Elsevier

Journal of Sound and Vibration

Volume 483, 29 September 2020, 115488
Journal of Sound and Vibration

Optimal finite locally resonant metafoundations enhanced with nonlinear negative stiffness elements for seismic protection of large storage tanks

https://doi.org/10.1016/j.jsv.2020.115488Get rights and content

Abstract

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.

Introduction

Metamaterials are entering the field of seismic engineering and other research areas with a variety of interesting structures. The two most prevalent concepts in the field of seismic protection are phononic crystals [1] and locally resonant metamaterials [2], where both are able to create the so called bad gap phenomenon. Band gaps signify frequency regions where waves cannot propagate through the material and are therefore able to provide new solutions to existing vibration problems. For the present work we focus on locally resonant materials, due to their ability to attenuate waves at wave lengths much greater than their unit cell size, which is a particularly important property for seismic metamaterials. To date, locally resonant materials have been used to conceive foundation systems [[3], [4], [5], [6], [7]] and wave barriers [[8], [9], [10], [11]]. While metabarriers have the advantage of being placed besides the structure, and can therefore be installed after the completion of the building, they can only attenuate surface waves. Metamaterial-based foundations on the other hand, can in principle attenuate any type of incoming wave, but have to be placed below the structure of interest, hence limiting their application to new buildings. The present work is concerned with foundation systems, which show a variety of different designs and applications in the current literature. A particularly interesting foundation was proposed by Cheng and Shi [5] who conceived a system tuned to the ground motion for the protection of nuclear power plants. Their foundation showed different band gaps for the vertical and horizontal direction, thereby addressing the vertical component of earthquakes. This is especially relevant for high consequence structures like nuclear power plants, since classical isolation systems, like concave sliding bearings, are not able to address the vertical motion [12]. Besides this, also Casablanca et al. [7] developed an interesting foundation based on concrete plates separated by Teflon sliding surfaces and verified its behaviour with laboratory experiments. Their experiments clearly depicted that these types of structures are feasible with common construction materials and can exert the band gap phenomenon. However, neither Cheng and Shi nor Casablanca et al. took the feedback from the structure into account. La Salandra et al. [4] on the other hand designed a foundation system and conducted a study on the most influencing factors on the attenuation behaviour. Two important findings shall be mentioned, namely, the influence of the stiffness and the non-negligibility of the feedback of the superstructure. Subsequently, Basone et al. [13] developed a foundation system based on their results and conceived an optimization procedure that can take a structure as well as an ensemble of expected ground motions into account. However, their design shows significant restrictions in terms of effectiveness due to the constraints given by the governing building codes (i.e. Eurocode 3 and 8, [14,15]). Besides this, an experimental study on the coupling effects between a tank isolated with a metamaterial-based foundation and a pipeline suggested that this type of foundation may provide a compromise between base shear attenuation and horizontal displacement [16], which is a property that cannot be obtained with classical isolation systems. Further worth mentioning is the work of Witarto et al. [17] who studied the application of metamaterial-based systems to small scale nuclear reactors; and the work done by Ungureanu et al. [18] who used auxectic like materials to protect high-rise buildings. Finally, a comprehensive review of seismic metamaterials including metabarriers as well as foundation systems was given recently by Mu et al. [19]. From their review one can clearly conclude that one of the most pressing problems of metamaterial-based foundations is the excessive size necessary to obtain a functional foundation. However, two advantages may become attainable through such foundations in future, namely: (i) attenuation of the vertical component [5] and rocking motions [20], which cannot be addressed by traditional base isolation systems [21]; and (ii) a compromise between base shear reduction and horizontal displacement [16]. One idea to improve the performance of a metamaterial-based system was proposed by Antoniadis et al. [22] who showed that a negative stiffness element (NSE) inserted in the resonator mechanism could potentially improve the system behaviour significantly. Note that this is not an effective negative stiffness as discussed in e.g. Ref. [23], but a composite spring system where the resulting force assists motion and does not oppose it. Note that Antoniadis et al. [22] included only a conceptual negative stiffness element that would exert the desirable amplification force, while a design for an actual mechanism that could be applied to a periodic structure was still missing. To date, most proposals including negative stiffness and metamaterials aim at the continuum level [24,25], while Morris et al. [26] conducted an experimental study on such a continuous metamaterial with buckling type instabilities and showed the energy dissipation capabilities of the structured medium. These proposals are interested mainly in the material level, and therefore, do not investigate the application to a structure or the inevitable nonlinear effect of an NSE on the band gap.

