Vibrational Analysis of Thin‐Walled Bars with Open Cross Sections

Natural frequencies and corresponding modal forms for thin‐walled bars of constant and open cross section are examined using the finite element method. Following the classical thin‐walled bar theory, warping and Saint‐Venant torsional rigidities are accounted for. A thin‐walled bar finite element with seven degrees of freedom at each node is adopted, and an explicit, consistent mass matrix of order fourteen is derived. Only small amplitude, linear vibrations are considered in the paper. A generalized eigenproblem is studied, and natural frequencies together with their corresponding vibrational modes are obtained. The convergence and accuracy of the method is tested based on some available closed‐form solutions and on other numerical results. Several examples illustrate the fast convergence of the method when applied to thin‐Walled bars of constant cross sections. Attempts are also made to use this finite element model to examine eigenproblems for thin‐walled bars with variable, open cross section. It is shown, that the method should not be implemented to analyze such bars since piecewise constant (stepped) finite element models may lead to erroneous results.