Keywords: auto-parametric systems, semi-trivial solution, dynamic stability, system recovery, post-critical response.

A strong vertical component of an earthquake excitation in the epicenter area very often represents the most dangerous condition leading to structure collapse resulting from auto-parametric resonance. This highly non-linear dynamic process caused in the past heavy damage or collapses of towers, bridges and other structures. In the sub-critical linear regime vertical and horizontal response components are independent and therefore in such a case no horizontal response component is observed. If the amplitude of a vertical excitation in a structure foundation exceeds a certain limit, a vertical response component loses stability and dominant horizontal response component arises [1]. This post-critical regime or the auto-parametric resonance [2], follows from the non-linear interaction of the vertical and horizontal response components and can lead to a failure of the structure. The seismic type broadband random non-stationary excitation can be particularly dangerous and amplify these effects.

In principle easily deformable tall structures are the most sensitive regarding effects of auto-parametric resonance (chimneys, towers, etc.). Therefore the structure itself is modelled as a console with continuously distributed mass and stiffness in order to respect the whole eigen-value and eigen-form spectrum. The subsoil model respects the vertical and rocking component of the response including internal viscosity of the Voight type.

A Hamiltonian functional is provided for the Lagrangian differential system linking the strongly non-linear double degrees of freedom part with multi-degree of freedom part modelling continuous console to the governing system. The system shows that horizontal and vertical response components are independent in the semi-trivial regime, which is linear in such a case. Their interaction takes place as a result of the non-linear terms in the post-critical regime only. Two generally different types of the post-critical regimes are presented in the paper: (i) the close neighbourhood of the stable state (area between the semi-trivial solution stability limit and the limit of irreversibility); despite the strongly non-linear response, the structure can regain the stable state, when excitation drops below a certain limit; (ii) if the response is beyond the limit of irreversibility, the rocking response component looses any periodic character and rises exponentially leading inevitably to a failure of the structure without a possibility of any recovery.

In principle the solution method combining analytical and numerical approaches is developed and used. Its applicability and shortcomings are commented upon. Some advice for engineering applications are given and some open problems are indicated.

1

J. Náprstek, C. Fischer, "Auto-parametric Stability Loss and Post-critical Behavior of a Three Degrees of Freedom System", in M. Papadrakakis et al., (Editors), "Computational Methods in Stochastic Dynamics", Springer, Chapter 14 , 267-289, 2011. doi:10.1007/978-90-481-9987-7_14

2

A. Tondl, T. Ruijgrok, F. Verhulst, R. Nabergoj, "Autoparametric Resonance in Mechanical Systems", Cambridge University Press, Cambridge, 2000.

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