Natural patterns of vibration of building frames

Abstract

World codes for aseismic design of structures prescribe empirical rules for determining the fundamental period of buildings approximately and no rules have been laid down for the higher mode frequencies. For preliminary designs, the structural engineer looks for handy methods for determining the fundamental and a few higher mode frequencies. Stodola's method of iteration with the dynamic and the inverse dynamic matrix, respectively, provide the fundamental and highest frequencies. Buildings in vibration can be categorized as “bending”, “shear” or of “linear” type. The frequency ratio patterns of buildings can be idealized as the classical ones for pure bending and pure shear cantilever beams. For the linear pattern, a rule for frequency ratio has been derived. Based on the frequency patterns, a frequency characteristics chart has been developed and is presented in the paper; it provides good trial frequency values for use in the Holzer technique for the determination of the intermediate frequencies. A non-dimensional parameter termed as the “nature parameter” connecting the fundamental and highest frequencies and the total number of floors, as explained in the paper, has some interesting properties relevant to the pattern of vibration and indicates the inherent behaviour of a building in vibration. The nature parameter of four well known buildings in the world and a number of research frames including those of U.S. researchers has been compared with the ideal patterns. Results of a computerized case study on 156 shear frames of 5 to 40 storeys, and various column stiffness patterns, indicate that the frequencies follow a defined pattern depending upon the storey stiffness and that a non-dimensional “period parameter”, connecting the total storey stiffness and mass, predicts exactly the fundamental frequency on the basis of some derived expressions. Frequency ratios for determining higher frequencies, mode shapes and mode participation factors also follow regular patterns and the values can be predicted from tables. The salient features are presented in the paper.