Free vibration analysis of cantilevered tall structures under various axial loads

Many cantilevered tall structures can be treated as cantilever bars with va riable cross-section for the analysis of their free vibrations. In this pap er, the differential equations for free flexural vibration of bars with var iable cross-section under various axial loads are reduced to Bessel's equat ions or ordinary equations with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the di stributions 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 t o derive the eigenvalue equation of multi-step bars subjected to more compl icated axial loads by using the transfer matrix method. One of the advantag es of the present method is that the total number of the finite elements (s egments) required could be much less than that normally used in the convent ional 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 t he natural frequencies of a television transmission tower calculated by the proposed methods are in good agreement with those computed by Finite Eleme nt Method. It is also shown through the numerical examples that the selecte d expressions are suitable for describing the distributions of flexural sti ffness, mass and axial loads of typical tall shear-wall buildings and high- rise structures. (C) 1999 Elsevier Science Ltd. All rights reserved.