Variational correctness and Timoshenko beam finite element elastodynamics

The finite element discretisation of the two-noded Timoshenko beam element for elastodynamics offers very interesting insights into the error analysis aspects of the formulation. In this paper, the relatively different order of convergence of the two spectra of the Timoshenko beam theory, and the extra-variational aspect of the use of reduced integration to free the element of locking, are investigated. The correct variational basis for finite element analysis of elastodynamic problems is presumed to originate from the principle of virtual work, with a simultaneous consideration of errors in both displacement and strains. A variationally correct element would lock; to make an element free of locking, some degree of variational incorrectness must be brought in. The present paper also demonstrates that reduced integration violates the virtual work principle which in turn causes the loss of boundedness of the finite element eigenvalues with the exact solution.