Free Vibration of Three-Dimensional Orthotropic Uniform Shear Beam-Columns with Generalized End Conditions

The free vibration analysis of asymmetrical three-dimensional (3D) uniform shear beam-columns with generalized boundary conditions (semirigid flexural and torsional restraints, lateral bracings, and lumped masses at both ends) subjected to an eccentric end axial load in addition to a linearly distributed eccentric axial load along its span is presented in a classic manner. The five coupled governing equations of dynamic equilibrium (i.e., two shear equations, two bending moment equations, and the pure torsion moment equation) are sufficient to determine the natural frequencies and modal shapes. The proposed model which is an extension of a 2D model presented previously by the writer includes the simultaneous 3D coupling effects among the lateral deflections, deformations of the cross section along the member (shear, torsional and rotational), the translational, rotational and torsional inertias of all masses considered, an eccentric end axial load in addition to a linearly distributed axial load along its span, and the end restraints. Deformations caused by shear forces and pure torsion are considered. The effects of axial deformations, warping torsion and torsional stability are not included. The proposed model shows that the dynamic behavior of 3D shear beam-columns is highly sensitive to the coupling effects just mentioned, particularly in members with both ends free to rotate. Analytical results indicate that except for doubly symmetric members with concentric axial loads and with perfectly clamped ends, the natural frequencies and modal shapes of 3D shear beam-columns are determined from the eigenvalues of a full $8\times 8$ matrix, rather than from the uncoupled equations of transverse (or shear-wave equations) and torsional moment equilibrium. Two comprehensive examples are presented that show the effectiveness of the proposed method.