It is worth noting that research work on nonlinear metamaterials is still limited and primarily concerned with weakly nonlinear resonant chains. A perturbation approach for the dispersion analysis of weakly nonlinear chains has been proposed by Chakraborty and Mallik [27], which clearly depicts that: (i) solutions to nonlinear wave equations are amplitude dependent; (ii) wave amplitudes influence their own propagation characteristics, the so-called self-action; and (iii) analysis methods in the presence of self-action often do not trace all solutions when more than one dominant component is involved. Another neat approach to calculating the band gaps for such materials relies on the harmonic balance method (HBM) as has been demonstrated by Lazarov and Jensen [28]. Banerjee et al. [29] on the other hand provide a comprehensive review of 1D metamaterials including materials with nonlinear oscillators. Both showed classical bi-atomic lattices with nonlinear oscillators, e.g. Duffing oscillator, pendulum, impacting resonators, and concluded that an increase in elastic nonlinearity, entails a shift and an elongation of the band gap. Based on the current state of the art, the present work conceives a new mechanism applicable to periodic structures, which is able to reduce the size of metamaterial-based foundations. In order to present a realistic application, a fuel storage tank was chosen as a superstructure and its feedback taken into account when designing and optimizing the foundation. Note that fuel storage tanks represent the most vulnerable and consequence intensive components of industrial plants during earthquakes, and that their seismic protection is still an ongoing issue [[30], [31], [32], [33], [34]]. The coupled Metafoundation tank system is analysed on its performance for various foundation heights and different levels of applied negative stiffness herein. Note that the practical application includes only a one layered foundation, while further analyses, carried out on the system considered as a periodic structure, shed light on the wave propagation in nonlinear negative stiffness enhanced materials.

The present work tackles three main research issues, namely: (i) Size reduction of metamaterial-based structures for seismic applications; (ii) development of an NSE that can be implemented in a metamaterial; and (iii) investigation of the inevitable nonlinear behaviour. The manuscript discusses these issues in the following order: Section 2 elaborates the structure, the foundation, and the mechanism and shows the simplified dynamic system used in the subsequent analyses; Section 3 shows the metamaterial-like behaviour of a periodic system with and without considering the nonlinear effect; Section 4 demonstrates an optimization algorithm for the optimal design of the foundation; Section 5 investigates the behaviour of the complete coupled and optimized structure under real seismic action; and Section 6 closes the paper with conclusions and future developments.

Section snippets

Description of the structure

The Metafoundation was initially conceived in Ref. [4] and later developed and designed according to common construction standards by Ref. [13]. The proposed foundation is based on steel columns that support concrete slabs, with resonators placed in between the columns, in order to provide the system with its locally resonant properties, see Fig. 1(a). Additionally, for an improved performance of the system, a new type of NSE is designed herein, which is then implemented by mounting it to the

Band gaps and wave propagation

On the one hand the effect of the NSE on the band gap behaviour is expected to be advantageous due to the amplification force, while on the other hand the effect of its inevitable nonlinearity is yet unknown. In this section the potential band gaps of the system will be investigated for the linearized as well as the elastic non-linear structure with parameters corresponding to the FULL system.

Optimization of the coupled system

As shown in previous publications [4,13], once the Metafoundation is coupled to a superstructure, the complete coupled system needs to be optimized. For this reason, we propose an optimization algorithm based on calculations in the frequency domain, which represents a simplification of the algorithm established in Ref. [13] and depends on the structure as well as the ground motion.

Behaviour of the system in the frequency domain

When running the optimization procedure on the coupled system subjected to the average PSD of the earthquakes, the PI can be computed and plotted for various frequencies and damping ratios for the resonators, as displayed in Fig. 11(a)–(d). Here, fR corresponds to the optimal frequency of the resonators, while ζR is the optimal damping ratio of the resonators, which are computed for the Metafoundation with 0%, 25%, 50%, and 99% of the maximal admissible NSE value, obtained from eq. (25). Note

Conclusions

In this work, a new type of NSE, based on a compression member in a stable snap through position, has been developed for the application to seismic metamaterials. The composite system showed enhanced wave attenuation characteristics and was studied on its fully nonlinear behaviour via time and frequency domain analyses. Due to the implemented NSE as well as the new type of established resonator chain, the system displayed a widening of the band gap and an amplification of the attenuation

CRediT authorship contribution statement

Moritz Wenzel: Conceptualization, Formal analysis, Investigation, Methodology, Writing - original draft. Oreste S. Bursi: Investigation, Conceptualization, Supervision, Funding acquisition, Project administration, Writing - review & editing. Ioannis Antoniadis: Investigation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 721816 for the first author. The second author acknowledges funding from the Italian Ministry of Education, University and Research (MIUR) in the frame of the “Departments of Excellence” grant L. 232/2016 and the SERA grant agreement no. 730900.

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