6. Finite Elements

This is the second chapter devoted to the approximate analysis of continuous systems. In the finite element method, the shape functions are defined within the element, once and for all, with the generalized coordinates being the nodal displacements and rotations. The method is also based on Hamilton’s principle; the element stiffness and mass matrices of the approximate discrete system are obtained by expressing the strain energy and the kinetic energy in terms of the generalized coordinates. This chapter considers the finite element formulation of a plane truss (made of bar elements) and of planar structures made of beams, including the geometric stiffness. The convergence of the finite element method is briefly addressed. The chapter concludes with a discussion of the methods for model reduction: Guyan method and Craig-Bampton method. A set of problems is proposed at the end of the chapter.