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## Main Question or Discussion Point

Hello, I am having somewhat difficulty understanding the concepts of rank deficiency and hourglassing in finite element methods. Essentially, I have been reading a book outlining this very briefly on half a page and I need a bit more information. As an example:

If we have a 2D elasticity problem with a bilinear element, the element stiffness matrix "k" will be an 8x8 matrix, composed of the integral of B(T) D B, where B is a 3x8 matrix and D is a 3x3 matrix. So far so good. In 2D, this matrix "k" is said to be of rank 5 because you subtract 3 rigid body motions from this 8x8 matrix. While I see that you can have these 3 motions, I don't quite understand why this reduces the rank of the matrix. Could anyone give me some insight into how this works? I know, this probably sounds ridiculously simple, but somehow I am having problems picturing it.

Secondly, when it comes to hourglassing: If you use one Gauss point only to integrate this "k"-matrix numerically, you have: k = 4J B(T)DB | xi, eta = 0, which makes sense. However, this k-matrix has rank 3, the same as the D-matrix, and I don't really know why it isn't 5, like before. This rank deficiency obviously then leads to 2 hourglassing modes. Does anyone have any advice of how to determine, not just here, but also in other cases, the rank of a stiffness matrix? That would be very helpful, thank you.

If we have a 2D elasticity problem with a bilinear element, the element stiffness matrix "k" will be an 8x8 matrix, composed of the integral of B(T) D B, where B is a 3x8 matrix and D is a 3x3 matrix. So far so good. In 2D, this matrix "k" is said to be of rank 5 because you subtract 3 rigid body motions from this 8x8 matrix. While I see that you can have these 3 motions, I don't quite understand why this reduces the rank of the matrix. Could anyone give me some insight into how this works? I know, this probably sounds ridiculously simple, but somehow I am having problems picturing it.

Secondly, when it comes to hourglassing: If you use one Gauss point only to integrate this "k"-matrix numerically, you have: k = 4J B(T)DB | xi, eta = 0, which makes sense. However, this k-matrix has rank 3, the same as the D-matrix, and I don't really know why it isn't 5, like before. This rank deficiency obviously then leads to 2 hourglassing modes. Does anyone have any advice of how to determine, not just here, but also in other cases, the rank of a stiffness matrix? That would be very helpful, thank you.