Analytical modelling of multi‐mass flexible rocking structures

Abstract

SUMMARYThis paper presents a new analytical model for describing the large rocking response of an elastic multi-mass structure resting on ideally rigid ground. Using the experimental results from a rocking steel column, the ability of the proposed analytical model to estimate the rocking and translational acceleration response under free vibration, pulse and earthquake excitations is evaluated. It is observed that the classical treatment of impact may result in an unrealistically large transfer of energy to vibrations. Therefore a new Dirac-delta type impact model that spreads the effects of impact over time and space is proposed. The use of a Dirac-delta model and accurate restitution factors play a pivotal role in prediction of rocking and acceleration responses. In order to characterize the nonlinear response better, a modal analysis of the linearized system is proposed. With this approach, the vibration mode frequencies and shapes during rocking action were determined. A comparison of analytical and experimental modal estimations suggests good agreement. The results emphasize that the vibration characteristics of several vibration modes are affected by rocking action, and these modes may be excited at impact.

Dynamics of rocking podium structures

“…In the presence of damping, the aforementioned equation needs to be correct only if damping satisfies the Caughey-O'Kelly condition [32], which is not the case. The free-standing columns respond to sufficiently strong ground motion excitation by uplifting and rocking.…”

A Centrifuge Investigation of Two Different Soil-Structure Systems With Rocking and Sliding on Dense Sand

“…On the contrary, the footings of the model RA stayed in position and therefore rocking was developed by that model with partial lateral sliding of the column tips into the grooves of the footings. During the large amplitude earthquakes and low frequency earthquake, the time-historey accelerations suggest an out of phase storey profile, which if combined with the existence of the impacts, it further suggests the development of an uplifted mode of vibration [18,30], driven by the input motion frequencies.…”

“…Structures idealized as rigid rocking blocks were studied by Housner [3] back in the 1960s. Since then, the overturning tendency and self-centering nature of rocking [4,5], the implementation of rocking with dampers to suppress swaying motion [6][7][8][9][10][11][12][13][14][15], the coupling of flexibility and rocking, [16][17][18][19] and the role of the higher modes [17,18,20,21] have been studied both from an experimental and analytical point of view, with a clear focus on the structural performance (structural rocking). In practice, while the base shear reduces due to structural rocking, excitation due to impacts at the interface of the stepping mechanism might amplify internal loads significantly.…”

Probabilistic Validation of the Housner Rocking Model

“…Even if the coe cient of restitution is relatively well predicted, or determined via experiments, the response of the system to a ground motion is so sensitive that predicting the whole time history is practically impossible [27]. This has generated a lot of discussion about Housner's approach to determine the energy losses during impact and has resulted in various di↵erent energy dissipation models [28][29][30][31][32][33][34][35][36][37]. Moreover, it makes practicing engineers hesitant to adopt rocking as an earthquake response modification strategy, e.g.…”