Vibration and stability analysis of non-uniform Timoshenko beams under axial and distributed tangential loads

Two sets of governing equations for transverse vibration of non-uniform Timoshenko beam subjected to both axial and tangential loads have been presented. In the first set, the axial and tangential loads were taken perpendicular to the shearing force, i.e., normal to the cross-section inclined at an angle Psi, while in the second set, the axial force is assumed to be tangential to the axis of the beam-column. For each case, there exist a pair of differential equations coupled in terms of the flexural displacement and the angle of rotation due to bending. The two coupled second order governing differential equations were combined into one fourth order ordinary differential equation with variable coefficients. The parameters of the frequency equation were determined for different boundary conditions. The exact fundamental solutions could be found by expressing the coefficients of the reduced differential equation in a polynomial form before applying the Frobenius method. Several illustrative examples of uniform and non-uniform beams with various boundary conditions such as clamped supported, elastically supported, and free end mass and pinned end mass, have been presented. The stability analysis, for the variation of the natural frequencies of the uniform and non-uniform beams with the axial force, has also been investigated. Moreover, the present work illustrates the frequency behavior of the beam under a tangential load. (C) 2000 Academic Press.