Frequency Response of Flag-Shaped Single Degree-of-Freedom Hysteretic Systems

Recognizing the importance of limiting residual deformations that are usually associated with the response of full hysteretic systems under seismic loading, a number of energy dissipating devices and innovative structural systems exhibiting a full recentering response have recently been developed. Although these systems encompass the highly desirable self-centering characteristic, they inherently have less energy dissipation capacity when compared to more traditional fuller hysteretic systems. Little information is available on the nonlinear dynamics of systems exhibiting this flag-shaped hysteresis. In this paper, the frequency response of single degree of freedom (SDOF) systems exhibiting the flag-shaped hysteresis is computed in a closed form using the method of slowly varying parameters. The solution is derived for two independent variables representing the postyielding stiffness and energy dissipation capacity that define all self-centering hysteretic systems. The hysteretic systems covered by this formulation range from bilinear elastic systems (with no energy dissipation) to full bilinear elastoplastic. The frequency response exhibits a softening branch with multiple solution regions. A relationship between the system properties and bounded response at resonance for a given amplitude of excitation is also derived. Further analysis is carried out to assess the stability of these multiple solution regions. It is found that both stable solution branches can be achieved depending on the nature of the excitation. An amplitude jump phenomenon is also present both in increasing frequency and decreasing frequency sweeps. The concept of theoretical versus likely response curve is discussed for less idealized excitation functions.