Evaluation of vertical dynamic characteristics of cantilevered tall structures

In this paper, cantilevered tall structures are treated as cantilever bars with varying cross-sectionfor the analysis of their free longitudinal (or axial) vibrations. Using appropriate transformations,exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one stepnon-uniform bar are derived by selecting suitable expressions, such as exponential functions, for thedistributions of mass and axial stiffness. The frequency equation of a multi-step bar is established usingthe approach that combines the transfer matrix procedure or the recurrence formula and the closed-formsolutions of one step bars, leading to a single frequency equation for any number of steps. The Ritzmethod is also applied to determine the natural frequencies and mode shapes in the vertical direction forcantilevered tall structures with variably distributed stiffness and mass. The formulae proposed in thispaper are simple and convenient for engineering applications. Numerical example shows that thefundamental longitudinal natural frequency and mode shape of a 27-storey building determined by theproposed methods are in good agreement with the corresponding measured data. It is also shown that theselected expressions are suitable for describing the distributions of axial stiffness and mass of typical tallbuildings.