Ruisong Xue and Hinke M. Osinga
Abstract
Earthquake-resilience is important for building design in areas prone to earthquakes. Building techniques have started to take advantage of geometric non-linearities so that structures, such as frames or bridges, are able to rock when an earthquake hits. Rocking reduces bending along structural members and limits the transfer of seismic energy, leading to far less post-earthquake damages. Whether the tilt angle of the rocking frame exceeds a prescribed maximum depends on the forcing properties. The boundary of such structural failure is organised by so-called grazing solutions, which reach but do not move beyond the design limit of the frame. Both theoretical and numerical analysis of such geometrically non-linear structures is commonly done based on a harmonic load with a fixed frequency and amplitude. Even though non-linear systems exhibit much more complicated behaviour than linear systems, the harmonic load is thought to comprise a worst-case scenario, provided the frequency and amplitude are chosen appropriately. In this paper, we design more realistic, non-harmonic forcings with broader frequency contents. Importantly, the energy, predominant frequency and bandwidth of these forcings can be controlled independently. We propose an aperiodic non-harmonic load along with a simplified form that super-imposes a very low-frequency amplitude modulation on the harmonic load; this simplified non-harmonic load provides sufficiently similar responses when compared with the broad-frequency aperiodic load, and allows for efficient and easy numerical investigation. We compare failure boundaries for harmonic and non-harmonic loads by computing curves of grazing bifurcation in the (predominant) frequency-amplitude plane for a class of rockable frames that are held together by pre-stressed steel tendons. Our results show that, grazing bifurcations occur earlier under non-harmonic loads. Furthermore, our findings are counter-intuitive: to minimise seismic failure, the frame should be designed as flexible as possible—this insight cannot be obtained by only considering harmonic loads.
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