A unified formulation of the piecewise exact method for inelastic seismic demand analysis including the P-delta effect

The non-linear analysis of single-degree-of-freedom (SDOF) systems provides the essential background information for both strength-based design and displacement-based evaluation/design methodologies through the development of the inelastic response spectra. The recursive solution procedure called the piecewise exact method, which is efficiently used for the response analysis of linear SDOF systems, is re-formulated in this paper in a unified format to analyse the non-linear SDOF systems with multi-linear hysteresis models. The unified formulation is also capable of handling the P-delta effect, which generally involves the negative post-yield stiffness of the hysteresis loops. The attractiveness of the method lies in the fact that it provides the exact solution when the loading time history is composed of piecewise linear segments, a condition that is perfectly satisfied for the earthquake excitation. Based on simple recursive relationships given for positive, negative and zero effective stiffnesses, the unified form of the piecewise exact method proves to be an extremely powerful and probably the best tool for the SDOF inelastic time-history and response spectrum analysis including the P-delta effect. A number of examples are presented to demonstrate the implementation of the method.