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. [1], we apply Buckingham’s Pi Theorem [2] 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” ( RN F).RN Fis 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. 1 Introduction In this paper we introduce and experimentally validate the most general theoretical framework governing the dynamic behavior and scaling of Multiblock Tower Structures (MTS) as dictated by the application of Bucking- ham’s Pi Theorem (BPT) [2]. BPT is a mathematically formal tool used in all branches of engineering and science to identify dimensionless groups of problem variables crucial in determining the response of a physical system when it is subjected to excitations in a particular environment. The ability of identifying dimensionless groups is particularly important for studying very complex systems where closed-form relations describing their behav- iors are not available. These groups allow for the prediction of dominant trends between crucial parameters. In addition, the identiﬁcation of the dimensionless groups, or dimensionless numbers, enables the design of “scaled” or “model” experiments, which can be used to indirectly, but accurately, study “prototype” system behavior by just observing the behavior of a model. This becomes possible, since knowledge of the appropriate dimensionless groups dictates the form of “similitude” or scaling laws and thus allows observations obtained at a small scale, in a controlled and highly instrumented environment, to be extrapolated to the often much larger engineering or geophysical scale without loss of accuracy. Historically, the above approach was extensively used in ﬂuid dynamics and in thermal sciences [3–5] where scaling enabled the use of wind-tunnel and water-tunnel testing driving the design of generations of aircraft and submarine systems at a time when large scale numerical computations were unavailable. A prominent example of dimensionless numbers used in scaling of ﬂuid mechanics and gas dynamics systems is the Reynold’s number (RN e:=v/lscript/ν) which represents the ratio of ﬂuid inertia to the viscus restoring forces and characterizes the complex 1 transition between laminar and turbulent ﬂow. Other examples are the Mach number ( MN:=v/c), which is a measure of the compressibility characteristics of ﬂuid ﬂow past an object moving at a certain speed, and the Froude number ( FN r:=v/√/lscriptg), which represents the ratio of inertial to gravitational restoring forces. In the above examples,vrepresents a measure of ﬂuid speed, cis the local speed of sound, νis the kinematic viscosity, /lscriptis a characteristic length while g, only involved in FN r, is the acceleration of gravity. In ﬂuids, FN rscaling is primarily used in analyzing hydraulic systems characterizing the nature of free-surface ﬂows or gravity wave formation in tsunamigenesis. It has also been used to scale the wave making resistance of similar ﬂoating bodies of various sizes. In contrast to ﬂuid dynamics and thermal sciences, the ﬁelds of solid mechanics and structures have only made limited use of dimensionless numbers and scaling, with notable exceptions such as the introduction of scaling laws governing ductility, instabilities, and fractures (Bazant et al. [6,7]). Other prominent exceptions to this rule can be found in structural dynamics and in earthquake seismology, where the use of just a few key dimensionless numbers has enabled the eﬀective testing of the resistance of buildings to earthquake excitation through scaled-down shake table experiments (extensively discussed by Harris and Sabnis [8]). For such testing, the most commonly used dimensionless number is the Cauchy number ( CN:=v//radicalbig E/ρ=v/c), wherevis a measure of the particle velocity of a solid or structure, ρis the mass density, and Eandcare representative moduli and elastic wave speeds of the material. The CNnumber represents the ratio of the inertial forces acting on various parts of the system to the elastic restoring forces provided by the system’s elasticity. In very limited cases, the Froude number (mostly in combination with CNscaling) has also been used to scale model/prototype systems and to test structures when gravitational restoring forces are also considered important (see Moncrarz and Krawinkler [9]). To understand the reasons for the widespread use of CNscaling in evaluating the seismic reliability of the built environment in modern times and to also explain the motivation of the current study, we now refer to Fig. 1. This ﬁgure pictorially describes the major types of large “monumental” structures that humanity has constructed in the last forty-six or so centuries [10–13], most of which have experienced and need to be protected from seismic excitation of various types and intensities. Figures 1(a) and (b) depict early monumental structures made of various quarried geomaterials such as limestone, sandstone, granite, syenite, alabaster and basalt as is the case with the Egyptian pyramids, or of limestone, sandstone or marble as is the case of ancient Greek or Roman temples. With time, and up to the past century, more monumental structures were built out of various combinations of rocks and raw and baked mud bricks as is the case in the damaged pagoda depicted in Fig. 1(c). In all cases shown in Figs. 1(a)-(c), the structures were characterized by the high density and rigidity of the constituent materials. These structures belong to the class of “discontinuous” or “discrete” gravitational structures in which a variety of nearly rigid elements are held together by either frictional or cohesive (mortar) restoring forces. These forces act at multiple interfaces and, in addition to gravity, provide the restoring forces and moments during seismic excitation. It is the competition between the seismically induced inertial forces with the various restoring forces (gravitational, frictional, cohesive) described above as well as the ability of the interfaces to dissipate energy, which ultimately determines the ability of these discontinuous structures to resist earthquake damage. This competition, along with the associated kinematic response of rocking, sliding, and its combinations have been studied in relation to either monolithic or segmented column systems by a number of authors (e.g., [15–25]), providing valuable insight regarding their dynamic response. Most notably, the behavior of single multi-spondyle marble columns have been shown to perform very well under seismic excitation compared to the monolithic equivalent as shown in a recent numerical study by Konstantinidis et al. [23] and analytical studies by Kounadis et al. [24, 25] and the