Fractal-like overturning maps for stacked rocking blocks with numerical and experimental validation

Abstract

A novel, compact mathematical formulation is presented to describe the dynamic rocking response of single and double block systems subjected to gravity and/or ground excitation. The derivation of the closed-form solutions for impact and motion is based on the Euler-Lagrange equation and the conservation of angular momentum, and combines all the different cases of possible block relative rotating and impact modes (16 in total) into a single set of equations without the need of transient expressions. The derived equations that describe the impact modes are the equivalent to the expression derived by Housner and depend on the angular velocity of the blocks before impact. The analytical model is integrated numerically via an ad hoc algorithm and its reliability & accuracy are verified after various self-consistency tests and comparisons with the literature. In addition, several shaking table experiments were conducted in EQUALS laboratory in Bristol, setup constructed to test free and forced rocking motion of single and double block configurations. The error margins of the measurements are determined, and the extracted data are in good agreement with the numerical results for most examined cases. The ideal Housner restitution coefficient of single block impact to a rigid base is adjusted to match experimental conditions, and it is found to be correlated with the block aspect ratio. The forced rocking of a two-block system is shown to exhibit numerous different response patterns depending on the excitation conditions. The integrated model is finally applied to produce normalised overturning maps for double block systems, subjected to singlepulse sine inputs, which uncover the existence of a fractal-type behaviour. This previously unsuspected trait of multi-block systems is reminiscent of the chaotic behaviour exhibited by a classical double pendulum and suggests that the risk of overturning can only be evaluated on a probabilistic sense.

Seismic response of slender MDOF structures with self‐centering base shear and moment mechanisms

“…Following the motion‐by‐motion comparison, the numerical and experimental results of the shaking table dataset were revisited with a focus on the statistical comparison of the Cumulative Distribution Function (CDF) of key response parameters that were discussed previously. The concept of statistical validation using the CDF plots has been applied to modelling of rocking structures that showed predictable rocking responses in the statistical sense 30–32 , 35–38 . Figure 5 presents the CDF plots used for the statistical comparison in this section.…”

“…The concept of statistical validation using the CDF plots has been applied to modelling of rocking structures that showed predictable rocking responses in the statistical sense. [30][31][32][35][36][37][38] Figure 5 presents the CDF plots used for the statistical comparison in this section. The numerical and experimental CDF plots were compared based on the Kolmogorov-Smirnov (K-S) distance 39,40 (i.e., the maximum vertical distance between the experimental and numerical CDF plots) and the relative errors (𝜀, defined as the absolute error between the experimental and numerical values divided by the experimental value) at the maximum horizontal distance, as well as at median and 90 th percentile of the experimental CDF plots, as listed in Table 1, and within each plot of Figure 5.…”

Finite element modeling of free‐standing cylindrical columns under seismic excitation

“…However, the seismic design problem is inherently stochastic, that is, one does not seek the response to an individual ground motion, but some statistic of the responses to a set of ground motions that define the seismic hazard. This observation allowed Bachmann et al 77 to revisit the validation procedure of numerical models: They tested a planar rocking structure under 600 ground motions and focused on the Cumulative Distribution Function (CDF) of the time maxima of each time history response. The CDF was both repeatable and predictable by the 1962 Housner model.…”

Shaking Table Seismic Experimental Investigation of Lightweight Rigid Bodies

“…In order to better observe the rigid block motion, the numerical studies were also combined with the improvement of the experimental knowledge of rocking systems [27-29. These studies were oriented to the identification of the role of an elastic foundation on which the blocks may lay [29], or to the assessment of the impact conditions [27,28]; few extensive experimental campaigns were presented to provide a large amount of data for possible comparison with numerical models [30]. The experimental campaigns usually converge in stating the non-repeatability of the experimental tests [31,32], as also testified by the failing of the numerical models to correctly reproduce experimental results [16,28,[33][34][35][36][37][38][39]; such a circumstance is mainly due to the negative stiffness of the rigid body systems, which led some authors to classify the rigid body oscillator as a chaotic system [40,41]. More recently, some authors interpreted the non-reproducibility of the experimental dynamic response of rigid body systems through statistical tools [42], leading to a reasonable prediction of their dynamic response.…”