Coupled bending and torsional oscillations of a modern skyscraper

Abstract

abstract The use of electronic digital computers for solving complex engineering problems has instigated significant changes in solution methods. Matrix algebra and numerical analysis techniques are now being utilized extensively in dynamic and static structural analysis. Seismic structural problems which were previously solved using approximate methods, because of the impossibility of “exact” numerical calculations, are now being solved using these techniques in conjunction with computers. Plresent day building codes advocate “static methods” of seismic structural anaysis where in the engineer is directed to use coefficients for determining design lateral loads. Empirical formulas are given for calculating T, the natural period of vibration of a structure. However, uncertainty exists regarding the empirical calculation of T for unsymmetrical structures subject to coupling between their bending and torsional vibratory motions. Dynamic coupling occurs when the center of mass and the center of rigidity of a structure are not coincident, resulting in motions, under seismic excitation, which may be of prime importance. The tallest structure in San Francisco, the forty-three story Wells Fargo Building, is subject to such coupling. The theoretical methods of analysis utilized in determining the dynamic characteristics of this uni-directionally symmetric structure are described in this paper. With three possible vibratory motions per story, two translational and one rotational, the resulting problem is seen to be a formidable one involving one hundred twenty-nine degrees of freedom. Using normal mode theory in conjunction with a matrix formulation of the governing coupled differential equations, frequency and vibratory mode solutions are obtained using a digital computer.