Plate Bending FEM Convergence Study Vs Theoretical Mc/I 7

[stressebookllc]

Dear Friends, I am in the process of double checking a simply supported 10" x 4" plate, supported at the 1.0" from the short edges. The edges are only constrained along TZ. A weak spring is used right in the middle to avoid any fatal errors. Then I used three different mesh sizes to look at the convergence of the normal stress. I have attached the model.

The plate is 0.25" thick and the applied load is 1000lb along -Z.

When we do a simple bending stress calc, Sigma = Mc/I, we get 48ksi.

In the model however (looking at the shell normal-X bottom output vector), it is close but higher than theory for the fine mesh, and the coarser mesh is closer to theory.

So I was wondering what you guys think about this. The goal is to use mesh convergence and hopefully get closer to the theoretical Mc/I at the middle of this plate.

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[JoshPlumSE]

I can't open your model, so I'm just guessing. However, the issue could be related to the support conditions. When you sub-mesh your model, you need to make sure that your boundary conditions are consistent with the new mesh.

Again, just a wild guess based on your description.

 

[stressebookllc]

Josh thanks for your quick response. The file was a FEMAP netral file with results. Let me know if it was too large or you can use another format.

There are three mesh size based groups in the model, the contour plot levels adjust automatically when you look at each active group, assuming you have done this in FEMAP.

I am not sure if I follow what you said but may be I am misreading it. The support conditions are the same for all three mesh sizes and they are based on geometry.
All associated nodes in each mesh size are constrained (the constraints and loads expand to the nodes from geometry) in the three different cases, no problem.

See the attached picture.

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[stressebookllc]

And here is the hand calc:

b = 4.0"
h = 0.25"
c = h/2 = 0.125"
I = bh^3/12 = 4*0.25^3/12 in^4

Mmax = PL/4 = 1000*8/4 = 2000 in-lb (simply supported beam, ROARKS)

Bending Stress = Mmax*c/I = 2000*0.125*12 / (4*0.25^3) = 48,000 psi

This should be the value that the FEM normal stress along X needs to match. Basically the top and bottom fiber normal stress Sigma X which.

Its not like the results are way off, but I am looking for a meaningful convergence of the FEM results with mesh size refinement, which I am not seeing in the FEM unless I am missing something.

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[structSU10]

So is the force applied as a point load on one node? That can lead to stress risers in FEM due to the singularity it creates. applying a load over a few nodes or an area could help the results. Also it appears your supports are at just two nodes on each end, is this things supposed to be supported by an edge? It should have Z restraint on all nodes along the line rather than at just two points. If the two points are correct, they will also cause the same stress riser effect due to the point support. It makes the plate effectively act as a two way member a little more, but that may not affect your solution at the center much.

 

[stressebookllc]

The constraints and loads are applied along the full width (4.0"), what you see there is FEMAP representation of an edge constraint and load.

The load is applied on the full width as well 250 lb/in, as a running load.

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[corus]

You still have a point load, albeit applied along a line. Your results will therefore be a combination of bending plus local singularity stresses at the load position. Plot the stress distribution as an XY plot of stress against distance to see this effect.

 

[stressebookllc]

So if I were to apply the load as a long beam 4 point bending test would then technically it should solve the problem. I will try that and post an update.. thanks all.

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[btrueblood]

Do you have a copy of Roark's Formulas for Stress and Strain handy? If so, read the preface to the chapter on beam bending theory, esp. where it discusses wide beams - you need to correct the predicted stiffness of the beam (I) to account for plate effects, essentially a wide plate in bending will "cup", or distort, along the width of the beam, and this affects the plate stiffness. In my copy (6th ed.), beams are Ch. 7, and the wide beam is discussed in section 7.11. You could also look in the section discussing rectangular plates.

Corus/s10 have a pretty good idea too, but I'd expect a point load singularity to give a pretty high stress, not just a few percent error.

 

[stressebookllc]

Yes I do.

Good points though. I did notice an hourglass type of contour, it makes sense now based on the cupping you mentioned.

