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Explicit Integration

One of the fundamental steps of the finite element method is to generate a stiffness matrix for the individual elements. These are then assembled together and a solution obtained for the whole structure.

In general, numerical integration is performed to obtain this (and other) matrices. For some element types (e.g. beam) it is possible to obtain them by using Engineering Beam Theory in conjunction with traditional techniques such as the "unit-displacement" or "Castigliano's" theorems.

The stiffness matrix for the Engineering beam family of elements is given in the LUSAS Theory Manual II (search for "7.2 Beam Elements" in the online help).

Because of the different integration schemes used, there are two sign conventions used for the beam elements. These are shown in the LUSAS Element Library Manual, Appendix D.  The last note suggests that, when looking at the second node of a GRIL element, positive moments indicate sagging. If looking at the first node, then a positive moment would indicate hogging. Thus it is necessary to investigate force output from explicit integrated beams (e.g. GRIL) with a knowledge of the direction of the local element axes (obtained via the MESH layer properties).

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