Free vibration of an unsymmetrical multistoried building modeled as a shear-flexible cantilever beam

A solution is presented for the undamped free vibration frequencies and mode shapes of a multistoried building with unsymmetrical floor plan, modeled as a continuous (distributed parameter), shear-flexible cantilever beam. The beam has constant mass per unit length and a linear distribution of elastic shearing resistance (stiffness) along its length. Three coupled, second-order partial-differential equations describe the rotational and lateral motions. Solution of the equations reveals the presence of “coupling modes,” which are dependent on the cross-sectional properties of the beam, and “z-modes,”, which are independent of cross-sectional properties. An exact solution is presented for the coupling modes, and approximate and exact solutions for the z-modes are obtained. Frequencies are given for different linear variations in elastic resistance, and are presented in dimensionless form. A discrete element analysis has been included for purposes of comparison. The results of this paper will be referred to in the accompanying paper entitled “Response Spectrum Solution for Earthquake Analysis of Unsymmetrical Multi-Storied Buildings.”