In the version I have it is Section 8.11, Beams of Relatively Great Width. Thanks for pointing it out. Its more complicated than I thought.

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[prex]

With your boundary conditions and load distribution there are no local stress risers (as it is apparent from the stress plot), nor there are "cupping" or distorting effects along the width of the plate, whose lines along the width remain straight (as it is also apparent from the figure).
What you are missing is that in plate theory (you are using plate theory in your model) you need to account for lateral constraint due to Poisson's ratio. The effective section moment of inertia has to be divided by 1-ν²=0.91 (for ν=0.3) with respect to the one for a beam with the same section.
So your stress is closer to 53,000 psi than to 48,000. Does this give a better picture of the convergence with finer meshes?

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[prex]

Sorry, I am wrong above: Poisson's effect acts on deflections only, not on stresses, and of course the deflections are lowered, not raised. However Poisson's effect is not completely null on stresses.
Timoshenko gives a reduction of 0.988 for max stress in a uniformly loaded plate with your proportions (b/a=2), as compared to a very narrow plate that has the same stress as a beam. Didn't find any treatment for a center load like yours.
So I can't explain your result, would need more data and results.


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[stressebookllc]

Actually there is an influence on the stress induced from the anticlastic curvature per ROARKS section 8.11.

There is the transverse bending stress induced due to the anticlastic curvature effect = poisson's ratio * longitudinal bending stress Mc/I

If rho is the radius of curvature after bending:

We have 1/rho = M/KEI

Where:
rho = radius of curvature
M = Max moment PL/4
K is the factor in the table in ROARKS based on poissons ratio Vs b^2/(rho*h)

b = width​

h = thicnkess​

I = b*h^3/12

I have 0.33 for Poisson's ratio

But how to calculate radius of curvature rho?

Once I have that, I could get the combined max moment and use it in Mc/I to calculate the maximum stress, that is the plan at least.

In the FEM, I could clearly see the anticlastic curvaure effect in the deformed shape. See attached.

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[prex]

As a first approximation you can take ρ=L²/8h (h being the deflection at center).

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[ShellsPlatesMeshes]

Not all finite elements for modeling of thin-walled structures are of the same quality. For example, some elements are prone to so-called "shear locking" effect, which may cause the model to fail to deliver reasonable results. The phenomenon that you have described may be just of the poor quality of the finite element itself. Other approaches are much more accurate; for example, a quadrilateral element based on Kirchhoff-Love approach delivers very good numerical results. You may want to have a look at the following webpage (there is also a number of examples with URL links):

http://members.ozemail.com.au/~comecau/quad_shell.htm

 

[stressebookllc]

The results are actually not unreasonable. Besides, for what I am trying to do, pretty much all reputable codes do a pretty decent job..

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[ShellsPlatesMeshes]

You explained (in the very first post) that coarse mesh delivers more accurate results than a fine mesh, which is strange (to say the least). My hypothesis has been that (a) reputable code(s) might have an inherent problem related to the quality of the mathematical approach used for the derivation of the stiffness matrices of corresponding plate elements. If you deny that hypothesis altogether, there is little you can do when trying to figure out: why such inconsistency exists, i.e. why coarse mesh seems to deliver a better solution.

 

[IDS]

The thing is the coarse mesh does not actually give a better approximation, it just happens to be closer (at the edge) to the average stress across the plate (48 ksi). I have re-run the analysis and plotted the XX stress across the plate for the 3 models (see attached). It can be seen that all 3 results are close to a parabolic distribution averaging 48 ksi, and the finer meshes are a significantly better fit than the coarse mesh, as would be expected.

The analysis was done in Strand7 with a Poissons Ratio of 0.25, but I would expect results to be close to identical from any other package (using the same PR).

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[stressebookllc]

Yeah, I was seeing a similar hour glassing shape of the contour. Thanks IDS, I like the way you plotted it along the width, simply yet so informative.
So in real life, there should be a higher stress at the edges than at the center due to the anticlastic curvature effect, but the Roark's table is for the average stress.

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