Static and dynamic stability of uniform shear beam-columns under generalized boundary conditions

The stability and dynamic analyses (i.e., the buckling loads, natural frequencies and the corresponding modes of buckling and vibration) of a 2D shear beam-column with generalized boundary conditions (i.e., with rotational restraints and lateral bracings as well as lumped masses at both ends) and subjected to linearly distributed axial load along its span are presented in a classic manner. The two governing equations of dynamic equilibrium, that is, the classical shear-wave equation and the bending moment equation are sufficient to determine the modes of vibration and buckling, and the corresponding natural frequencies and buckling loads, respectively. The proposed model includes the simultaneous effects of shear deformations, translational and rotational inertias of all masses considered, the linearly applied axial load along the span, and the end restraints (rotational and lateral bracings at both ends). These effects are particularly important in members with limited end rotational restraints and lateral bracings. Analytical results indicate that except for members with perfectly clamped ends, the stability and dynamic behavior of shear beams and shear beam columns are governed by the bending moment equation, rather than the second-order differential equation of transverse equilibrium (or shear-wave equation). This equation is formulated in the technical literature by simple applying transverse equilibrium at both ends of the member “ignoring” the bending moment equilibrium equation. This causes erroneous results in the stability and dynamic analyses of such members with supports that are not perfectly clamped. The proposed equations reproduce as special cases: (1) the non-classical vibration modes of shear beam-columns including the inversion of modes of vibration (i.e. higher modes crossing lower modes) in members with soft end conditions, and the phenomena of double frequencies at certain values of beam slenderness (L/r) and (2) the phenomena of tension buckling in shear beam-columns.