Why do we need integration points in FEA?

[pukb]

Hi

Why are displacements calculated on integration points but not directly over the nodes. The whole purpose of discretizing a structure in FEA is to have fewer degrees of freedom, then why add integration points when there are well defined nodes.

Also, can somebody please explain having integration points over thickness and how it is taken care while solving a problem in FEA.

 

[AlephZero]

To form the element stiffness matrix, you need to integrate some function (related to the internal strain energy of the element) over the element volume. Except for simple elements with simple geometry, this is done numerically. In principle you could do this using something like the trapezium rule or Simpson's rule to do the integral using only nodal values, but other integration rules like http://en.wikipedia.org/wiki/Gaussian_quadrature are more efficient and/or more accurate.

For example, Gauss-Legendre integration with n points give the correct results for polynomials up to order (2n-1), and good approximate results (which can be interpreted in terms of least-squares fitting a lower order function and integrating it) for higher order polynomials.

The integration is always over the volume of the element. For shell or beam elements, if the material properties vary through the thickness of the element (e.g. layered composite materials) you may need to integrate for each layer separately. For isotropic materials and linear problems, you can usually do the integration through the thickness of a shell or over the area of the beam analytically, and then integrate over the area of the shell or the length of the beam numerically.