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]

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.

 

[Biggadike]

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?

 

[ivymike]

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?

 

[Biggadike]

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.

 

[ivymike]

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

 

[Biggadike]

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).

 

[Biggadike]

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.

 

[GregLocock]

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

 

[Biggadike]

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.

 

[Speedy]

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

 

[Biggadike]

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.

 

[Kiranpatel]

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

 

[Biggadike]

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.

 

[prex]

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

 

[Biggadike]

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.

 

[gunsmith]

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

 

[Biggadike]

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

 

[gunsmith]

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

 

[Biggadike]

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!