My Previous Research on the Rocking Problem
During my Masters program in the Structural Engineering, Mechanics and Materials (SEMM) at U.C. Berkeley and the summer after graduation, I did extensive analytical research on the dynamic response of unanchored rigid blocks subjected to ground shaking.
When subjected to strong earthquakes, a variety of rigid structures, such as concrete radiation shields, heavy equipment, and slender massive stone structures, can set into rocking motion and occasionally overturn. The need to understand and predict rocking response has motivated a number of studies on the rocking response of rigid blocks, but not until recently has it been established that the rocking response of rigid blocks under forced excitations is a fundamentally nonlinear problem with a multivalued solution. For instance, a rigid block may actually survive overturning due to a ground motion with acceleration larger than the minimum acceleration required to overturn it. It has also been shown that there is a finite range of excitation frequencies of the input for which the previously used linear approximation gives erroneous results even for very slender blocks.
Working under the guidance of professor Nicos Makris, I studied the dynamic behavior of rigid blocks subjected to ground motions with various kinematic characteristics. An in-house developed algorithm that solves the nonlinear equations of motion was used to create graphs that further improve our understanding of the rocking response. For example,
Overturning acceleration spectra for blocks subjected to cycloidal (trigonometric) pulse ground excitations. These are plots of the overturning acceleration amplitude that is needed to cause overturning of a block of given geometry versus pulse frequency. These plots show the multivalued essence of the response by indicating safe and unsafe regions (regions where overturning does not occur and regions where it does).
Rocking spectra. Just like response spectra show the maximum response quantities of a single-degree-of-freedom oscillator (of given viscous damping) as a function of its natural period, rocking spectra show the maximum rotation (uplift) and angular velocity of a rocking rigid block (of given slenderness) as a function of its inverse frequency parameter. As the response spectra that correspond to different damping values for a SDOF oscillator are different, so are the rocking spectra that correspond to blocks with different slenderness values.
My research demonstrated that the rocking spectrum is a distinct and valuable indicator of ground shaking due to earthquakes since it reflects kinematic characteristics not identifiable by the response spectrum. Moreover, since there are several types of structures in the real world that behave more like rocking blocks and less like flexural SDOF oscillators, a more comprehensive examination of an earthquake’s potential for damage should include not only the response spectrum but also the rocking uplift spectrum.
Rocking of rigid blocks is a complex nonlinear problem and engineers have naturally attempted to simplify it. In 1978, Priestley et al. presented a practical methodology to estimate rotations due to rocking motion. The Priestley et al. methodology is based on the assumption that “it is possible to represent a rocking block as a single-degree-of-freedom oscillator with constant damping, whose period depends on the amplitude of rocking,” and proposes that rocking uplift can be estimated by an iterative procedure on a standard acceleration or displacement response spectra (corresponding to an “equivalent” viscous damping value).
The approximate procedure proposed by Priestley et al. has been adopted without sufficient scrutiny by the design guidelines document FEMA 356. Prestandard and Commentary for the Seismic Rehabilitation of Buildings.
The oversimplification of a really complex problem however needs to be treated with caution. Having worked to great extent on the problem of rocking, the Priestley et al. assumption and methodology based on it appeared to be fundamentally flawed. The two dynamical systems have inherently different mechanical structures. The free-vibration response of a SDOF is described by trigonometric functions whereas the free-vibration of a rocking block is described by hyperbolic functions.
Over the summer I decided to work with professor Makris on examining the validity of the approximate methodology. Using the Priestley et al. and the FEMA 356 procedures (which only differ slightly from each other), rocking spectra where developed and compared with rigorous rocking spectra obtained by the direct numerical solution of the equations of motion. It was found that the results obtained by the two different methods were very different.
We concluded
that the Priestley et al. and FEMA 356 simple design
approaches are alarmingly misleading, in particular, for small, slender blocks.
Our findings were recently published in a