Basic FE Theory

[sengkannan]

Reading literature there seems to be two methods of deriving element unaveraged stresses at nodes
1) Interpolating results from gauss points
2) Directly using nodal displacements.
In simple explanation how are stress derived in both cases.
If you are after stress within the element i.e. centre is interpolating from gauss point more accurate than interpolating from node point stress derived directly from displacements.

can anyone recommend easy to understand references.
Thanks

 

[apldmch]

The finite element method uses a strain-displacement matrix where the calculated nodal displacements are used to calculate strains at the gauss points. Stresses at the gauss points are calculated by multiplying the gauss point strains by the elasticity matrix. The nodal stresses are typically averaged from the elemental gauss point stresses.

 

[ThomasH]

It might give more to post the question in the Finite Element Analysis Engineering Forum but I'll give the explanation a try.

As I understand your question the two altenatives you give are actually the same in my opinion(but I can be wrong).

The typical FE-process is as follows:

Calculate the element stiffness matrix. Usually this is made by numeric integration on the element using for example gauss quadrature.

Since the typical model consists on more then one element you the assemble the element matrices to a global stiffness matrix.

Apply boundary conditions (loads and constraints) to the equation system K*p = P.

Solve p = K^(-1)*P to get the nodal displacements p.

Now you have all the nodal displacements and can, using the element stiffness matrix, calculate the stresses in the gauss points. This I would say is your 2) option.

A typical software does as result give the stresses in the nodes and element center.

The nodal stresses can be exprapolated from the gauss points or interpolated in some manner.
The center stresses are interpolated from gausspoints or for some reduced integration schemes, the gauss point can be the center point.

To my knowledge the is no method to direcly calculate nodal stresses from nodal dicplacements, at least not for 2d or 3d elements. If i'm wrong about this, please correct me.

Regards

Thomas

 

[sengkannan]

Thanks for your input.
My question relates to the three options available in Nastran for calculating element nodal stress BILINEAR, CUBIC and SGAGE (the last two are similar).
From reading Nastran notes the BILINEAR uses results at gauss points and extrapolates it to the node points where as the CUBIC uses the grid displacements and rotations to curve-fit a cubic equation to develop the element strain/curvature distribution.
Am I correct in saying the CUBIC option works directly with the nodal displacements only. Any simple explanation on how the element strain/curvature distribution is developed.
I ran two models using the two options, the nodal stress were different as expected but the centroidal stress were the same, I assume the centroidal must be based on gauss point stress extrapolated to the centre. I wonder why they do not extrapolate the nodal stresses to the centroid for the CUBIC option, is the gauss point stress the most accurate/reliable.

 

[ThomasH]

I must say that that was a more "in depth" question than I first expected. My interpretation of the notes is that for all three methods the gauss points are the base points.

BILINEAR means the corner(gridpoint) and center stresses are computed using strain gage approach with a bilinear exprapolation.

SGAGE means that stresses are computed using a strain gage approach.

CUBIC finally works in the same manner as SGAGE except it uses a cubic bending correction.

Also, nodal stresses are extrapolated using stresses from the adjoining elements.

And to answer your final question: yes, gauss point stress is usually considerad the most accurate.

Maybe this answeres your question to some extent.

Thomas

 

[sengkannan]

If you are referring to the notes in the quick reference guide I think it gives a poor explanation of the three methods I should have mentioned the notes I am referring to were obtained from the MSC site. I have attaced the notes bellow, let me know what you think.
Thanks for your contribution

"What is the difference between the three corner outputs, BILINEAR, CUBIC and SGAGE? (bilin,
cubic, sgage)

Answer: BILINEAR is the default CORNER method. This method uses the results at the gauss points
to create a linear function that is then used to extrapolate the results
out to the grid points. In the case of a linear varying moment in an cantilevered
beam engineering model, since the QUAD4 element is an almost constant strain
curvature element, the gauss results are constant curvature in each element,
thus the linear function is a constant curvature for each element, thus the CORNER
strain curvatures are constant in an element. In the cantilevered beam problem,
corner stress output using the bilinear method will cause a step function
from the cantilever point to the load application point, which is obviously a
coarse approximation of the correct linear stress distribution of the engineering model.

The CUBIC method works from the grid displacements. It uses the grid displacements
and rotations to curve-fit a cubic equation to develop the element strain/curvature
distribution. In the case of a linear varying moment in the cantilevered
beam engineering model, since the grid point rotations vary across the element,
the curve fit gives gives the correct engineering result of a linearly varying
strain curvature across the element, which translates to a linear varying stress.

The SGAGE method is similar to the CUBIC method in that grid displacements are used,
but in-plane strains and curvatures are calculated independently. First strains
are calculated in the u and v and diagonal uv directions at each grid point.
The state of in-plane strain at the grid point is calculated using rosette strain
gauge equations. Grid strain curvatures are done similarly. The resulting
grid point strain and curvature states can then be changed to forces and stresses
at the grid points. The BILINEAR option is the default because it is more stable in all cases, but as is
shown in the linearly varying moment case, it can much less accurate than the
CUBIC method. The CUBIC option is very accurate if the mesh is fine enough to
give accurate displacements. However, coarse meshes can cause the cubic curve-fit
to be more inaccurate than the BILINEAR technique. The SGAGE technique is similar to the CUBIC method, but not as accurate. Testing indicates
it may be obsolete."

 

[ThomasH]

First I would say that the qrg is perhaps not the best source for information in this case. At least not compared to the information you now have given.

OK, my "interpretation" is as follows:

BILINEAR means that it uses the gauss point stresses and the bilinear element base functions to extrapolate nodal (grid point) stresses. That is also my interpretation of the qrg.

CUBIC, now I have to admit that I'm not as sure what happens so this is more of guessing. Stresses are calculated in the gauss points. Then the grid displacements are used to develope the strain/curvature distribution. Based on this distribution the stresses are extrapolated to nodes.

SGAGE seem to be more of a mix because it uses bilinear approximation for part of the calculation. But, as they say, it may be obsolete.

I must also say that this thread has become a lot more interesting then I first thought.

Best Regards

Thomas

 

[sengkannan]

Thanks for participating, I shall keep you informed of my findings.