100 years. Only the clamping conditions, material properties, beam dimensions or force could be wrong.

Andreas

RE: Beam calculation doesn't add up.

(OP)
Can you give me some more data? I can't reconsile your equation with mine without knowing the parameters you are using.

I assume:

f=?
P=?
L=length
E=Youngs mod
I=Second moment of area

RE: Beam calculation doesn't add up.

You are using the following parameters:

l=95 mm
b=100 mm
h=1.2 mm

I=b*h^3/12 (=14.4 mm^4)

E=201000 N/mm^2  

(should be between 179000 and 216000 for spring-steel)

P=19 N

f=P*l^3/(E*I*3)  (=1.876 mm)

That is the whole mechanism You need for the calculation.

Andreas

RE: Beam calculation doesn't add up.

(OP)
Yes,
That is the equation I used and the one I got. My deepest apologies if I've given the wrong values at any point in this discussion. I'm juggling between two sets of figures - the original design and the adjusted design. The actual error is still only a factor of two. The original values were calculated as:

width = 50mm
depth = 1.2mm
length = 95mm
Force = 19N
Deflection = 3.75mm

The measured deflection was 6.75mm

Sorry if I jumbled the values, I'm trying to do too many things at once!

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

(OP)
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!

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

(OP)
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.

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

(OP)
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.

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

(OP)
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.

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

(OP)
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.

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

(OP)
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...

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

(OP)
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.

RE: Beam calculation doesn't add up.

(OP)
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

RE: Beam calculation doesn't add up.

(OP)
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

RE: Beam calculation doesn't add up.

(OP)
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.

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

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.

RE: Beam calculation doesn't add up.

(OP)
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.

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

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

RE: Beam calculation doesn't add up.

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).

RE: Beam calculation doesn't add up.

(OP)
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.


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