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Vibration modes and equivalent models for flexible rocking structures

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Acikgoz, S 
DeJong, MJ 


Predicting the displacement and force response of flexible rocking structures to ground motion is important for their assessment and design. Insofar as practical, it is desirable to use simple mechanical models to make these predictions. However, the complex coupling between rocking and vibration makes accurate predictions with simple models difficult. In this paper, the use of semi-coupled equivalent models to approximate the dynamic response of multi-mass structures rocking on rigid ground is evaluated. These equivalent models feature a two-degree of freedom coupled rocking oscillator to describe the interaction of rocking and the first mode of vibration, and uncoupled linear elastic oscillators to describe higher mode vibration response. To evaluate these equivalent models, the modal components of the dynamic response of multi-mass structures are first determined. These components highlight the critical influence of the excitation of vibration modes at impact. Then, further investigations are carried out by comparing equivalent model simulations to recent shake table tests and multi-mass analytical model simulations. These comparisons reveal that the equivalent models can capture the rocking response accurately for a realistic range of displacements, if a new ground acceleration scaling term is adopted. However, the uncoupled linear elastic oscillators do not consider excitation at impact and consequently, the equivalent models do not capture the acceleration response adequately. Therefore, on the basis of the analytically identified modal components, a further modification that improves the equivalent model acceleration predictions is proposed and validated.



modal analysis, equivalent model, rocking, uplift, shake table

Journal Title

Bulletin of Earthquake Engineering

Conference Name

Journal ISSN


Volume Title


Engineering and Physical Sciences Research Council (EP/N021614/1)
Engineering and Physical Sciences Research Council (EP/K000314/1)
Engineering and Physical Sciences Research Council (EP/L010917/1)
Engineering and Physical Sciences Research Council (EP/I019308/1)
The first author would like to express gratitude for funding from Trinity College, Cambridge Overseas Trust and the Suna and Inan Kirac Foundation.