Evaluation of vertical dynamic characteristics of cantilevered tall structures

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