Optimal Finite Locally Resonant Metafoundations Enhanced with Nonlinear Negative Stiffness Elements for Seismic Protection of Large Storage Tanks

Metamaterials represent a new trend in the field of seismic engineering. Their capacity to attenuate waves at the superstructure level is highly desirable and sought after in recent years. One of their main drawbacks to date, is the excessive size of the necessary resonators and, consequently, the uneconomic design they require. In order to tackle this problem, we apply the concept of negative stiffness to a metamaterial-based foundation system and analyse the potential improvements such a mechanism may have on the metamaterial as well as the coupled structural behaviour. Since negative stiffness is a property that cannot be achieved through conventional measures, a novel mechanism, designed for the implementation in periodic metamaterial-based structures, is proposed herein. The inevitable nonlinearity of the mechanism will be discussed and taken into account, while the advantages of the negative stiffness element (NSE) will be treated analytically and verified numerically. Additionally, through an optimization in the frequency domain and nonlinear time history analyses (THAs), the performance of the system coupled with a fuel storage tank is elaborated. With only 50% of the theoretically allowable NSE value, the foundation system could be reduced to 1/3 of its size. Furthermore, the nonlinear effect of the device has proven to diminish the band gap of the periodic system, which led us to introduce nonlinearity parameters that can help avoid the strongly nonlinear range. In sum, this article tackles three problems that are intertwined: (i) reducing the size of metamaterial-based structures; (ii) the design of a mechanism that exerts a negative stiffness in a periodic structure; and (iii) the study of the inevitable nonlinearity of NSEs and the subsequent effect on the metamaterial behaviour.