### ANALYTIC INTEGRATION OF THE STIFFNESS MATRIX OF A 8-NODED PLANE CUADRILATERAL SUBPARAMETRIC FINITE ELEMENT

#### Abstract

Computer Algebra Systems (CAS) are powerful tools to obtain analytical expressions for many engineering applications in both academic and industrial environments.

The explicit (analytical) for of the stiffness matrix of an 8-noded plane-elasticity finite element is obtained and described in this paper. Six basic formulas to generate the stiffness terms were obtained by using MAPLEã. The formulas are function of the cartesian coordinates of the element, as well as a function of the elastic parameters.

Once the formulas were obtained, some algebraic manipulation were done in order to optimize the analytical expressions. The reduction in CPU time by using analytical integration instead of classical Gaussian-numerical integration was very promising, leading to over 50% saving in CPU time. Also, as expected, analytical integration leads to a more accurate results than numerical integration does.

A sensitive analysis was done, showing that analytical integration is useful even in the case of very distorted finite elements.

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