Free Vibrations of Thin‐Walled Bars with Open Cross Sections

An application of the finite element method to the theory of thin‐walled bars of variable, open cross section is presented. A thin‐walled bar is considered as a special case of the membrane shell with internal constraints (Vlasov's and Wagner's assumptions). The bar is divided into elements along its longitudinal axis, then the shell midsurface of the element is approximated by arbitrary triangular subelements. Displacements of the element are represented by polynomials of the third degree, and both an equivalent stiffness matrix and a consistent mass matrix are obtained. An eigenvalue problem is analyzed using standard and generalized forms. Convergence of the method is discussed and numerical examples are presented for I‐beams of constant and variable cross sections.