Applied Numerical Mathematics 47 (2003) 59–73 www.elsevier.com/locate/apnum The composite element method applied to free vibration analysis of trusses and beams Marcos Arndt, Roberto Dalledone Machado ∗ , Mildred Ballin Hecke Federal University of Paraná, Numerical Methods in Engineering Graduated Program, Caixa Postal 19011 – CEP, 81531-990 Curitiba, PR, Brazil Abstract This work deals with an enrichment technique of the finite element solution to the free vibration problems, called the composite element method [J. Sound Vibration 218 (1998) 619, 659, Key Engrg. Mater. 145–149 (1998) 773]. The enrichment of the solution space is obtained combining the FEM and the high accuracy of closed form solutions from the classical theory. The analytical solutions must be in accordance with some special boundary conditions in such a way they do not change the nodal values of FEM and, also, they must be the solutions to the frequency equation. The CEM can be improved using two types of approach: h and c-versions. The h-version, the same as FEM, consists of the refinement of the element mesh. The c-version is the increase of degrees of freedom related to the classical theory. Truss and beams elements are used in order to verify the numerical efficiency of the CEM. Some examples are presented and the frequencies and mode shapes of vibration obtained by CEM are compared with the FEM solution, and also the classical theory. The numerical results have shown that CEM is more accurate than FEM with the same number of total degrees of freedom. The CEM is also more accurate to determine higher frequencies than the FEM, except for the last ones. The results suggest that, for higher frequencies, numerical instabilities are presented.  2003 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Composite element method; Vibration analysis; Finite element method 1. Introduction Nowadays, buildings are becoming taller and slender, and machines, automobiles, planes are made with stronger and lighter new materials. The dynamic effects are getting much more significant. In most cases, it is essential that a vibration analysis be used in the structural design. * Corresponding author. E-mail addresses: rdm@cesec.ufpr.br (R.D. Machado), mildred@cesec.ufpr.br (M.B. Hecke). 0168-9274/$30.00  2003 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/S0168-9274(03)00064-3