Summary An argument of engineers and researchers against the use of rocking as a seismic response modification technique is that the rocking motion of a structure is chaotic and the existing models are incapable of predicting it well. This argument is supported by the documented inability of rocking models to predict the motion of a specimen excited by a single ground motion. A statistical comparison of the experimental and the numerical responses of a rigid rocking oscillator not to a specific ground motion, but to ensembles of ground motions that have the same statistical properties, is presented. It is shown that the simple analytical model proposed by Housner in 1963 is capable of predicting the statistics of seismic response of a rigid rocking oscillator.

Summary In order to use rocking as a seismic response modification strategy along both directions of seismic excitation, a three‐dimensional (3D) rocking model should be developed. Since stepping or rolling rocking structural members out of their initial position is not a desirable performance, a rocking design should not involve these modes of motion. To this end, a model that takes the aforementioned constraint into account needs to be developed. This paper examines the 3D motion of a bounded rigid cylinder that is allowed to uplift and sustain rocking and wobbling (unsteady rolling) motion without sliding or rolling out of its initial position (i.e., a 3D inverted pendulum). Thus, the cylinder is constrained to zero residual displacement at the end of its 3D motion. This 3D dynamic model of the rocking rigid cylinder has two DOFs (three when damping is included), making it the simplest 3D extension of Housner's classical two‐dimensional (2D) rocking model. The development of models with and without damping is presented first. They are simple enough to perform extensive parametric analyses. Modes of motion of the cylinder are identified and presented. Then, 3D rocking and wobbling earthquake response spectra are constructed and compared with the classical 2D rocking earthquake response spectra. The 3D bounded rocking earthquake response spectra for the ground motions considered seem to have a very simple linear form. Finally, it is shown that the use of a 2D rocking model may lead to unacceptably unconservative estimates of the 3D rocking and wobbling seismic response. Copyright © 2017 John Wiley & Sons, Ltd.

The rise in the frequency of anthropogenic earthquakes due to deep fluid injections is posing serious economic, societal, and legal challenges to many geo-energy and waste-disposal projects. Existing tools to assess such problems are still inherently heuristic and mostly based on expert elicitation (so-called clinical judgment). We propose, as a complementary approach, an adaptive traffic light system (ATLS) that is function of a statistical model of induced seismicity. It offers an actuarial judgement of the risk, which is based on a mapping between earthquake magnitude and risk. Using data from six underground reservoir stimulation experiments, mostly from Enhanced Geothermal Systems, we illustrate how such a data-driven adaptive forecasting system could guarantee a risk-based safety target. The proposed model, which includes a linear relationship between seismicity rate and flow rate, as well as a normal diffusion process for post-injection, is first confirmed to be representative of the data. Being integrable, the model yields a closed-form ATLS solution that is both transparent and robust. Although simulations verify that the safety target is consistently ensured when the ATLS is applied, the model from which simulations are generated is validated on a limited dataset, hence still requiring further tests in additional fluid injection environments.

Rocking motion is sensitive to the boundary and initial conditions of a rocking structure, making experiments nonrepeatable. Thus, the claims that numerical rocking motion models are not only inaccurate but that all rocking structures behave unpredictably. Hence, rocking is not used as a seismic design approach. This paper revisits the issue of rocking motion unpredictability. Seismic behavior of structures is inherently stochastic, because the loading is stochastic. Therefore, the question of interest is not whether models can predict the seismic response to a single ground motion, but if the statistical characteristics of the ensemble of responses to a set of ground motions that define the seismic hazard can be predicted. For this purpose, a rocking podium, which is a three‐dimensional structure comprising an aluminum slab supported by four tubular steel columns, was tested on a shake table excited by two sets of 100 consistently generated ground motions. It was found that the cumulative distribution function (CDF) of the experimentally obtained displacements is statistically stable. Next, a blind prediction contest was organized. The contestants were invited to predict the CDFs of the slab lateral displacement. They were able to predict the slab displacement CDF relatively well. Both finite element and discrete element modeling approaches were used, but no clear pattern emerged as it was found that the performance of either approach depends on the input parameters used and the assumptions made. It was also observed that the contestants who did not use Rayleigh damping in their models produced better predictions.

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