# When does reduced integration lead to artificial zero energy modes in stiffness matrix?

This question relates to the topic of locking free finite element development.

In the case of application of reduced integration to global stiffness matrix for the Timoshenko beam element with quadratic shape function of field variables, I get an integrand up to 4th degree in case I calculate this integrand with 2 Gauss quadrature points should I expect additional zero eigenvalues to the resultant matrix?

To generalize, under what condition do I get additional zero eigenvalues when I integrate an $$n$$ degree polynomial function with reduced integration of Gauss quadratures?

• Given that you have a beam, you have to be careful with the shape functions. When you have rotational field variables hermitian shape functions will be used, which are of a higher order. Feb 13 '20 at 11:24
• Yes, I am aware of that. Hermitian should be used to balance the ansatz spaces and also represent the curvature in the case of linear moment distribution. However, this question was just for the educational purpose. In my understanding, the reduced integration should introduce additional zero energy modes, however as I see in some examples it is not always the case. I wonder when it is the case and when it is not. Feb 21 '20 at 18:32