Dynamic analysis of axially loaded cantilever shear-beam under large deflections with small rotations

The modal analysis and seismic response of a vibrating cantilever, with or without tip mass and rotary inertia, are investigated in this study using a shear deformable beam model and including the effect of vertical load. Based on existing approaches, an original method is proposed that does not use fourth-order uncoupled equations to determine modal deflection and rotation. In fact, the approach presented herein transforms the second-order coupled system into a first-order system which can be solved more easily using matrix algebra and Laplace transform. Furthermore, the proposed form allows a straightforward demonstration of orthogonality conditions, that is, the problem is self-adjoint, and the solution in the case of forced response using modal superposition. In addition, even if the solution presented herein is applicable only to the cantilever with a tip mass and rotary inertia, the scope is general, and the approach can be applied to shear deformable beams with other boundary conditions. Finally, the seismic response by modal superposition is shown, and some examples are proposed and discussed for the case of uniform or continuously varying cross-sectional properties.