Preliminary design method for rocking and negative stiffness systems

Abstract

This paper proposes a simple analytical system that can be used to describe the dynamics of Negative Stiffness Bilinear Elastic (NSBE) systems, and consequently design them in a simpler manner. The NSBE oscillator is a mathematical idealization, which can be used to describe rocking structures with or without flexible restraining systems or curved extension at their bases. The paper defines the characteristic quantities to make the bilinear system and actual rocking structures equivalent. A simpler “equivalent” system to describe the behavior of NSBE systems is proposed: The equivalent system is the Zero Stiffness Bilinear Elastic (ZSBE) system, which is a bilinear system with zero stiffness in the second branch. The ZSBE system is useful and simpler because it needs one parameter less than the NSBE to be defined. The paper proceeds by defining the “Equal Displacement” and “Equal Energy” rules that provide estimates of the maximum displacement of the NSBE based on the response of the ZSBE. Using a simpler system to predict the response of a more complicated one, is a concept similar to the RμT relations that provide estimates of the response of bilinear yielding systems based on the response of an equivalent linear elastic system. However, the method should not be confused with the approach of FEMA 356: it does not resort to a linear elastic system but to the ZSBE. Finally, the preliminary design of a real rocking structure is presented, as a case of study. The paper compares the response predicted by the proposed methodology to the one predicted by a more accurate numerical analysis.