Summary The response of a rigid rocking block is traditionally described by its tilt angle. This is a correct description, but this paper suggests that describing rocking via displacements is more meaningful, because it uncovers that two geometrically similar blocks of different size will experience the same top displacement, provided that they are not close to overturn. The above is illustrated for both analytical pulse excitations and for recorded ground motions. Thus, the displacement demand of a ground motion on a rocking block is only a function of its slenderness, not of its size. This reduces the dimensionality of the problem and allows for the construction of size‐independent rocking demand spectra.

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This work studies the dynamics of the Negative Stiffness Bilinear Elastic (NSBE) oscillator. Such a mathematical idealization can be used to describe deformable rocking systems equipped with restraining tendons or with curved extensions of their bases. First, this paper establishes the characteristic quantities of the bilinear system to make it equivalent to the actual rocking structures. Then, it proceeds by proposing a simpler "equivalent" system that can be used to study the behavior of the NSBE. The equivalent system is not some linear elastic oscillator but a bilinear elastic system with zero stiffness of the second branch: the Zero Stiffness Bilinear Elastic (ZSBE) system. ZSBE is useful because it needs one parameter less than NSBE to be defined. Next, "Equal Displacement" and "Equal Energy" rules that provide estimates of the maximum displacement of the NSBE based on the response of the ZSBE are defined. The concept is similar to the RμΤ relations that provide estimates of the response of bilinear yielding systems based on the response of an equivalent linear elastic system, with one major difference: it does not resort to a linear elastic system but to the ZSBE. The proposed methodology is applied on the FEMA P695 ground motions scaled at three different levels. The results show that ZSBE is a good proxy of NSBE and, hence, indicate that an exhaustive study of the ZSBE is useful for the design of rocking structures.

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This paper presents the shake table test results of a novel system for the design of precast reinforced concrete bridges. The specimen comprises a slab and four precast columns. The connections are dry and the columns are connected to the slab by an ungrouted tendon. One of the tendon ends is anchored above the slab, in series with a stack of washer springs, while the other end is anchored at the bottom of the column. The addition of such a flexible restraining system increases the stability of the system, while keeping it relatively flexible allowing it to experience negative post‐uplift stiffness. It is a form of seismic isolation. Anchoring the tendon within the column, caps the design moment of the foundation, and reduces its size. One hundred and eighty‐one shake table tests were performed. The first 180 caused negligible damage to the specimen, mainly abrasion at the perimeter of the column top ends. Hence, the system proved resilient. The 181st excitation caused collapse, because the tendons unexpectedly failed at a load less than 50% of their capacity (provided by the manufacturer), due to the failure of their end socket. This highlights the importance of properly designing the tendons. The tests were used to statistically validate a rigid body model. The model performed reasonably well never underestimating the median displacement response of the center of mass of the slab by more than 30%. However, the model cannot predict the torsion rotation of the slab that was observed in the tests and is due to imperfections.

This paper presents uniform risk spectra for zero stiffness bilinear elastic (ZSBE) systems. The ZSBE oscillator is a bilinear elastic system with zero post‐“yield” stiffness that satisfactorily predicts the response of different systems with negative lateral stiffness (e.g., free‐standing or restrained rocking blocks). It can be described by a single parameter; thus, it is simpler to produce its spectrum. Using the ZSBE proxy, this paper provides the uniform risk spectra for sites in six locations in Europe. The spectra are constructed using two distinct intensity measures (IMs): peak ground velocity (PGV) and peak ground acceleration (PGA). The efficiency of both IMs at different ranges of displacement demands is discussed and analytical approximations of the spectra are proposed.

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This paper proposes a simple analytical system that can be used to describe the dynamics of Negative Stiffness Bilinear Elastic (NSBE) systems, and consequently design them in a simpler manner. The NSBE oscillator is a mathematical idealization, which can be used to describe rocking structures with or without flexible restraining systems or curved extension at their bases. The paper defines the characteristic quantities to make the bilinear system and actual rocking structures equivalent. A simpler "equivalent" system to describe the behavior of NSBE systems is proposed: The equivalent system is the Zero Stiffness Bilinear Elastic (ZSBE) system, which is a bilinear system with zero stiffness in the second branch. The ZSBE system is useful and simpler because it needs one parameter less than the NSBE to be defined. The paper proceeds by defining the "Equal Displacement" and "Equal Energy" rules that provide estimates of the maximum displacement of the NSBE based on the response of the ZSBE. Using a simpler system to predict the response of a that provide estimates of the response of bilinear yielding systems based on the response of an equivalent linear elastic system. However, the method should not be confused with the approach of FEMA 356: it does not resort to a linear elastic system but to the ZSBE. Finally, the preliminary design of a real rocking structure is presented, as a case of study. The paper compares the response predicted by the proposed methodology to the one predicted by a more accurate numerical analysis.