Stability and vibration analysis of axially-loaded shear beam-columns carrying elastically restrained mass

Classical shear beams only consider the deflection resulting from sliding of parallel cross-sections, and do not consider the effect of rotation of cross-sections. Adopting the Kausel beam theory where cross-sectional rotation is considered, this article studies stability and free vibration of axially-loaded shear beams using Engesser’s and Haringx’s approaches. For attached mass at elastically supported ends, we present a unified analytical approach for obtaining a characteristic equation. By setting natural frequencies to be zero in this equation, critical buckling load can be determined. The resulting frequency equation reduces to the classical one when cross-sections do not rotate. The mode shapes at free vibration and buckling are given. The frequency equations for shear beam-columns with special free/pinned/clamped ends and carrying concentrated mass at the end can be obtained from the present. The influences of elastic restraint coefficients, axial loads and moment of inertia on the natural frequencies and buckling loads are expounded. It is found that the Engesser theory is superior to the Haringx theory.