This research is a preliminary study to evaluate the seismic force of nonstructural components (secondary system) beyond the elastic limit of steel buildings. The objective of this paper is to propose a convenient evaluation method of higher harmonics in a periodic response of an SDOF system with bilinear hysteresis subjected to sinusoidal ground motion.
Firstly, numerical examples are demonstrated to show how the higher harmonics in a floor response affects the acceleration response of the secondary system. Two types of floor acceleration are considered. One is waveform obtained by nonlinear time history analysis and the other is a sinusoidal waveform having the same amplitude. The result shows that secondary response to these floor acceleration are significantly different if the frequency ratios of the secondary system to the input motion are close to or coincide with odd numbers.
Secondary, this critical higher harmonics are formulated by specified dynamic characteristics of the building and its peak displacement. In order to simplify the formulation, floor acceleration waveform in the steady state condition is approximated using piecewise linear waveform defined by a post-yield stiffness ratio and a ductility factor. A closed form of Fourier coefficients of the higher harmonics is derived with respect to the approximated waveform. Furthermore, the solution is simplified by neglecting higher terms in the exact solution with respect to the post-yield stiffness ratio for practical application.
Finally, the proposed solution is compared with the result obtained by discrete Fourier transform in time history analysis. The maximum ductility factor is assumed to be 100 in a numerical example. The post-yield stiffness ratio is in the range of 0.01 to 0.3, depending on the type of structures. The ninth higher harmonic is analyzed at most. The results show a good accuracy of the proposed solution in a wide variety of ductility factors.
