Implications of Buckingham’s Pi Theorem to the Study of Similitude in Discrete Structures: Introduction of the RFN, μN, and SN Dimensionless Numbers and the Concept of Structural Speed

Motivated by the need to evaluate the seismic response of large-capacity gravity energy storage systems (potential energy batteries) such as the proposed frictional Multiblock Tower Structures (MTS) recently discussed by Andrade et al. (2021, “Seismic Performance Assessment of Multiblock Tower Structures As Gravity Energy Storage Systems,” ASME J. Appl. Mech., Submitted), we apply Buckingham’s Pi theorem (Buckingham, E., 1914, “On Physically Similar Systems; Illustrations of the Use of Dimensional Equations,” Phys. Rev., 4, pp. 345–376) to identify the most general forms of dimensionless numbers and dynamic similitude laws appropriate for scaling discontinuous multiblock structural systems involving general restoring forces resisting inertial loading. We begin by introducing the dimensionless “mu-number” (μN) appropriate for both gravitational and frictional restoring forces and then generalize by introducing the “arbitrary restoring force number” (RFN). RFN is subsequently employed to study similitude in various types of discontinuous or discrete systems featuring frictional, gravitational, cohesive, elastic, and mixed restoring forces acting at the block interfaces. In the process, we explore the additional consequences of inter and intra-block elasticity on scaling. We also formulate a model describing the mechanism of structural signal transmission for the case of rigid MTS featuring inter-block restoring forces composed of elastic springs and interfacial friction, introducing the concept of “structural speed.” Finally, we validate our results by demonstrating that dynamic time-histories of field quantities and structural speeds between MTS models at various scales are governed by our proposed similitude laws, thus demonstrating the consistency of our approach.