FEA Stress recovery triangular element

[Katha90]

Hi,

I am currently writing a FEA program. I cannot find any information on how to extrapolate stresses from Gaussian points to the corner nodes.

The mesh constists of six noded triangular elements. The stresses are calculated on four Gaussian points. The shape functions are in terms of natural coordinates.

Does anyone know how the stress recovery for triangular elements work?

Kind regards
Katharina

 

[btrueblood]

There are a lot of methods, from none (plot the gaussian stresses and ignore the nodes), to simple linear interpolation, to constructing n-degree polynomial spline fits, and finally, using the shape functions to compute the nodal stresses. According to the first reference below, interpolation gives better results than the shape function method.

http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/IFEM.Ch28.d/IFEM.Ch28.pdf

http://imechanica.org/node/10439

http://eprints.qut.edu.au/13842/2/13842.pdf

 

[Katha90]

Thanks for your reply.

I would like for the simple linear interpolation, but ..

I checked the references. For reference 3, the number of Gaussian points and nodal stresses must be equal (I think), which is not the case.
I know how to do a linear interpolation for cartesian coordinates but not for natural coordinates. For a quadratic element the interpolation would be very easy.

Can anyone help? Does anyone have an example?

 

[btrueblood]

Are you sure your triangular elements have 4 Gaussian points? Only 3 should be required...but I don't know your element formulation either.

 

[Katha90]

Triangular elements can have either 1,3,4 or 7 Gaussian points. I picked 4 Gaussian points as the integration with 3 GP was not as correct as with 4 GP.

But I could just use three Gaussian points and use the shape functions for the extrapolation, as I still don't know how to do a linear extrapolation with natural coordinates (it should be fairly simple though).

Thanks for your help!

If anyone else knows anything about stress extrapolation, I am happy about any hint.

 

[ShellsPlatesMeshes]

It is unlikely that you will find a published reliable algorithm. The problem is that the stresses are discontinuous between elements, which is the consequence of the fact that the FEA is an approximate method by itself, based on the minimization of one or another functional. Normally, the displacements are continuous between the elements and do correspond to the reality; the stresses correspond to the 2nd or even 3rd derivative of the displacements (the normal stresses due to bending and the transverse shear stresses, correspondingly). Numerical differentiation is an unstable process, tending to deliver somewhat ridiculous results for the 3rd derivatives (and sometimes even to the 2nd derivatives) where there are just small fluctuations in the displacements field.