Understanding Guass Points, Shear Energy, Hourglassing & Shear Locking

[pukb]

i am facing difficulty in understading the following terms in fea

1. Gauss points and integration points
2. zero shear energy elements
3. Hourglassing
4. shear locking

Please provide information about the four points in a much simpler way compared to what is presented in internet or books.

 

[afreiden]

I am by no means an expert here, but I hopefully at least conceptually (hand-wavy) understand your four terms.

1. Gaussian quadrature is the way that numerical integration is done in FEA. I believe it can give an exact solution (of the integral) with sufficient number of integration (gauss) points.

2. This isn't a common term, so I googled it and saw the term being used at least once in the context of plates. Maybe it is referring to the kirchhoff-love plate theory (an extension of the commonly used euler-bernoulii beam theory), where shear strain is ignored in the derivation. In FEA, Reissner-Mindlin plate theory (an extension of the so-called Timoshenko beam theory) is what is used, and you have shear strain. Maybe the phrase "zero shear energy" is referring to the shear strain that is taken to be zero in the Kirchhoff formulation but not in the Mindlin formulation.

3. With a lot of integration points ("Gauss" points -- see "1"), and a coarse mesh, you can get shear locking. This phenomenon makes your structure artificially stiff. To understand why, read the paragraph of page 87 (and look at the associated figure) here: http://www.scribd.com/doc/59724360/65/Volumetric-locking
I suppose that shear locking is a problem in FEA analysis of plates (recall "2" regarding the presence of shear strain under deformation in Mindlin Theory) of a certain aspect ratio, if that interests you. One solution to the shear locking of brick elements is to use less integration points (i.e. use "reduced integration" elements instead of "fully-integrated" elements).

4. The problem with using less integration points is "hourglassing." This phenomenon makes your elements distort in a manner such that two adjacent elements form the shape of an hourglass (image search it), which can lead to nonsense. To see why this occurs, consider a 2D quad element with a single integration point in the center. There are 8 degrees of freedom (two translations at each of the four nodes). There are 3 rigid body modes (two translations and a rotation). There are only 3 stresses for the single integration point (two normal and a shear). We can see that 8 minus (3+3) leaves us with 2 "hourglass" modes. One possible solution to the problem of "hourglassing" is to go to fully-integrated elements, but you'd better use a fine mesh (see "3"). The computational expense of using a fine mesh and fully-integrated elements is the main reason why you don't always want to so.

Hope that helps (and isn't too hand-wavy)