Free vibration analysis of cantilevered tall structures under various axial loads

Abstract

Many cantilevered tall structures can be treated as cantilever bars with variable cross-section for the analysis of their free vibrations. In this paper, the differential equations for free flexural vibration of bars with variable cross-section under various axial loads are reduced to Bessel's equations or ordinary equations with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the distributions of stiffness and mass as well as for the axial forces acting on the bars. The general solutions for free flexural vibration of a one-step bar with variable cross-section subjected to simple or complex axial loads, including concentrated and variably distributed axial loads are presented first in this paper. Then the general solutions of one-step bars are used to derive the eigenvalue equation of multi-step bars subjected to more complicated axial loads by using the transfer matrix method. One of the advantages of the present method is that the total number of the finite elements (segments) required could be much less than that normally used in the conventional finite element methods. The numerical example 1 demonstrates that the calculated fundamental natural frequency of a 27-storey building under the actual axial loads is closer to the measured field data than that computed without considering the axial forces. The numerical example 2 shows that the natural frequencies of a television transmission tower calculated by the proposed methods are in good agreement with those computed by Finite Element Method. It is also shown through the numerical examples that the selected expressions are suitable for describing the distributions of flexural stiffness, mass and axial loads of typical tall shear-wall buildings and high-rise structures.