Stability and free vibration analyses of orthotropic 3d beam-columns with singly symmetric section including shear effects

A complete beam-column classification and the corresponding characteristic equations for the stability and undamped natural frequencies of 3D orthotropic Timoshenko beam-columns with singly symmetric closed section and with elastic end connections subjected to an eccentric end axial load are presented and derived using three different approaches. The first two approaches are those by Engesser and Haringx that include the shear component of the applied axial force proportional to the slope (du/dx and dv/dx in planes xz and yz, respectively) and to the angle of rotation of the cross-section (θx and θy in planes yz and xz, respectively) along the span of the member, respectively. The third approach is a simplified formulation based on the classical Euler theory that includes the effects of shear deformations but neglects the induced shear component of the applied axial force along the member. The proposed methods and characteristic equations are capable of determining the critical axial loads and undamped natural frequencies of beam-columns with elastic end connections. Four comprehensive examples are included that show the effectiveness and simplicity of the proposed method and the results obtained are compared with experimental results available in the technical literature. It is shown that: (1) the natural frequencies and critical axial loads of beam-columns increase as the shear stiffness GAs, the degrees of fixity and lateral bracings at the ends of the member increase; (2) the natural frequencies calculated using the three approaches are identical to each other when the applied axial load is zero; (3) the critical axial load in compression using the Engesser approach is lower than the one obtained using the Haringx approach; (4) the critical axial loads in compression are highly affected by the degree of flexural fixity at the supports, but those in tension are not affected much; and (5) the Haringx approach is the only one among the three approaches capable of capturing the phenomena of tension buckling observed in seismic isolators.